Mathematics - Student's Book
89
• MATHEMATICS YEAR SIX STUDENT'S WORKBOOK
Transcript of Mathematics - Student's Book
•
MATHEMATICS YEAR SIX
STUDENT'S WORKBOOK
WEEK 16
WEEK 17
WEEK 18
WEEK 19
WEEK 20
WEEK 21
WEEK 22
WEEK 23
WEEK 24
WEEK 25
page 176
Addition and subtraction of fractlons ..................... ... . ....... . 122
Axial symmetry: construction of symmetr1caJ figures ................ 124
Proportionality: percentage ................................... ,....... ... 127
Perimeter of a circle ............. , .... ............ ,..... . .. ........... .. . . 128
Division 01 0 decimal by 0 decima l .... ........... ...... ............. . 130
Division at decimal numbers: exercises and problems ..... ........ 131
Parallelepiped rectangles: development and construction ....... 132
Area: agricultural measurement units .................................. 134
Proportionality: average speed ........................................ .
Cubes: other developments and construction .. ,.. , ............ .
SUrface area of rectangular prisms and cubes ..... ... ............. .
Multiplication and division of measurements of duration ..... .... .
CyUnders: development and construction ......... .. . .... ...... .
Area of discs ............ .. ....... ...................... . . ..... .. ,., .. , .. ,'
Area of cylinders .................................. ....... .... . .......... .
Multiplication of a whole number by a fraction ..... . ............. .
Proportlonoll1y: plan and scale ... , .... ........ ..
Proportlonall1y: plan and scale ... ....................... .
Volume of cubes and rectangular ptIsms ........... ....... ......... .
Volume of cylinders ............... ........................................ .
Rote of flow ........ .
Relationship between capacity and volume .......... . . . .... .... . .
Dens/1y ......................................................... .
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140
142
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150
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Division of a whole mrnbef by 25. SO. 75 ..... ... ........... .. ... ... ... 166
Revision .................................................................... . 167
Revision 168
Revision 170
•
MATHEMATICS YEAR SIX
STUDENT'S WORKBOOK
AVANT-PROPOS
Lo collection des manuels de mothemotiques de la CONFEMEN (Conference des Mlnistres de I'Education des pays ayont le fran<;ais en portage) temoigne de raboutissement d 'une volonte politique et d 'un effort soutenu des pays membres vers 10 mise au point de materiels pedagoglques communs, performants et adaptes aux rea lites et besoins socloeconomiques et culturels des populatlons locales.
Ce manuel est le shdeme d 'une collect ion couvrant les six annees de I'ecole prlmalre.
11 est con~u pour un travail indlviduel de r eleve. Les recommandatlons sont consignees dans le guide pedagogique correspondont. Le c ahler et le guide constituent par consequent un tout Indissoclable.
Les exerclces du Itvre de releve sont p resentes de fa<;on progressive p our una me-me notion. 11 est done recommande de respecter I'ordre dans lequel its sont d isposes.
On prendra egalement soln de lire et d ' expliquer les consignes aux eleves avont de les loisser trovoiller librement dons le manuel.
Nous esperons que cet ouvroge que nous ov~ns voulu moderne et aHroyant sera un outll opprecle pour I' apprent issoge des mathematlques en slxieme annoo.
© CONFEMEN
BP 3220 Dokor-Senegol AU rights reserved
LES AUTEURS
The reproduct1on of any extract from this book by whatever means is strictly prohibited.
ISBN 92-9133-059-0 4"' editIOn revtewed and adapted (1S8I\I 2-87344-131-3 1· edtIon)
page 2
0/1997/3125/53 (0/1993/3125/22 1· edition)
WEEK 8
WEEK 9
WEEK 10
WEEK 11
WEEK 12
WEEK 13
WEEK 14
WEEK 15
Simple fractions Addit10n and subtraction of decimal numberS ............ .
Trapeziums - Construction ............................................. .
Units of d uration - Conversions ................ .............. .. .
DIvisibility by 3 and b y 9 . . . . . . .. . .. . ......................... .
Per1meters - Revision ................... . .. , .................. .
PrIme numbers - Decomposition of a number .................... .
BIsecting line - Construction ..... . ..................................... .
Units of capoc ity - Conversions . . . . . . . . . . . . . .. . ........................ .
Equal fractions . Proportionality and rule of three ................. , ............. , •.. .....
Multiplic ation pf a decimal by a INhale number ....... .. ... ... . .
Parallelograms and diamond-shapes .... ..... ..... ... .. .... .
Units of area - Conversions ....................... , ............ '
Multipllccrnon of a decimal by a decimal .. .... ....... .. .. . ......... .
Parallelograms and diamond-shapes - Construction .. ............ .
Area of squares. rectangleS and triangles ........................... .
Addition and subtraction of measurements of duration ......... .
The rule of three ... ..... . ...... ............................................ .
Enlargement and reduction of shapes ....................... .. ....... .
Area of parallelograms and trapeziums ............•.. " ...... . ...... .
Area of d iamond-shapes ........... .. ................... ... .......... .
Division of whole numbers (quotient to the nearest 0.1. 0.01 , 0.001) ........................ . .... .
Circ le · Disc . . .. .... .... .. ......... ... .. .... .. ... ....... ... .
Decimal value of fractions - Decimal frac tiOns ......... .
DivIsIon of a decimal by a lNhole number .................. .. ..• . ....
Polygons: regular hexagons .. ... .................................. .
Area of complex figures ................................................. .
COmpar1son of tracttans ................... ... .. ... .................. .... . DMsIon of a W'hole number by a decimal ........................... .
Perc entages - Interest rote - Capttol ........................... .. .... •.
AxfaI symmetry . ...... ........................ .... ... ............ ......... .•.
Area of complex figures .... . ......................................... .
58 60 62 64
66 67
68
70 72
74 76 78
80
82
84 86 8&
90 92
94
96 98
100
102
104
106
108
110
112 114 116 118 120
page 175
WEEK I
WEEK 2
WEEK 3
WEEK 4
WEEK 5
WEEK 6
WEEK 7
page 174
CONTENTS
large numbers ....... .
Addition and subtraction of whole numbers .....
Convex or concave polygons ..... .
Measurement of angles - Using 0 protractor
Multtpllcation of whole numbers ...................................... .•
Angles In Isosceles or equilateral triangles ....................... .
Comparison of angles ........... .
Comparison and ranking of whole numbers ................. ........ .
Operations with whole numbers .......... . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . Calculation of perlmet9fS
Roman numera ls ........ .
DMsk>n of whole numbers (whole quot1ent) .................... .
Proportionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .• . ........ .
Construction of the height lines of trkJngles ............. .
Perimeter - Calculation of one dimension .....
Division of whole numbers: particular cases .................... .
Proportionality: graphic representation .......................... .
RIght-angled isosceles triangle - Construction ............ .
Units of length - Conversions ............................... .
Decimal numbers .......... .
6
8
10
12
14
16
17
18
20
22
24
26
28
30
32
34
36
38
40
42
Proportionality ........ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Revision: geomehy ..... . .......................... ... .................. .
Order of decimal numbers .... ........ . ................ ... .... .
Murtples and dMso!s: dMslbll1ty by 2. 5 and 10 ............ .
46
48
51
Proportional shares ............ . .......................... ........ . ........ 52
Trapeziums ...
Units of weight - Converstons .
54
56
PREFACE
The series of CONFEMEN (Conference of Education Ministers from countries
which have French as one of their languages) mathematics manuals testifies to
the political will and sustained efforts of member countries to perfect common
teaching resources which are both effective and adapted to the
SOCioeconomic and cultural realities and needs of the local population.
This workbook is the sixth of a series which covers the six years of primary
education.
It is designed to encourage the student's individual activity. Advice is gathered
in the corresponding teacher's guide book. The workbook and the teacher's
guIde should be regarded as an indivisible whole .
The student's workbook exercises are designed to cover a given notion In a
progressive manner. It is therefare recommended to follow the order in which the
exercises are set out.
The teacher will endeavour to read and explain the guidelines before the
students begin to work alone in the workbook.
We hope this workbook will be appreciated as an attractive and modern tool for
mathematics learning.
THE AUTHORS
page 3
DOge 173
page 172
The student's workbook Is reusable.
students should theref()(e not use It
for writing do.......-n their answers
to the questionS In the exercises.
page 5
Large numtHm
LET'S FIND OUT
Look ollhese numbe~: 47 109 063; 942 693 417; 24 768 259 310. Write them In the numeration table below:
bllHons millions thousands
h I u h I u h I u
- Read these numbers aloud. - Look at Ihese: 47 109 063 = 47 000 000 + 109 000 + IIJ + 3
simple units
h I
47 109 063 = (47 x I 000 000) + 009 x I 000) + (6 x 10) + 3 - Break down the other numbers In the some way. - What do the digits 3. 4. I. 7 represenl? - Wrfte these numbers in words.
LET'S LEARN ABOUT IT
u
In decimal numeration each digit has a precise value. This value Is determined by c lass and by order. There are several classes (simple units. thousands, millions and billions) and three orders In every c lass (units, tens, hundreds). This Is shown In the following numeration table:
billions millions thousand. IimpMJunlts
h I u h I u h I u h I u
4 7 I 0 9 0 6 3 9 4 2 6 9 3 4 I 7
2 4 7 6 8 2 5 9 3 I 0
• The number 24 768 259 310 Is written: twenty·four billion seven hundred and sixty-eight million two hundred and tttty·nlne thousand three hundred and ten.
NOTES
.............................. ... . " .. ... ... " .. , ................ ....... ........... -.... ... .... ...... ... .
........ ...... ................ ............ .. -................. ........... .............. ....... .... ... .... .
, .... .... .. .................... - ... ..... .. " ... , ... ... .................................. -........ ... . " . . .
... . . .. . . . .... . . . . . ..... . .. . . .. ... . . . . ... . .... .................. .... ... ....................... ...... .... .
.................. .... . .... ... .. ... ........... ....... ..... .. ............... .. ..... .... ... ......... ... .
......... .. .............. .. ....... ....... ....... ...... .............. ... ...... .... ... ........... .. .... ... ..
.... ..... ..... ..... ........ . ............ .. ........... .. ............. ....... ..... .... ...... ..... .... .....
........ .... .... ..... ..... .. .... .. ........ .. .... ......... .................................... .......... ...
......... ........ ........ .... ............ .. ................... ..... ............. ...... .. ....... ........ ...
.. ................. .. .. ............... .. ..... ................................... ............... ... .. ... . .
page 6 page 171
Rev&lon
SOL VING PROBLEMS
1. Wnte down all the simple ~actions ablaned ~am using the digits 2. 3. 5. 7 as both numerator and denominator. - State the decimal value fOf each of these fractions.
Ind icate the tractions with an exact decimal value and those wfth an apprOximate decimal value.
- Rank these fractions in Increasing order.
2. Draw the diamond-shape B 3. Draw an equilateral A
4.
ABCD. Mark In the toongle ABC. Draw In the height lines AH. 81( and Cl.
diagonals AC and BD.
Check that each of the diagonals Is on axis of A sym metry of th e diamond-shape.
C Check that each height line Is on axis C '------>8 of symmetry of the triangle.
D
Draw 0 circ le wtth centre 0 and radius R = 3 cm. Mark a diameter AB In this circle. From a poklt H on this diameter. extend the perpendicular to AB which intersects the
0 circle at C and D. R - What kind of ITIangle is OCD?
- What Is AB In relation to the triangle OCD? - What Is AB In relation to the c irc le wfth centre 0 and rad ius R? - How many axes of symmetry are there in 0 circle?
5 . On 0 plan to the scale of ~. 0 piece of land Is
represented by a right-angled triangle wtth the sides of the .-r1ght angle measuring 3 cm and 5 c m. % 7fIJ (XX) VI has been paid for this land.
What Is the price per ore? ~c~
ScOIe: ~
6. Jaseph and his brother decide to go to the next village. Joseph sets out at 2.40 pm. walking at on overage speed of 4.5 km/hOUr. His brother sets off an his bicycle at 3.10 p .m. and catches up with hin at 3.20 p .m. • How long was Joseph walking before his brother caught up with him? · At what distance from his starting point Is Joseph rejoined by his brother? • What was his brother's average hourty speed?
page 170
LET'S PRACTISE
1. Write the follo'W'lng numbers in the numeration table; 79 464849; 4007325; 921 479341; 102 548 878 421; 21 006 007 003; 473639321 418.
billions millions thou","", slmpkt units
h t u h t u h
2. Read aloud and write in words: 423938 706 524; 17912310 416;
4008705 101 ; 2206 010 083.
t u h t u
3. Write the following numbers in digits, then break them down and show them os the sum of muHfples of 1 D, 100, 1 (lX) ...
• seventy·three billion two hundred and twelve million six thousand and eight; - eight billion six million thirty-seven thousand four hundred and eighteen; • one billion seven hundred and thirteen million eight hundred thousand one hundred
and twenty-nine.
4. Write in digits the number Immediately before and atter the following numbers: - elghty-eight thousand; - two hundred million.
5. in the number 43 408 329 287: a) what Is the dig~ showing;
- tens of millions? - tens of thousands? - tens of billions?
b) what Is the number of: - thousands? - hundreds of millions? - units of billions?
6. Write down the following numbers: (4 x 1 (XX) COl) + (7 x 10 (XX) COl) + (3 x 100 (xx)) + (9 x 1 COl) + (5 x 10) + 8 (342 x 1 (XX) (XX) (xx)) + (977 x 1 (XX) (xx)) + (108 x 1 COl) + (3 x 10) + 2
page 1
AddItfon and subIractton 01 Whole numbBts
LET'S FIND OUT
1. A breeder sells sheep for 326 (Xl) VT and goats tor 179500 VT. Wtth port of this money. he buys cattle for 268 (XX) yr. Ask questions and soNe the problem.
2. Using addition and subtroctlon. wOO: out the followng calculations h one go: 2 500 + 6 250 + 1 500; (250 (Xl) + 100 (Xl) - (125 (Xl) + 100 (Xl)
Explain your steps.
3. Give the approximate value of the result of the following calculations to the nearest hundred. then to the nearest thousand:
Work out the exact results.
LET'S LEARN ABOUT IT
325472 + 296 319 946 612 - 754 679
1. The breeder has made the sum of 328 CO) + 179 50').
After the purchase of the cattle. he stJl has 507 500 - 268 CO).
To cany out these operot\on:s. wrtte down the digits In the correct columns and odd up collrnn otter 11
column, beghnlng with the sJmp" urIfs and not 326 (Xl)
forgetting to cany over when necessary. + 179500
2. To help you work: out a sum more qulcidy. you con 507500 change the order of the terms and re-group them to
make the calculation easier. Example: 2 500 + 6 250 + 1 500 '" (2 50') + 1 5(0) + 6 250
= 4 (Xl) + 6250 = 10 250
507500
-266 (Xl)
239500
I· To _out a._ence'llJlt?l<~, you con _.~Of t1ad .l;~ I lite IQn'Nt ,...""., 10. t:iOIh Mtms cl ",. __ •• "ICe. , .
Example:
In the subtraction 250 ())) - 125 CO) you can either subtract the same number: (250 (Xl) - 100 (Xl) - (125 (Xl) - 100 COl) = 150 (Xl) - 25 (Xl) = 125 (Xl)
or. a ltemoftvely, odd the some number. (250 (Xl) + 100 (Xl) - (125 (Xl) + 1 00 COl) = 350 (Xl) - 225 (Xl) = 125 (Xl)
3. 325 472 to the nearest hundred Is: 325 500 296 319 to the nearest hund'ed Is: 296 300 325 472 + 296 319 to the nearest hundred is: 325500 + 296 300 = 621 600 To the nearest thousand, 946 612 - 754 679 Is: 947 (Xl) - 755 (Xl) = 192 (Xl)
Revision
SOL VING PROBLEMS
1. Every traveller going to France must hove French curency. The exchange rate is 1 French franc = 20 VT. Zak who is going to Paris has exchanged 200 CO) VT. How much did he receive in French francs? How many VT must he exchange to receive 1 £XX) f rench francs? Claudia is going to Montreal (Quebec). The exchange rate Is 1 Canadian dollar = 100 VT. Claudio exchanges 200 (XX) VT. How many Canadian dollars does she recetve? How many Canadian dollars would she receive for 1 500 french francs?
2. On 0 triangular piece of land with a base of 72 m and a height of 49 rn , 0 rectangular shed Is put up , measuring 28.5 m long and 17.8 m wide. The rest Is left as arable land. What Is the arable area?
Rectangutar shed
3. The triangle A Drow a tr1angle MpQ with the some ABC has the angles as ABC but with the sides MP foll o wing and pQ twice as long os AB and BC. dimensions: Compare the lengths of the sides AC AB = 8 cm B and MQ. AC = 6 cm BC = 10 cm c
4. Drow a square ABCD with the side AB '" 8 cm. A B
Beside It draw a square MNOP with MN - 4 cm. - Compare the lengths of the diagonals AC and NP. - Draw a third square EFGH with EF '" 2 c m. - Compare the lengths of the sides EF and AB. - What can you say about the perimeter and the area of the
squares MNOP and EFGH In relation to the square ABCD? D C Check your answer by doing the calculations.
5. Chartle cuts out of cardboard 9 figures. 6 of which [> JoO are equilateral triangles with the some dimensions and 3 of which are superlmposable diamond-shapes. the sides of which hove the same
6 measurements os the sides of one of the equilateral triangles. wtOCh of the p ieces can you arrange to form a
[> 6 0 regular hexagon? (There Is more than one solution!)
page 169
LET'S PRACTISE
Complete this takings sheet from a large store by doing the additions both up and down and across. The amounts are given In yr. Total takings a re to be checked In both directions.
1998 Cosh regllter Cosh register Cash register Daily total
1 11 111
4 Mach 49874 71432 32400 .......................
5 Mach 37428 54 642 21214 .. ..... . ....... .... .. 6 March 24237 61004 30 317 ... ..... . ... .. .. ... ...
7 March 4203 61 300 28509
8 March 63 500 84 870 43 800 . . . . . . . . . . Total . . . . . . . . . . . . " . .. ... ......... ..... ..... . . . . . . . . . . .
SOL VING PROBLEMS
1. Two housewives buy a fish together costing 350 VT per kg and weighing 3 600 g. Calculate the purchase price of the fISh. After gutting and cleaning It, they each take a share. The waste from cleaning weighs 450 g. One of the housewives takes 1 400 kg of the c leaned fish and the other one takes the remainder. How much must each pay?
2. A truck driver delivers 2 ())) kg of scrap iron to a factory, made up of 78 bars of Iron and a rail. Each bar weighs 23 kg. What is the total weight of the bars? What is the weight of the rail?
3. 396 labourers are employed on the construction of a road. In a fortnight they work 92 hours. Each labourer is paid an hourly rate of 265 VT. How much does he earn In a week? What is the total amount eamed by all the labourers In a fortnight?
4. A motorist sets off with a full tank. After 250 km he notices that he has used up 1- of
his petrol. He fills up and pays 2 400 VT for petrol costing 120 VT the litre. Calculate: - how much petrol he buys; - the consumption of petrol per 100 km; - the capacity of his petrol tank; - how fa r he can travel on a full tank.
5. Milk yields 3 % of its weight as cream and cream yields 25 % of Its weight as butter. A farmer has 12 cows. Each gives an average of 9. 12 litres of milk per day. One litre of milk weighs 1 .033 kg. - What weight of cream does the former p roduce every day? - What percentage of the weight of milk is represented by the weight of butter
obtained?
page 168
Addition and 5ubtracffon of whole numbers
LET'S PRACTISE
1. In the following additions, break down one of the terms and then reassemble the numbers obta ined to make the calculations easier: Example: 286 + 104 = (286 + 4) + 100 1 312 + 98; 114 + 1 116; 1 519 + 160 + 151; 1457 + 13 + 140
2. Work out the follO\'IIng . comparing the calculations and the results: 20- 8= ... (20+10)-(8+10)= .. . 53 - 9 = ... (53 - 3) - (9 - 3) = .. .
256- 7=... (256- 6)-(7-6) = .. . 96 - 49 = ... (96 + I) - (49 + I) = .. .
32 1 - 33 = . (321 + 7) - (33 + 7) = .. .
3. Add or subtract the some number to both terms to quickly calculate the following differences: Example: 100 - 36 :::: (lOO + 4) - (36 + 4)
= 104 - 40 = 64 147 - 17; 158 - 39; 342 - 78; 596 - 58; I 837 - 49; 639 - 129
4. Find an approximate value, then work out the following sums and differences: 948 523 + 4579 6321 + 849 124 569 + 72 426 127945 + 67 + 47 625 59677 1 - 196746 848439 - IQ 956 340 02 1 - 176452 39000 - 22132
5. Complete: 8 7 I 6 2 4 I . 8 8
+ 3 2 I 7 9 . + 2 7 0 8 3 580
2 543 . 8 3 7 6 6 8 0 5 4 492
SOLVING PROBLEMS
1. During the day, the cashier in a shop has taken In 3 200 VT , 10 170 VT and 2 210 VT. What sum of money has he in the till at the end of the day?
2. To buy a pair of trousers costing 2 350 VT, NeD g ives the shopkeeper a 5 (0) VT note. How much change must the shopkeeper give him? .
3. A lorry travels from Cotonou to AbidJan, a distance of 1 234 km. The first day It covers 317 km, the second day, 487 km. What distance has It already covered? What distance is It from Abldjan?
4. The tanker contains 8 750 ~ of petrol. It delivers 4 200 Q, then 2 300 ~. A third customer wants 2 450 ~. Can it fulfi ll this o rder?
5. A shopkeeper has three remnants of material on display. The first is 1.2 m 45 cm .Iong, the second 6 m 75 cm and the third as long as the other two combined. What IS the length of the third remnant?
page 9
Convex or concave polygons
LET'S FIND OUT
1. Here are some polygons.
~~'O~A£I ~<2>00 g
1 __ PO_'ygo_n._--LI _ _ con_vex_---tI __ con_cav_e ~I 2. How can you tell the difference between convex polygons and concave polygons?
3. In the table below, classify the polygons according to the number of sides.
Number of sides
Polygons
Tnong,"
LET'S LEARN ABOUT IT
1. Convex
Quadrilateral
Concave
Pentagon Hexagon Octagon
2. In a convex polygon, none of the sides, even when extended, cuts through the figure. In a concave polygon, a t least
Polygons a , c, 1. g, L j, k, I b.d.e, h one of the sides, when extended, Intersects the figure.
3. Triangle Quodrlloteral Pentagon Hexagon Octagon
Number of 3 4 5 6 8 sides
Polygons 0 f, g, k, I b, d,I.J c, h e
page 10
Revision AII--l L------------jW-J SOL VING PROBLEMS
1. For 8 hours of work . a labourer earns 2 160 VT. What will be his wages this week If he
has worked 6 days at the rote of 7 h 30 mln per day?
2. A match stick measures 5 cm. There are 50 matches In a box. If the matches were placed end to end, how many boxes would be needed to cover a distance of
240 km?
3. From New York to Washington, the distance Is about 230 miles. Express this distance
in kilometres. One mile equals 1.f:£:R km.
4. Two thousand years ago, Eratosthenes has measured the distance from Asswan to Alexandrkl in Egypt and found it to be 5 0CXl stades. Given that a stode = 157.5 m, what, in kilometres, is the distance between Asswan and Alexandria?
5. Adam takes 144 mangoes to market. He sells three quarters of them. How many
mangoes has he sold?
6. lowering a bucket into a well requires 21 tums of the handle. What Is the depth of the well?
~ ) I )))) ~
roller of 20 c m In diameter
7. 78.65 m has been used from a roll of wire measuring 165 m in length. Given that this wire weighs 0.065 kg per metre, what Is the weight of the remaining wire?
8. A wooden sklrtlng board costing 800 VT per metre Is Installed In a square room measuring 4.50 m x 4.50 m. What will be the cost jf labour Is 1 350 VT and there are
two doors in the room, each 105 cm wide?
page 167
DIvIsIon of a whole number by 25, 50, 75 iJ!I-J '--------'--------JW--J QUICK SUMS
I. Calculate: 2. Calculate: 12 + 25 = . 320+25= .. . 25+50= ... 102 + 50 = ...
60+25= ... 328+25: ... 125 + 50 = .. , 525+50= ... 72+25= ... 960+25= ... 75+50= ... 1 525+50= ... 84+25::: , .. 1 200 + 25::: .. , 110 + 50 ::: ... 3CXXl+50= ...
100 + 25 = ... 18CXJ+25= ... 246+50= .. . 9CXXl+50= ...
Example: Example:
32+25= 32 1( 4 =032)(4 = 100 . 1.28 42 + 50 = 4~~2 _ 0.42)( 2 = 0.84
3. Calculate: 15 + 75 ::: .. . 96+75 = ... 522 + 75 = ...
45 + 75 = ... 153+75= ... 681+75::: ...
121 + 75 = ... 312 + 75 = ... 963+75 = .. .
Example: 12 + 75 = ~ = 12 x 4 = 4 101 4 :::.J2.. = 0. 16 1(0 )( 3 300 100 100
QUICK SUMS
- By doing calculations with the digits given, find the following numbers:
Example 1: wtth the digits 3, 4, 5, find the number 256.
3 x 4 = 12 12x5=60 6Ox4=24O
4 x 4 = 16
240 + 16 = I 256 I
- Find the numbers: 512 wfth the digits 4,5,6 483 with the digits 4. 5. 7 318 wfth the digits 3, 4, 5 245 with the digits 3, 5, 7 113 with the djg~s 3. 5. 6 923 wtth the digits 7, 5, 6
1 028 with the digits 7. 2. 9
page 166
Example 2: with the digits 3, 4, 9, find the number 726.
4 x 3 = 12 12 x 9 '" 108
108 + 4 = 27 27 x 3 = 81 81 x 9 "" 729
729-3 =~
Convex or concave polygons
LET'S PRACTISE
1. Find the concave polygons from the figures below.
2. Draw a convex polygon with four sides. 4 . • Reproduce this 0 Then by changing only one angle, turn it Into a pentagon.
_ --"co:::n"'c::o::v:.::e"po=ly"g"'o::n.::. ___________ --j • Separate it into two figures with
3. Draw a convex polygon with five sides. the some vertl-Change it into a concave polygon by changing ces as the pentagon. only one angle. What are these two figures? Do the same INith a six-sided polygon.
SOL VING PROBLEMS
1. Draw a line AB 6 cm long. Draw a perpendicular line d to AB in its middle 1. On d measure lengths on each side of I: IM = IN '" 4 cm. What type of polygon is AMBN?
2. Draw 0 convex quadrilateral ABCD. Extend AB from B and place a point E on this extension. Join up EO. This will cut through BC at point F. What types of polygons ore AEFC and AECD?
3. - Cut out two superlmposable tT1ongles, as shown:
- With these two triangles, form: 0) a convex quadrilateral; b) a concave quadrilateral;
- Draw these quadrilaterals.
4. Study the figure ABCDEF.
,~---;>, F 1) How many sides has It? What's It called?
B
o
c
2) What type of figure is it? 3) Join AD. What type of figure is ABCD? 4) Join OF. What type of figure Is ABCDF? 5) What type of figure Is AFEC? 6) DIvIde the figure ABCDEF Into two convex
quadrilaterals. Name these quadrilaterals.
page 11
Measurement of angles - Using a protractor
LET'S FIND OUT
This is 0 protractor:
Using your protractor, measure these angles:
A
LET'S LEARN ABOUT IT
The protractor IooI<s like a half circle
It has 180 dMsions. Each division is called a degree. A degree Is represented by the symbol o.
The angle ABC measures 450•
page 12
B
What geometric figure does it look like? What is a protractor used for? How many divisions has It? What is each division called? ABC is an angle. measurement?
What is its
c
The protractor is used to measure angles
Measurement of the angles shown:
A B c
L~ ________________ ~ ____________ ~~ LET'S PRACTISE
1. Complete: 25 Q or 25 dm' of pure water weigls 25 .. . 62 rn' or 62 ... of pure water weighs .. . 7 hI or 7 ", of pure water weighs ... 6.76 Q or ... dm3 of pure water weighs .. . 3,04 r or ... cm" of pure water weighs .. .
SOL VING PROBLEMS
2. Complete: 2.25 f Of ... dm" of pure water weighs .. . 17 do' or ... dm3 of pure water weighs .. . 24 h or , .. rn" of pure water weighs .0. S hI or ... dm' of pure water weighs ...
1. The weight of water contained In a parallelepiped-shaped reservoir 4 m long and 3.5 m wide Is 2.5 tons. What Is the height of the reservoir?
2. What is the volume of objects which. displace:
when Immersed in a container of water
1 I ; 5 I ; 41 ;
34 hI ; 124 dot;
6 dl ;
19 dl;
1 I ' ~ .
15 dl ; 140 dl .
3. Twenty-elght bars of a luminium hove 0 total volume of 18 dm' and 0 totol weight of 46 600 g . Calculate the volume of one bar. the density and the relottve density of the metal.
4. What is the mass of a bar of Iron of 0.842 dm'. given that the relative density of Iron Is 7.6?
5. A sheet at zinc has a volume of 12.6 dm3• It takes 37 similar sheets to cover a
p latform. What Is the weight of this zJnc covering? The density of zJnc Is 6.86 g/cm3
•
6. A piece of Ice weighs 41.4 kg. Calculate Its volume. given that the density of ice is 0.92 kg/dm'.
7. A stone Immersed in a vase at water disploces 700 g of water. GNen that the stone alone weighs 182.4 dog, calculate Its density.
8. A barrel contains 135 Q of oM with a density at 0.81 kg/dm'. Calculate the total weight of this barrel fun at 011. given that the barrel when empty weighs 65 kg.
page 165
LET'S FIND OUT
study the sketches:
1 2
250 cm'
P~. of Iron
/ ObselVe the weights.
\ When the piece of iron is immersed in the water, the water overflows Into the graduated tube. Why?
What Is the weight of the piece of Iron?
The volume of water measured In the tube Is 250 c m3. So what is the volume of the
piece of iron? • Using the weight aOO the volume of the piece of iron. find the weight of 1 c m3 of Iron . • ,..The displaced water has a volume of 250 cm3,
What is its weight, given that the weight of 1 cm3 of water Is 1 g7 - Compare the weight of the piece of iron and the weight of the displaced water.
LET'S LEARN ABOUT IT
- The piece of Iron causes elevation of the water, which then overflows Into the graduated tube.
- The volume of the displaced water Is 250 cm3; It Is also the volume of the piece of
Iron. - The weight of the piece of iron Is: 1 kg + 500 9 + 400 9 + 50 g ::::: 1 950 g.
- The weight of 1 cms of Iron Is in grams: 12: ::::: 7.8
- We say that the density of the Iron Is 7.8 g/cm3.
- On the other hand: • 250 cm3 of water weighs 250 g; • 250 cm3 of iron weighs 1 950 g.
The iron therefore weighs 12;:} '" 7 ,8 times more than same volume o f water.
We say that the iron Is 7.8 times denser than the water and that 7.8 is the relative denstty of the Iron with respect to water.
page 164
Density of the Iron: 7.8 g/cm3
Re.mive denstty of the iron: 7.8
"'" dfIosIt!f /$ expreosed os g/cm' and If>e _e dfIosIt!f /$ exfX8$Md by file sanHI nun>IJM but _ any unit.
Measurement of angles - Using a protractor
LET'S PRACTISE
1. Measure the angles Fond M, in the figures shown,
2. Draw the following angles:
SOLVING PROBLEMS
49" • 58" . 79" . 100 . 64" • 90".
1, Measure the angles In these triangles. Work out the sum of the angles. A ,..-----, B 0,,----, .
c F -----.....',.G Q
What conclusion do you draw?
T
J
2, What Is the measurement of the angle formed by the two hands of the clock when it is: 16 h. 15 h. 12 h. 13 h. 14 h ?
3. Draw a right-angled triangle and mark the angles A, B and C.
- Cut It out and make two other copies, A
A - Arrange the three triangles so that the angles A, B and C (:») have the same vertex S,
- Measure the angles A. Bond C wtth your protractor, then measure the
S C S C
angle S formed by the Joining up of S A~ 8 the three angles.
- Compare the measurement of the angle S with the sum of the measurements of the angles A, B andC.
S CV
page 13
Multiplication of whole numbeis ~ L----------J8--J LET'S FIND OUT
1. Bars of soap ore to be packed in a rectangular cordbOard box. Seven con be placed widthwise , 9 lengthwise and 5 heightwise.
, , , , :> , , , , , , ,
. __ --.J •• _. .. .--
1 1 1 11-;::!I.lJ 1 1 1 ~.- qoots
_ How many bars of soap c on be placed horizontally across the bottom o f the box? _ How many identical layers will be needed to fill the box? - How many bars o f soap c an be packed in the box? _ Show other ways 01 calculating the number of bars which may be packed In the
box. - What do you notice about the arrangement of the fact()(s In the different
multiplications obtained?
2. A cook buys meat for the week twice. The first time he buys 13 kg meat at 350 VT per kg, the second time 25 kg at the same price. Work out the amount paid by two different methods.
LET'S LEARN ABOUT IT
1. The number of bars of soap a t the bottom of the box: 7 x 9 or 9 x 7. Five layers a re needed to fill the box. The total number of bars of soap:
,
(7x9) x5 or (7x5)x9or (5x9)x7
In mulllpllcalloll, /he cxdet 0I-/he _ may ~ cIIanged and /hey may ~ ~ at ... _ _ ling /he ... ,,11.
2. The total price of the meat may be calculated in two ways: - Total expenditure for the whole week:
(350 x 13) + (350 x 25) ()( 4550+ 8 750 = 13300
- Total weight In kg of meat bought for the week: 13 + 25 = 38 Price in votus of 38 kg : 350 x 38 = 13 300 Amount paid, 13 300 VT.
page 14
Relationship between capacity and volume ~ '----------lr#/-J LET'S PRACTISE
1. Complete: 2h@= ... dm3 4.5 cD :: ... ems l /8dQ", ... ems 5dQ= ... ems 1.7m3 = ... M 0.1 cD= ... ems 3 Q = .. , ems t rn' = '" Q 0.01 ,= ... ems
2. Express In litres. then In 3. Express in centilitres: 30 ems; 8.45 dm3; 14 500 mm'; 18.7 dm3
; 537 ems, decilitres: 8 m3; 15.75 m3
;
3.7 m3; 0.95 m3; 4. Express In rn' and In ems:
0.082 m3; 47 em3;
251 em3; 2.7 ems,
50 hI; 20 300 hI; 45 daD; 750 I; 20.3 hI.
SOL VING PROBLEMS
1. A cube-shaped box has an edge of 3 m. What Is Its capacity in dm3 and ht'l
2. Three cisterns have respective capacttles of 27 m 3,
255 hI and 32 450 I. Express In heclolttres and Ittres the total capacity of the three cisterns.
3. A truck transports 36 m3 of petrol. This is siphoned off into vats of 1 CO) ,. How many vats con be filled?
4. A reservoir with rectangular walls has on interior volume of 168.75 dm3
.
How many hectolltres does It contain
when It Is t full?
5. A reservoir In the shope of a rectangular prism measures 1.50 m long, 1.20 m wide and 0.85 m high. What Is Its capacity?
6. A pond has 0 capacity of 1.265 m3.
73 bucketfuls of 1 do' ore token from It. Calculate In litres the volume of water remaining In the pond.
8 . A reservoir can contain 15.75 m3 water.
! are filled . Calculate:
- In hQ, the volume of water It contains; - In Q the volume of water still required
to completely fiR it.
7. A top d ischarges 2 d' of water per second . What Is, In dm3
, the volume of water d ischarged in one minute , In one hour?
9. A water tank Is In the shope of 0 cube with on edge of 15 dm. What Is its volume In m~ It Is filled with water to supply a family who use 337.5 , of water per day. How many days will the supply last?
page 163
LET'S FIND OUT
Matthew pours a full bott1e of water into a cube-shaped box wtth an edge of 1 dm.
---- -~ ~ , , " , , , , , ,
" , , , , , . -.............. " . \.
Set the equtvalenc es out In a converslon table.
LET'S LEARN ABOUT IT
Mot1hew Is going la discover that 1 lire completely fills the cuoo.shaped box with on edge of 1 dm.
I , litre c~ 10 a CubIc ~. I 1 t= 1 COJmt .. l dm3 = 1 COJ c m'
lcm3 =lmf
1 rn' = 1 OCXldm3", 1 COOf
What will he discover?
To what unit of volume does 1 litre correspond?
Do the following conversions:
11 r = .. . m~ ; 1 r", ." dm3 = .H cm'
To what unit of capacity does 1 m3
cooespond?
F=
------ ------1 litre - 1 dm'
ldm
Table at conven:lon and correspondence
m' dm' cm'
h I u h t u h I u
hI dol I d l c l ml
1 0 0 0
t 0 0 0
1 0 0 0
page 162
Multfplicafion of whole numbers
LET'S PRACTISE
1. Regroup the fac tOf'S to make the c alculations easy: 3 x 2x6 = .. ,: 2)(7)( 4= .. . : 4 ><7x5 = ... : 3x7)( 3= .. ,: 2 )(5x7 )(5= ...
2. Calculate in two ways: 2 x (7 + 5) = ... ; 3 x (5 + 9) = ... ; (5 + 10) x 12 = ...
3. Write In the form of 0 sum of two products: Example: 17 x 15 = (1 0 x 15) + (7 x 15)
23 x 12 = ... ; 20 x 153 = ... ; 36 x 25 =
4. Without working out the sums, compare the following using the signs >, :: or < and explain your method: 7 x 9 and 9 x 8 ; 250 x 6 and 25 x 60 ; 42 x 3 and 6 x 21 35 )( 9 and 7 x 50 ; 400 x 8 and 200 x 17 ; 60 x 125 and 120 x 6
SOL VlNG PROBLEMS
1. A breeder exports 1SO sheep at 7 CO) VT each. If 25 die in transtt, how much will he make?
2. Brian plants 30 rows of lettuce. with 24 in each row, and Jocob plants 6 rows of 120 each. Who has planted more?
3. A rectangular garden measures 65 m )( 40 m. It Is enclosed by fencing costing 725 vr per metre. What Is the cost price of the fencing. if the gate is three metres wide and costs 1 5CO VI?
4. At a fair. Joke sold 485 sheep for 6 700 VT each. 237 goats for 4 450 VT each. ond 129 heads of cattle for 43 650 VI 0 piece.
0) What are the takings for each group of animals? b) What total amount does the breeder make from the sole?
5. To renew her wardrobe. Kylie buys 3.75 m of poplin at 500 vr per metre and 2.50 m of dimity at 800 VT. Calculate her total expenditure. To make a blouse, she needs 1.25 m of material and for a d ress 1.50 m. How can she best use the ma terial to get 2 d resses and 2 blouses out of It?
page 15
LAng _ _ Ies_In_._Isosc __ ele_S_0I_eq_ ul_Im_8_ral_lrIa_ngle _ _ S _ __ ---"-----j·1IJ LET'S FIND OUT
Situation 1 : SHuatlon 2 : This is an isosceles triangle with the base This is an equilateral triangle: BC,
A
B C - Using your protractor, measure the angles
A. Band C and fill out the table:
Value In degrees
LET'S LEARN ABOUT IT
Situation 1 :
- The triangle ABC is an Isosceles triangle .
A B C
I Value in degrees 40" 70" 70"
I''; anllOsc_ tnang/e' /he ",,~ a( I the bt;IstI havf1 the .$Qf)'J6 vaIut!. <.j
- If an Isosceles triangle is also a rlght-angled triangle, what is the measurement of the acute angles?
page 16
E
f G
- Using your protractor, measure the angles E, F and G and fill out the table:
Value In degrees
Situation 2 :
- The triangle EGF is equilateral.
I E
I F
I G
I I Value in degrees Iff Iff Iff
I,n;';' ~~, each of the I anQte. 1ileaJ .... 6(} degrees.
Rate of ffow 1fI1Il '------------lW-J LET'S FIND OUT
In order to fill a swimming poot with a capacity of 780 m3 the tap has been left open for 24 hours. - What average volume of water flows through this tap In 1 hour, 3 hours, 12 hours?
How long will it take to discharge 520 m3 of water? - How do you call the volume of water flowing from the tap during a unit of duration? - Express it In: dm3 per mln (dm3/min);
cm3 per s (emJ/s).
LET'S LEARN ABOUT IT
- Total of m3 discharged from the tap: In 1 h = 780 = 32.5' in 3 h = 780><3 = 97.5' In 12 h::: 780)( 12 = 390
M · M' M
- 32.5 m3 in 1 h or 32.5 m3/h Is the rate of flow from the tap.
- This can also be expressed as: dml/rnin, Q/rnln, c rnl/s, etc. The calculation below shows that the volume of water discharged is proportional to the duration of flow, so you can use a table of proportionality to make the calculation easier.
Dwatlon of ftow 24 1 3 12 b-In hours k~ 32.5 (. 32.V
VoflM'l'l8 of water discharged 780 32.5 97.5 390 520 ~ ...-/ In m'
To calculate the rate of flow, the duration of flow and the volume of water discharged, It Is easier to use the properties of proportionality.
The duration in hours needed to discharge 520 ml is g::; = 16.
SOL VING PROBLEMS
1. A hand pump discharges 1.5 litres when the handle Is activated once and this Is dane three times In 5 seconds. - What Is the rate of flow per minute? - This pump is used to empty a tank of 225 litres.
How long will it take to empty It?
2. A pump is used to fill the petrol from a tank into a petrol tanker. After 26 minutes pumping, the tanker Is full. - What in litres is the capacity of the tanker, given that the pump discharges
150 O!min? - If a pump with a flow rate of 70 O!mln is used, how long will It take to empty the
tank?
page 161
LET'S FIND OUT
look at this cylinder wfth a radius at 3 cm and a height of R 5cm.
Remembering how you calculated the volumes of the cube h and the parallelepiped (rectangular prism), how can you
calculate the volume at a cylinder using the base and height measurements?
LET'S LEARN ABOUT IT
The volume of the cylinder Is equal to the area of the base multiplied by the height.
B: area of the base
Calculating the area et the base of a cylinder: the base is a disc, therefore B=nxRZ
Calculating the volume of the cylinder:
B = 3.14 x 3 x 3 = 3.14 x 9 B = 28.26
The area of the base is therefore: 28.26 cm2
.
LET'S PRACTISE
SOL VING PROBLEMS
3m
Sm
R:""3cm
so cm
120 cm
V=Bxh V=28.26xS V= 141.30
The volume of the cylinder Is therefore: 141 .30 cm3
.
42dm
37dm
29dm
Sm
1. A cylindrical vat has a 2. Calculate the volume of a drum radius of 1,4 m and a height of petrol with a diameter of of 2.5 m, 46 cm and a height of 6,2 dm, ~L What Is Its volume in dm3?
• "
page .60
Comparison of angles
LET'S PRACTISE
1. Construct an equilateral triangle with sides of 8 cm. By cutting out and folding It, compare the three angles of the triangle.
2. Andy has constructed two triangles and measured their angles. Check the values with your protrac tor and state . true. or . false_,
Triangle 1: Triangle 2: 300.& . 90" & . 40" . 500
• True or false? True or false?
2
3. Cut out an Isosceles triangle ABC, so that AB = AC. Now fold it a long the height stemming from A What can you soy about the angles B and C?
4. - Us i ng a set A 5. Draw a line AB and draw the square, check perpendicular line to AB in its that the angle A B middle. Place a point C on this is a right angle. perpendicular and complete Using your pro- the triangle ABC. Measure the tractor, compare angles A and B with your the angles Band protractor. C.
• What kind of triangle Is ABC? • What Is a triangle like this called? C
6. Draw a line AB 4 cm long. On the same side of the line at A and B construct two angles of 50 degrees.
h ~ Extend the sides of the angles to the point of Intersection C.
· What kind of trlangle is ABC? A 4cm B
7. Construct two equilateral triangles with 8. Draw an isosceles right..angled trlangle. sides of 3 cm and 5 cm respectively. Measure the acute angles.
· Compare the angles of these two - What value have they? triangles. • Are the acute angles In an isosceles
- Who' can you soy about their right-angled triangle always equal? Do measurements? they always measure the same number
- Can you state the some thing for any of degrees? other equilateral triangle?
page 17
"
Comparison and ranking of wfloIe numbers
LET'S FIND OUT
This Is an extract from the State Budget (in vatus). • Education ....... . .... . .. . • Health . ......... ... . . . ..... . .. . .... . • Post and Telecommunications . • Armed Forces . • Culture .... . .... ... . .. .. .
- Which sector has the biggest/smallest budget? - Which budgets are:
• larger than 1 OX! (XX) OCXJ VT? • smaller than 1 0Xl (XX) !XX) VT? • between 500 OCO cx:o VT and 2 OOJ CXXl (XX) VT?
13645 316 975 1923810406
828790 305 7 026 oo:J oo:J
52531561)
_ Compore the budgets two by two, using the signs < or >. - Arrange them In Increasing then decreasing order. - Round each budget to the:
• nearest thousand; • nearest million; • nearest hundred of millions.
_ Put the budgets between the two nearest billions. - State the total sum of the budgets to the nearest billion.
LET'S LEARN ABOUT IT
1. - wtth two whole numbers. the larger is the one with more digits. We can wnte 13645 316 915 > 1 923810 406.
_ If two whole numbers have the $OITl8 number of digits, we begin comparing them from the ktft. Example: compare the numbers 7 826 853 427 and 7 923 810 406 . • The first digit is the same: 7; that doesn't tell us anything. so we go on to the next
digit: 8 < 9. • The refore we can write: 7 826 853 427 < 7 923 810 406 .
2. We can round 0 number up or down to the nearest ten, hundred. thousand. Example: the number 6 980 768 may be rounded: • to the nearest upper ten: 6 980 770;
this Is rounding up; the number obtained is larger than the g iven number.
• to the nearest lower ten: 6980 7t1J; this Is rounding down; the number obtained is smaller than the given number.
In the some way. we con round 0 whole number up or down to the nearest hundred. thousand. million ...
Volume of cubes and rectangular prisms
LET'S PRACTISE
1. Complete the table: 2. Calculate In dm" the volume of cubes
Edge Volume of the cube with following edge measurements:
5em 4dm; 30 cm; 80 cm; 2m; 0.7 m: a.05m; 3.8 m ; 15 cm.
0 ,) dm
45 cm 3, Calculate In rn' the volume of cubes
37 m wtth following edge measurements:
4m: SOdm; 200 cm; 84 dm 35.6 m: 1.7 dam; 3.05 dm.
4. Complete the table:
Area of bole I w h V
14 cm 5em 8em
24 m2 60dm SOdm
)m Um 600
7em 4 cm 80 mm
5. Complete this table,
...... "'boIe 14 cm2 30 cm' 9m' 150 mm2
h 5 dm 20dm 0.2 m 40 cm v
SOL VING ProBLEMS
1. How ma~rg~m3 can you fit into 0 square with sides of 1 m and 0 height of 1 dm. then a het ht of 8 dm?
2. What Is the volume of 0 block 17 cm 3. A cube-shaped c rate wtth an edge of long, 10 cm wide and 9 cm high? 0.65 m Is filled wtth rice. What is the
price of the rice If It costs 40 oc:o vr per cubic metre?
4. A ~ardener has to transport some manure In 0 cube shaped container with an edge of .80 m. He ma~es 29 trips. • Calculate In dm the volume of manure transported. ~ ~ has to pay 75 VT every time he goes fOf manure. How much will he hove to
In al?
5. A dormitory measures 24 m long, 12 m wide and 2.5 m high. • What Is Its volume? • If each student needs 15 m3 of air. what numbef of students should sleep in the dorm~OI'(1
page 18 page 159
Volume of cubes and rectangu/aI pd8m$
LET'S FIND OUT
1. This Is a cube wtth an edge of 3 cm:
- How many cubes with an edge of 1 cm are there on the bottom layer?
- How many layers with a height of 1 cm could ftt into this c ube?
f - Calculate the total number of cubes of cm3 in this cube with on edge of 3cm.
LET'S LEARN ABOUT IT
1 .• The Ioyer shown contains 9 c ubes with an edge ot 1 cm (9 cm">.
. Three layers c an be tttted helghtwlse. • The to tal number of c ubes with on
edge at 1 c m Is 3 x 3 x 3 - 27 .
The volume of the cube Is:
V- e. e • •
page 158
2. This rectangular prism has the dimensions 5 cm, 2 cm and 3 cm:
.-/ .-/
I I I I y - How many m3 are there on the
bottom layer shown In the plan? - How ma ny layers 1 c m high are there
heightwlse? - Calculate the total number of cubes
of 1 cm3 which can be tttted Into the rectangular prism.
- How can you calculate the volume of the rectangulor pnsm given Its dimensions?
2 .• The layer shown contains 10 c ubes with a n edge of 1 cm (10 cm">.
((rrrEO . The number o f Ioyers helghtwlse Is 3 . • The total number of cubes Is:
3x5x2=30 This resutt represents the product of the three dimensions of the rectangular prism.
The volume of the rectangular prism Is:
I V - ,xw x h I I x w Is the area B of the base, from which:
Comparison and ranking of whole numbers
LET'S PRA CTISE
1. Using the signs <. > or =, compare the following numbers: 12016927 and 12022058; 19783 219319 and 17783 219 315; (24 x 1 OCO CXXl) + (67 x 10 OCO) and 24 670 CXXl .
2. What is the largest 5-digit number? What is the smallest 6-digit number? Which is the greater of these two numbers?
3. Look at the following numbers: 93 159; 3 ti::R; 215282; 11 835; 27069312; 150 308; 73083; 71 818; 152809. a) Classify them In the table below'
Numbers < 19600 21 560 < numbers < 93 500 Numbers> 152 908
b) Write these numbers in decreasing order. c) Round the smallest number up to the nearest hundred. d ) Round the largest number d own to the nearest thousand.
4. look at the number 476 382. • Write down 011 the numbers you con obtain by Inserting the digit 9 between two
neighbouring numbers . • Write the numbers obtained in increasing order .
SOL VING PROBLEM
Four fishing boots mode the following c atches: Boat one : 12625 tons; boot two: 110 120 tons: Boat three: 9 615 tons; boot four: half of boot two's catch. 0) Which bOat caught the most fish? b) Round each catch up to the nearest thousand tons. c) Without actually working it out, give an estimate of the quantity of fISh c aught by
the four boots.
page 19
operations with whole numbers
SOL VING PROBLEMS
1. A cook earns 50 (XX) VT per month. Every month he pays 8 (XX) VT rent, 3 0:::0 VT for
electricity, 1 800 VT for water and 10 CXXl VT for food. He wants to buy 0 moped In Instalments and would need to make a monthly
payment of 25 OXJ VT. _ Con.he undertake such a purchase? _ If so, how much can he save?
2. A lorry driver leaves Santa with 0 load of 6 489 kg. In Pusei. he unloads 2 750 kg and
takes on 1 870 kg. _ What Is the new weight of the load? Is the lorry lighter or heavier than when it set
out? - What's the d ifference?
3. Andy is in debt. He has paid back 11 COJ VT. then 5250 VT. He still has 3 OCO VT left
to pay. o What was the amount of Andy's debt?
4. A merchant first sells 24 metres from a roll of c loth. He then sells 16 metres less than
the first time. o What length of material does he sell the second time? The third time, to flf1ish the roll. he sells 13 metres more than the second time.
o What length at material does he sell the third time?
o How much material was there In the roll?
5. A car costs 2400 OCO VT. The dealer suggests credit payments spread over three
years, with quarterly payments at 280 OCO VT. o What will be the better type of payment tor the customer, cash or c redit?
page 20
SOL VING PROBLEMS
1. On a map to the scale of -2 - '- what is the distance between two towns 45 km apart? 5000J
2. The top of a table Is a rectangle 240 cm x 100 cm.
Draw it to the scale of _1 . ...!.. _,_ 10 50' 100 '
Compare the area of each pion to that of the table.
3. A rectangular garden measures 20 m )( 15 m. Make a plan to the scale of -'- . 500
Draw up a table showing this situation and calculate the dimensions from the plan of the garden.
4. A rectangular field measures 240 m x 93 m.
Make a plan to the scale of 3 ~ .
By how much do you need to multiply the perimeter of the plan to find the actual
perimeter at the field?
5. Make plans of a rectangular yard to the scales of ..1.... 2 I 100 ' 100' 100 '
The yard measures 30 m x 18 m.
Draw up a table to help you calculate the dimensions of the yard on the plan. Draw the three plans.
Does the form of the rectangle change? What about Its dimensions?
6. Jake has bought two maps of the same country. The first one is to the scale of
50~ and the second to the scale of SOOlClXJ .
o Two towns In this country are separated by a distance of 30 km. What In cm is the
corresponding d istance on each of these maps?
• Joke measures 25 cm on the first map and 5 mm on the second. To what actual distances do these measurements correspond?
page 157
1'IopoItfonaIIt: plan and ICoIe
LET'S FIND OUT
1. An architect askS 0 draughtsman to draw him 0 plan to the scale of 5 ~ for an
airport. The runway Is to be 3 km long and 100 m wide. The draughtsman draws up a plan on which the runway is 30 cm long and 1 cm wide. Draw up a table showing what the architect wants. Draw up a table showing the draughtsman's plan. Has he kept In mind the requested scale? If not , what scale has he been working to?
2. The model of a car to a scale of ~ measures 15 cm In length.
Draw up a table of proportionality for this situation. What is the actual length of the cor? The width of the cor Is 1.45 rn, what will this be in the model?
LET'S LEARN ABOUT IT
l.
Actuc:i dimensions 50CXl 3 OOJ 100 Actual dimensions
'nm 'nm
Olmenstons on the 1 0.6 0.02 Dimensions on the
pkm In m pion in m
100
0.01
Table illustrating Table Illustrating
3 OOJ
0.3
the arc hitect 's request the draughtsmm's work
Closer examination of the tables shows that the draughtsman has made an error: 1
0.01 = 100 + 10 CO) or 100 x lOM . , The scale used by the draugtsman Is 100CXl .
2. ~ ActuaI_ 20 ? 145
In cm 0do Dirnenl60N et the fTIOdeI 1 15 ? In cm
\@J
Actual length In cm: 20 x 15 = 300 .
Width o f the model In cm: 145 )( ~ = 7.25 .
page 156
Operations with whole numbers ~ '------------j8-J SOL VING PROBLEMS (continued)
6. A IOfry Is transporting 400 crates each containing 12 boffies of drink. At the Esperonce restaurant. it delivers 124 crates and collects 72 crates of empties. At the Welcome restaurant, it delivers 72 crates and collects 32 crates of empties. - How many crates of full bottles are still to be delivered? - How many empties hove been collected ?
7. An agricultural cooperative buys 0 plough for 180 (XX) vr and 0 tractor whic h costs 2 260 OCXJ VT more than the plough. - How much does the trac tor cost? - What Is the total expenditure?
8. The local sporting newspaper appears each week and costs 100 VT. There are 48 Issues per year. The subscription Is 3 8(X) VT annually and 1 200 VT quarterly. - What Is the best value for the reader, to buy weekly, or subscribe quarterly or
annua ly?
QUICK SUMS
Do the fo llowing additions, breaking down the terms and regrouping them:
36+ 14 : ... 34 + 46= ... 53 + 39= ... 84+ 9: ...
53+20::: ... 76 + 26 = ... 69 + 18: ... 35 + 15: ...
18 + 27 : ... 45 + 38= ... 44 + 16 : .. . 38+ 12 : ... 43 + 37 = ... 56 + 26 = ... 56 + 29:::: ... 27 + 37 : ... 62+28= ... 78 + 25 = ... 73 + 17: ... 63 + 28= ...
page 21
calculation of perimeters
LET'S FIND OUT
look at these polygons:
A 15
~OB~ A(-:----f
D<:;:t i--~~ B C E 35
30 D A 3 B
J~l • Write down the perimeter of the polygon in the form of the sum of the lengths of its
sides . • What Is the perimeter of a polygon equal to? _ How many sides have the polygons 1? 2? 3? 47 57 67 17 _ What kind of flQures are the polygons 5 and 67 _ Show a simple way of calculating the perimeter of the polygons 5, 6 and 7? _ Calculate the perimeter of the polygons 1, 5 and 6.
LET'S LEARN ABOUT IT
_ To find the perimeter of the polygon ABCDEF. you add up the lengths of each of the sides: AB + BC + CD + DE + EF + FA
"': ".""...,.,., of Q ~ " <>qJQI1o ",. ..." of,,!! /egIh5 "! ~ --
I Polygons
_ Figure 5 is 0 rectangle. Figure 6 is a square. Figure 7 Is an equilateral triangle.
Here ore easy ways of calculating the perimeter of some polygons: _ Perimeter of rectangle 5: P '" 2 x (f + w) with I = length and w = width _ Perimeter of square 6: P = 4 x s with 5 := side _ Perimeter of the equilateral triangle 1: P = 3 x a with a = side of the equilateral
triangle. _ Calcula tion of the perimeter of the polygons 1, 5 and 6 is as follows:
Perimeter of the polygon 1: p:::: 15 + 25 + 35 + 30 + 20 + 50 = 175 Perimeter of the rectangle 5: P :::: 2 x (7 + 3) :::: 20 Perimeter of the square 6: P = 4 x 3 = 12 Perimeter of the equilateral triangle 1: p:::: 3 x 5:::: 15
L-VoMne __ '_: unJIs __ W_._d_'m'_._Cl_rn'_._mrn'_) __ ~_-=-_~~ LET'S PRACTISE
1. Complete: 5 m3
:::: ... dms ldm3 = ... mm' 1 cx:o dm3 = ... m3
3.2 dm3 = ... cm3 lcm3 :::: ... mm3 15 (XX) m3 = ... dm3
0.72 dm3 = ... cm3 l m'::: ... cm3 15 (XX) cm3 = ... dm3 - .. . m3
2. Fill in the conversion table:
0.6 m' 150 (XX) cml 100 cm3 64 cml
2.65 dm3 25 dml
7.3 m3 247 mm3
9 (XX) dm3
3. Complete the following: 586 dm3 + ... ::: 1 m'... 415 cm3 + ... = 1 dml 6 cm3 + 721 cm' + ... ::: 1 drn' ... + 68 dml :::: 1 rn' 315 dm3 + ... dm3 :::: 1 m' 34mm3 + 15cm3 :::: .. . mm3
SOL VING PROBLEMS
1. What number do you need to multiply 125 cm' by to obtain the
2. Dad has c leared 4 m' kg of boulders from his field. He takes them to the side of the road in a
following: wheelbarrow with a maximum capacity of
125 dm'? 125 dm'.
125 m'? How many trips must he make?
1.25 m'? 3. Given that the volume of a brick Is 1.458 dml, 0.125 ml? calculate the volume of 2 350 bricks. 12 5(X) dml?
4. The 75 students at the school in Kyanza have been vaccinated against diphtheria. Each student received 3 Injections of 2 cm'. Calculate the volume of vaccine used at the school.
5. How many ampoules of 5 cm3 can be filled from 2.5 dml?
6. A mason builds a wall with 8 OOJ bricks III each with a volume of 1.200 dml. What's the volume 01 the wall. given that he uses 3.10 m3 of mortar between the bricks?
7. What shoud be the volume of a classroom for 60 students so each student has 2 ml of olr?
page 155
Calculation of perimeters ~ "-----------------j8-J LET'S FIND OUT LET'S PRA CTISE
~ ~ • How many dm3 ore
1. Complete this table concerning five different rectangles:
Width 75 cm 143 m 58 em 42 m ? there In a cube with an edge of 1 m?
Length 97 cm 228 m 198 cm ? 32,5 cm Perimeter ? ? ? 216 m 119,50 m
/he voIt.me of ~j c.- wIih ari~ - How many c ubes with an edge of 1 cm are Of 1 an Is ~,tfe there In 0 cube with
cul>lcdoctn.ttw (t/Iri) an edge of 1 m? 2. Anni9 trims a square cloth measuring 125 cm x J 25 cm with braid which costs 220 VT
per metre. How much will she spend? ,
- _ do you col"'" _ ofa ,~ - How many cubes with
cube wItt! C1f/ ec.g& of 1 cm? an edge of 1 mm are , there in a cube with
- woof do you call"'" voIt.me ofa an edge of 1 cm? c.- wItt! an ec.g& of 1 (TVTI?
- Draw up 0 table for - _do you cdI"", - of~ll the conversion of the , c.- wItt! ,n ec.g& of' ",l" u n It s 10 r t he . " . ' , . measu remen t 01
3. A rectangular garden measures 24 m 4. The principal of 0 school organizes 0 race long and 16 m wide. Fencing it in will around a sports field measuring 87 m long cost 850 VT per metre. How much will and 49 m wide. When the students have it cost? done three laps how much distance will
they have covered?
5. The measurements of the sides of the polygon ABCD are: AD = 8 cm; AB = 4 cm: volume.
1 em' H im' BC = 5 cm and CD = 5 cm. Work out the perimeter.
V ~ SOL VING PROBLEMS
1. - A polygon ABCDEF has the following 2. - A field is 76 m wide. The length Is 27 m
LET'S LEARN ABOUT IT dimensions: more than the width. AB = 4 cm BC = 3 cm CD = 5 cm - What is the length of the field? DE = 2 cm EF =8cm FA = 6 cm - Work out the perimeter.
- Work out the perimeter.
3. Look ot this parallelogram ABCD:
B~ ~D - Which sides of the parallelogrom are equal? - Calculate its perimeter. - Demonstrate a simple method of calculating the perimeter of a
"m parallelogram.
Attb. wftt) an edge o f A c.-..."" an '. li' 1 m has.a votme of 1 rrf ec.g& of 1 cm has (cublcnielTe) . a 1dJme..of 1 crrf A eubi. wItt! an .;ag.. of '
(cubic cenft)'I91T9) , mm has a _ of , mrrf . _'" . (cubIc_i,!1
rrf, rXd. Crrf. mrrf ere II1Ifs tor """C' ' rneostKement of 1dJme" I,
4. The figure ABCD is a rhomb (diamond-shape): B
What can you say about its sides? A<>C -- If AB measures 5 cm, c alculate the perimeter of the rhomb and
demonstrate an easy way of· doing this .
Conversion table m' Om' cm' mm'
1 ml = 1 COJdml 1 0 0 0
1 dm3 = 1 (XX) c m3 1 0 0 0 ...1)_
5. - A field consists of a rectangle Joined to a j
I square, as ittustrated, The rectangle measures
I ,
276 m long and 75 m wide, and the side of , the square is 92 m long.
, , Calculate the perimeter of the field.
,
1 cm3 = 1 ())) mm3 1 0 0 0
1 m3 = 1 (0) Cl)) cm3 1 0 0 0 0 0 0
1 "'" II1Ifs of __ DId ~" t,y ,cm,· £ 1 , ,
page 154 page 23
Iloman numerals ~ L------------'------'-----------,Ii#-J LET'S FIND OUT
1. These two clocks show the same time.
Clock A ClOck B
• What time Is It? Wrfte the time in the usual numerals.
- Wrfte the usual numerals corresponding to the symbols on clock 8.
- Read all the numbers written In Roman numerals on dial 8.
- Indicate the three Roman characters at the base of the Roman numbers written on dial 6.
2. Study the numbers on dials A and B. - Draw up a table showing how the numbers from dials A and B correspond. - Starting \o'J1th the numerals on dial B, work out the rules for writing as Roman
numerals the numbers: a) to the right of the characters V and X; b) to the left of the characters V and X.
- Use the rules to read the numbers)(N, XIII , XVII, XIX , XVI .
3. - Loak closely, XXIV . XXII • lOOM • XXIX . XL • III XC XCII 24 22 36 29 40 52 90 92
- Work out the rules fOf writing numbers as Roman numerals.
4. - These ore the basic characters in the Roman numet'ol system:
I I v C M
5 100 1 IXXJ
- Using the rules we have worked out aoove, read the numbers: XL • XLV. XC . XCIII . DL . DCX • MCM .
- Write the following os Roman numerals: 46 • 37 . 53 . 94 . 125. 1 200 .
LET'S LEARN ABOUT IT
Note
Before adopting the decimal numeral system. certain peoples used Roman numerals. They ore still used on some clock foces. and on inscriptions on monuments ... They are also used to Indicate the order of succession of kings and queens. chapters of books, centuries ...
page 2A
PropottIonaIJty: plan and scale
LET'S LEAliN ABOUT IT (continued)
2. The graphic scale used In situatk>n 2 tells us that 2 c m on the pion represent 10 km In reality or that 1 cm on the plan represents 5 km (or 500 CXXl cm) in reality. That
graphic scale corresponds to the numerical scale soo'OC(J .
The straightest d istance between two points on the ground Is obtained by measuring the length of the segment Joining the two points on the map and by mutflptying this number by the numerical scole's denominator.
Example: If two towns A and B are 9 cm away on a f:«J'OCXJ scale mop, on the
g round the distance between the two towns Is 9 )( 500 (XX) = 4 500 CO) cm ,.. 45 km.
LET'S PRACnSE
1. On your mop 1 cm represents 2. On 0 pion 5 mm represent 1 m on the 1 OX) cm In reality. ground. Gtve the scale of that pion. What Is the scale of the mop? Calculate the real distance COfTesponding to
a distance of 3.5 cm on the pion.
3. 18 cm on 0 mop represent 30 km. Wrthovt calculat1ng the scale. indicate the real distance represented by 3 cm. What is the scale of this mop?
4. You are given a map with a scale of 25~ . What real distance Is represented by
1 cm on the map? (Give your answer In cm.) How many centimetres on the mop do you need to represent 5 km on the ground?
5. Write In digits the scales: one 6. The school's gate Is 4 m wide. On the plan it Is hundredth, one millionth, one represented by 4 cm. twentieth . one fifth . one five What Is the scale of that plan? thousandfh.
7. A playground is 0 rectangle 40 m long and 20 m wide. On 0 pion It Is represented by a rectangle of 8 cm by 4 cm. What Is the scale of the plan?
8. And the numerical scale corresponding to each of the graphic scales: 0 5km 0 20 km 0 12km L-..J I I I I I I I
9. What length represents 0 distance of 2 km on a map whose scale Is 2OO1cm ?
page 153
ProportIonality: plan and sca/6 1I{Il ~---------jrItI-J LET'S FIND OUT
1. Elleen wants to make 0 pion of her classroom which measures 11 m long and 9 m wtde. She decides to reduce the actual dimensions by dividing them by 100. - Work out the dimensions of the c lassroom on her plan. o FlII in the results in the table:
DimensioN on the pion 9 ~ In cm
Actual dimensioN 0 100)
Incm 900 1 100 100
_ Haw do you go from the actual dimensions to the dimensions on the plan? - Is it a proportion table? If so. find the proportion coefficient. - Fill In the lost column In the table.
Explain what It means. - Con you work out the sca le for Elleen's plan? Draw this pion.
2 . The scale of 0 map Is not always represented by 0 fraction. Very often 0 graphic
scale Is used.
_ What Is meant by the graphic scale 0,-, _---'-_1--'0 ,km ?
_ What do you hove to do to find the direct distance between two towns A and B1
LET'S LEARN ABOUT IT
1. Let's work out the dimensions of the plan in cm and fill In the table: I 100 900 1'00 - 11 ; 100 = 9.
- The lost column Is completed by ~~ = 1.
- To wOO< out the dlmensk:lns on the plan, divide the actuol dimensions by 100. this is a sltuatJon InvoMng proportlonallty.
_ The lost column means that 1 cm on the plan represents 100 cm In reality. this shows the rekJtk)n between the dimensions of the plan and the corresponding octuat dlmensklns, expressed In the same unH. this relation Is the scale of the pion,
written os 1~ and read as «one hundredtf'l.-.
- In doily life. we often encounter scales: 1 100 1 1 1
50 • 100 ' 20 OO'J ' 50J OO'J ' 1 CO) CO) ...
page tS2
Roman numerals
LET'S LEARN ABOUT IT (continued)
1. The relationship between Roman numerals (dlal B) and the usual or Arabic numerals (dkJl IV Is set out In this table,
Dial • I 11 III IV V VI VII VlII IX X XI XII
0101 A 1 2 3 4 5 6 7 6 9 10 11 12
_ You'll hove noticed that all the numbers on dial B are written uslng the characters: 1= 1 .V=5.X = 1O .
2. From dial e, we c an woO< out the rules for using Roman numerals. - A c harac ter placed on the right of another which Is equal to It or larger. Is odded
to the first. Examples: 11 = 1 +1= 2 VI=5 + 1 =6
_ A charac ter ~aced on the left of another which Is larger. is subtracted from the second . Examples: IV= 5 - 1 = 4
IX =1O-1 = 9 _ Any character placed between two larger ones will be subtracted from the one on
the right. Examp,,, XXIX = 20 + (10 - 1) =20+9::29 - The some character may be used three times Of more.
Examples: III = 1 + I + 1 :: 3; XXX:: 10 + 10 + 10 = 30
3. Based on these rules: - these may be rood os: XIII = thirteen; XXIV = twenty-four; XXIX = twenty-nine; - these may be written as: 27 = XXVII; 53 = UII; 125 = CXXV ...
LET'S PRACTISE
1. Write the following as Roman numerals: 7 . 60 . 19 • 34 . 47 . 72 . 120 . 210 • 600 . 930 . 1 S43 .
2. Write the following In Arabic numerals: LXII . VIII . CD . CVIII . MCC . XlC . Put them In decreasing order.
3. Wrfte down the followtng numbers from the smallest to the largest: Cl • MCDlXX • MM • XXXVIII .
4. Compare these numbers: XXXVI and Xl; XLIII and LV; MCC and MCX; MCXC and MCD; CMl V and CMXV; CCDXlI and CDVl .
page 25
L ________________________ ~~, . 0Iv/$I0n of whole numbers (Whole quotients) fiI-J
LET'S FIND OUT
1. A wholesale merchant p repares 2 793 bottles of 011 for dispatch In boxes of 12.
• How many boxes can he completely fiU?
According to Matthew: .Wdhout doing 0 dMslon, I know that the quotient of 2 793 dMded by 12 must be close to the quotient of 2 790 d Ivided by 10 I.e . 279. If win be a number of three digits .•
• Wrlte down and work out this division. Explain the various stages. Was Matthew
nght?
2. Another wholesale merchant wants to dispatch 17856 bars of soap. He packs them In boxes of 12. These boxes ore then transferred to crates which can hold 12 boxes.
How many crates will he be able to deliver?
LET'S LEARN OUT ABOUT IT
Finding out the number of digits in the quotlent of 0 division Is the first step needed to
begin lhe division.
1. i- -R : 2709 3 . ' • •
(0) : -24: .~
, .- - -3'9- ' " " --.---' ,
• • (b) 1-36'
• ,~-. : ' 3:3 ' _ ,J ___ , .--.
(c) :-24 : .--. : 9' --- -- ,
1 2
232
a) look at the hundreds digit: dMde 27 which Is the number of hundreds in 2 793 by 12. The remainder Is
3. b) To find the digit for the tens, brtng down 9 beside 3.
DMde 39 by 12. The remainder Is 3.
c) To find the digit for the simple units, brtng down 3 beside 3. The remainder Is nine.
2793 • (1 2 ' 232) + 9
~ ~ ~ ~ dividend d ivisof quotient remolnder
In division, the remainder is always less thon the divisor.
2. in this problem, there ore 12 bars of soap in a box and 12 boxes In 0 crate, so this means there ore 144 bars of soap in a crate (12 x 12 = 144). Divide 17856 by 144 to
wOO< out the number of crates. Do this in the same way as in problem 1.
page 2.
17856
dividend
• (144
divisof'
124)
quot1ent
+ o remoindet'
MuIIfpIIcatfon of a whole number by a fraction
LET'S PRACTISE
L Woo out,
5x.1.. · ~x9 ' 2.L x s ' 5 ' 3 "00 '
-E-x3 . 11 • ~x3 ' 22 x....!... · 9 . 9 .
7 x 36 . 15 •
8 25 x 100 .
2. Complete:
... x ~ = 238
; 17 51
''')(5=5";
3. Calculate:
the ~ of 42 ' 11 3 . the 100 of 22;
..@ x 4 ' 2. x 4 ' 9 x.L· 12 x.l. · .£! x 5 . 5 • 3 • 10 • 4 • 25 •
15 x .l. . 18 x ..§.. . 15 x ..!. . ..!. x 5 ' 11 x 3 . 10 • 3 • 10' 12 • 15 •
11 33 "')( "'6="'2 ;
12 ... x 6= 6
4 the 100 of 75;
4 the .L of 102' the.!! of 144' the '5 of 120; 17 • 12 .
the ~ of 6<1; the ;, of 140; the ~ of 279.
SOLVING PROBLEMS
1. A cyclist has to cover a course of 2. The capacity of a racing car's fuel tank 112 km. After two hours cycling, he Is 93 Q. Atter an hour rac ing, the tank is
notes that he has done j of the only ~ full.
distance . How much petrol Is left In the tank? What distance has he already covered? What distance remains?
3. A bicycle covers a length ~ metres with one tum of the pedals.
What's the distance with 3, 9, 27 and 45 turns of the pedals?
4. Shelled peanuts weigh ; of the weight of the unshelled nuts. What wtll be the
weight of the yield fiom 455 kg of unshelled nuts?
The unshelled peanuts are sold fo< 250 VI per kg. sI1elled tor 570 VI per kg. Shelling comes 10 eo VI per kg. A producer has 325 kg of UnsheHed peanuts fo< sale. To make the best profit. should he sell the nuts shelled. or unshelled? Justify your answet'.
page 151
Multiplication of 0 whole number by 0 fraction
LET'S FIND OUT
1. A litre of 011 weighs 16 kg.
What Is the weight of 6 litres of all?
2, A drum contains 12 litres of petrol; ~ ore removed for the c or.
How much petrol has been removed?
3. How ore you going to set about these calculations? State the rule for:
• multiplying a number by a fraction; • taking a froctlon from a number.
LET'S LEAliN ABOUT IT
1. There are several ways of calculating the weight In kg of 6 litres of 011. - You can odd the weight In kg of each of the 6 litres of 011: 999 9 9 Q ~ 10+10+10 + 10+10+10= 10 or 5.4kg
• You con mt.itip/y the weight in kg of one Itre of 011 by 6:
9)(6 6)( 9 9 x 6 54 54k TO = TO =----;0 = 10 or . 9
2. In this case, the contents of the drum must be divided Into ttYee ports and then two of these ports are token:
2 l2lc2 24 (12 + 3) x 2 = 12 x "3 = -3- = "3 = 8
Thus, ~ of 12 litres are equal to 8 litres. 2 "3
~
" • ~i' ..
1 ro lIndo -. do -. - "'" ~by "'" --01""' 1 frocffon. /hen mufIPIY "'" _ by "'" __ .
page ISO
121
Division of whole numbers (whole quot/ents) ~ '---------- ------jlil----J LET'S PRA eTISE
1. Find the number In figures for the whole quotient of the following numbers and do the divisions:
584 divided by 7 ; 15779 divided by 97;
1 795 divided by 18 ; 367 737 dMded by 251 ;
2 793 dMded by 24 ; 848 274 dMded by 412.
2. During the course of the business week a salesman sells 125 pencils for 3 125 VT . What is the price of one pencil?
3. A village has 7 669 trees for planting in 361 Identical rolNS. How many trees will there be In each row? How many trees will there be left over?
4. The school canteen has 54 275 VT in Its till . A meal costs 45 VT. How many meals must have been sold to give the total in the cosh register?
5. A truckdriver has to transport 136 bogs of millet and 120 bags of nee. He can carry only 48 bags every trip. How many trlps wtth a full load does he need to make and how many will there be left to carry for the lost trip?
6. The perimeter of 0 square Is 456 m. What Is the length of one slde of the square?
7. The amount of 75 CO) VT is being dMded between two people in such 0 way that the share of one Is twice that of the other. What do the two people each receive?
8. The perimeter of a rectangular piece of land Is 490 m. The length of land is 4 times Its width. Whot ore the length and the width of this piece of land?
page 27
PrOportIonal/ly
LET'S FIND OUT
1. A metal rod is 12 m long and weighs 36 kg. What is the weight of rods o f the following lengths: 8 m . 5 m . 4 m. 3 rn , 2 m and 1 m? - set the results out in table form.
2. A square with 5 c m sides has a surface area 01 25 cm2, What is the surface area of a
square with 6 cm sides? Also work out the surface area of the following squares. with 7 cm, 8 cm. 9 cm and 10 cm sides. - set the results out in a table. _ Compare this table with the one for the rods previously and Indicate which one
shows proportion.
3. Look at the following and without calculating the coefficient complete the proportion table'
Number ot litres of petrol 2 3 ... 5 .. .
Price In VT 200 300 400 .. . tJ:JJ
Explain how you work It out.
LET'S LEARN ABOUT IT
1. Situation 1 is one which deals with proportion because fa find the weight of each rod the number of metres was multiplied by the some number each time: 3. 3 is therefore the coefficient of the proportion (3 kilograms per metre).
2. Situation 2 is not on example of proportk>n because there Is no fixed coefficient to show the relationship between the length of a side and the area surface of a square.
3. Situation 3 Is an example of proportion.
5=3+2
I I @ I
Number of litres of petrol 2 3 4 5 6
Price In vr 200 300 400 500 tJ:JJ
l @ @
500 = 200 + 300
To complete the table without resorting to the coefficient of the proportion, the following properties should be used: 4 = 2 x 2; therefore 4 li tres of petrol is 2 times that of 2 litres: 2 x 200 = 400. Similarly, 5 = 3 + 2; therefore the price of 5 litres is that of 3 litres plus that of 2 litres: 300 + 200 = 50). Also, 6 = 4 + 2 or 6 :: 2 x 3. The price of 6 litres of petrol Is therefore: 400 + 200 = 6OC) or 300 x 2 = 600 or 200 x 3 = 6CX) .
page 28
Area of cylinders
LET'S PRACTISE
1. Complete the table:
x • 3.14
Radiu. 01 the base (cm) 14 9 2 SO
R )C R (cm')
Height of the cylinder (cm) 28 16 12 130
Area of both bases (cm')
Area of lateral lace (cm')
Total area of the cylinder (cm')
2. Find the total area of each of these cylinders:
A 8 C D E
• S cm 7 cm 4cm O.Srn 37 dm
h 10 cm 9 cm 200dm SOdm 400 cm
A,ec
3. Calculate the area of a cy1inder with a height of 6 m and a diameter of 4 m.
4. The lateral area of 0 cylinder Is 62.8 m2 and Its height Is 10 m. What Is its radius?
SOL VlNG PROBLEMS
1. A silo 5 m high hos a diameter of 3.50 m . It is covered with sheet metol which overshoots by 20 cm 011 round.
• What Is the area of the silo?
• What Is the weight of the sheet metal cover If 1 square decimetre weighs 1.7&1 kg?
2. To maintain his lawn. a gardener uses a roller with a hOllow c ylinder which he fills with water. The cylinder has 0 radius of 20 cm and Is 70 cm long. Calculate In cm2 how much sheet metal was needed to make the roller.
page 149
Area of cynnders
LET'S FIND OUT
Calculating the area of a cylinder.
~dl~) 11 B 7 base
h height lateral lace
c
/ ~
A f "\
o
Cylinder dl~
R=3 cm ~ h=4cm Development of the cylinder
1. What is the length of rectangle ABeD? Calculate Its area.
2. Calculate the area of the discs (bases).
3. Calculate the area of this cylinder.
4. How do you calculate the total orea of 0 cylinder?
LET'S LEARN ABOUT IT
1. The length of the rectangle ABCD is equal to the perimeter (circumference) of the disc.
AD=2X7txR AD::: 2 x 3.14 x 3:: 18.84 cm
The area of the rectangle ABeD or the lateral area of the cylinder may therefore be calculated In cm2 as:
2 )( 7t )( R )( h = 18.84 )( 4 '" 75.36
2. The area of one disc or base In cm2 is: 7t x ~ '" 3.14 )( 3 )( 3 '" 28.26 The area of two bases In cm2 is: 2 )( 28.26 :: 56.52
3. To find the total area of the cylinder. add the area of the two bases and the area of the lateral face. The total area A In cm2 Is: A :: 56.52 + 75.36 :: 131.88
4. The total area A of a cylinder is given In the following formula:
I A • ~ :.;. hi + (2 ua:,:..x Rl I N.B. The area of the rectangle ABCD Is called the lateral area of the cylinder.
page 148
Lh<_~ ______ 'Hy ________________ ~ ______ ~.~ LET'S PRACTISE
1. Complete the following proportion tables:
Number of bread loaves
1 7~ 1 2 1 2;~ 1 10 1 12 17 ~ 1 Price In vr
Number 0' litres of syrup 2 3 4 5 10 18
Weight of sugar In 9 500 750 3000
SOL VING PROBLEMS
1. For a letter weighing 25 9 you need to buy a 20 vr stomp: for 0 30 9 letter a 25 vr
stamp is needed: for 35 g, 0 35 VT; for 0 40 g. a 40 VT; fOf 45 g, a 50 VT and for
50 g, a 60 VT stamp.
Set out the table showing corresponding weight and postage.
Is this a proportion table?
2. A tap dispenses 30 litres o f water In 2 minutes. How many litres of water flow out In
3 minutes, 5 min. 30 mln, 45 min. 1 hand 1 h 30 mln?
3. To make 2 kg of confectionary cream, Nick uses 8 eggs, 1 litre of milk, 400 g of sugar
and 250 g of flour. What are the quantities needed to make up 4 kg of cream, 6 kg, 8 kg, 10 kg . 12 kg
and 20 kg?
Set out In a table. What quantity of cream In kg can be made from 80 eggs, 10 litres of milk, 4 kg of
sugar and 2.5 kg of flour?
page 29
of file height lines of j'ria"o/."s
LET'S FIND OUT
SltuaHon 1: Situation 2: Situation 3:
A
C~B F
C D~E B
Construct the height lines A straight line goes from C - Which is the height line storting from A, Band C. to line AB intersecting it at that Issues forth from E? - What do you not ice? right angles at H. - Which Is the height line
- What Is the position of H In that issues forth from F? relation to line AB? _ Draw the height line from
- What does CH represent D. for the triang le ABC? - What do you notice
- Draw the height line BK. concerning these three - Draw the height line AG. height lines? - What do you notice?
LET'S LEARN ABOUT IT
Here we show the construction of the asked for height lines:
Situation 1: Situation 2: Situation 3:
Ace G F~ --h---~ - I B 0 E
B
- The three height lines intersect on the inside of the triangle and at the same point.
:H I', ' I '<>,'
I " " I ' K '
I ' "
I -' , I," • I, '/, '
,~ ," ,
- Point H Is outside AB because angle A is an obtuse angle.
- CH represents the height line Issuing from C.
- The three height lines intersect at the same point outside the triangle.
- The height line Issuing from E is also side ED.
- The height line from F is also side FD.
- The three height lines intersect at point D which is the vertex of the right ongle.
I n;e heigIit _ oI.a ~ ~ ~'a!. tt.:altim..,;i.;t I
page 30
Area of discs
LET'S PRACTISE
1. Fill out the table:
A • C 0 E F
Diameter In cm 12 34
Radius In cm 10
R IC Rin c m2 64
Area of the disc In cm2 26 314
2. Calculate the area of a d isc with a 3. The perimeter of a disc Is 12.56 m. radius of 8.6 m. Calculate Its area.
4. And the area of the shaded portion of the figure (annulus).
R1 = 4 cm R2 = 5 cm
5. Calculate the radius of a disc which surface a rea Is 452.16 m2
.
6. A circular pond has a perimeter of 45.53 dm. Calculate Its radius and Its area.
SOL VING PROBLEMS
1. The radius of the large circle is 7.6 dm.
- Calculate the area of the large circle. - Calculate the area at the small circles. - Calculate the area of the shaded portion.
2. If the radius of a disc is doubled, how many times larger does Its area become?
3. Calculate the area of a disc given which 4. The perimeter at a disc Is 28.26 m. area Is 12.56 m2
• Find Its area.
page 147
AreiJ of discs
LET'S FIND OUT
Look at the figure shown In the diagram. The diameter of the disc Is 10 cm.
A r---~~~F~~----,B
I . Calculate the area of ABCD then the area of EFGH (formed by four right-angled Isosceles triangles).
2. What con you say about the areas of ABCD. EFGH and about the disc with centre O?
3. How can you calculate the area of 0 disc?
4. Do the exercise again, this time with R = 2 cm, R = 3 cm, R = 7 cm. D L---~~H~~---- C
LET'S LEARN ABOUT IT
1. Calculate the area of the square ABCD: the side of ABCD Is equal to the diameter of the disc (EG), so 10 cm.
The area of ABCD Is therefore 100 cm2,
2. Calcutte the area of the square EFGH:
A B
E 1----------1 G
The square EFGH Is formed by the four right-angled Isosceles triangles EFO = FGO = GHO = HEO.
The area of the triangle EFO Is 12.5 cm2, since 5; 5 = 12.5
The area of EFGH is 50 cm2, since 4 x 5; 5 = 50
The area A of the disc is between of 50 cm2 and 100 cm2:
50 < A < 100 or 2 x 25 < A < 4 x 25 The a rea of the disc A is equal to 25 multiplied by a number. This number, beween 2 and 4, is apprOximately 3.14 The area of the disc Is: 3.14 x 5 x 5 N.B. 3.14 is the number 1t (pI),
3, The area A of the disc is given In the following formula:
(n .1I')
R Is the radius of the disc. 1t = 3.14 approximately.
page 146
F
E 1E--.C:::::f'0~--jj G 5cm
H
Construction of the height lines of triangles
LET'S PRACTISE
1. Construct a right-angled triangle which has sides measuring 4 cm and 3 cm meeting at right angle A. Draw the height line and measure tt.
2. Using compasses draw a triangle ABC. Side AB = 3 cm. AC = 4 cm and BC = 5 cm. With the aid of a set square draw the three height lines.
3. Using compasses construct on equilateral triangle with 6 cm sides. Draw the three height lines with a set square. Measure them and compare their lenghts. What do you notice?
4. Construct an ordinary triangle ABC which has an obtuse angle at A. Extend the two sides of the obtuse angle. Drop a perpendicular line from vertex B down to line AC and another perpendicular line from vertex C to fine AB. Extend these two perpendicular lines so that they Intersect at I. Draw fine AI lN11ich Intersects BC at H. Check that AI Is perpendicular at BC. What does AI represent for the triangle ABC?
5. Construct two triangles ABC according to the measurements Indicated below:
AB BC AC AH 1" triangle 5 4 7
2nd triangle 6 3 4
Draw the height line AH for both triangles, measure them and complete the table.
SOL VING PROBLEMS
1. You want to draw 0 triangle ABC, Rrst draw side BC making It a cm long. Next construct the third vertex In such a way that height fine AD is 5 cm long.
2. Construct on Isosceles triangle ABC in such a way that angle B measures 120" and AB = BC. Draw the height lines of triangle ABC which intersect at point P. Measure AP, PC. AC. What do you notice? What kind of triangle is APC?
B
~ A C
page 31
Perimeter - Calculation of one dimension
LET'S FIND OUT
1. Here is a rectangle, with 2. Look at this SQuore. Its 3. This polygo n Is an o perimeter of 60 m. perimeter is 80 m. equilateral triangle. Its
20 m
D perimeter is 45 m.
D 6 - Work out the half - Calculate the side of • Calculate one side of
perimeter. this square. this equilateral - Work out the width. - How do you calculate triangle . - How do you work out one side 01 a square - How con you find the the width of a when you know its side of an equilateral
rectangle when you perimeter? triangle when you are given the perimeter know Its perimeter? and the length?
LET'S LEARN ABOUT IT
1. ~ In the example above, the perimeter is 60 m. So the half perimeter Is 30 m: 60+2=30 p
- =I+w 2
<t perimeter = l ength + width)
~ The width of a rectangle is equal to the half perimeter less the length: 30 - 20 = 10 (width: 10 m)
20 m Er---J E ~ L----.J ~
20 m
P = 20 + 10 + 20 + 10 = 60 p
- =20+ 10=30 2
The length of a rectangle Is equal to the half perimeter less the width:
I ='!"" -w 2
The width of a rectangle Is equal to the half perimeter less the length:
w . '!""- I 2
2. The side of the square in metres Is 60 + 4 = 15 The side 5 of a square is equal to its perimeter dMded by 4.
3. The side of the equilateral triangle in metres is 45 + 3 = 15 The side s of an equilateral triangle Is equal to the perimeter divided by 3.
page 32
Cylinders: development and construction
LET'S PRACTISE
1. Copy the development shown here. 0 2. Make the development of 0 The diameter of the bases is 3 cm. cylinder wtlh the bose Then construct the cylinder. I I diameter 5 cm and the You can odd tabs to help you with height B cm. gluing.
U Then assemble.
3. Tem wants to make a cylindrical box from 4. Make the development of a cardboard. The diameter Is 4 cm and the height 8 cylindrical box, with a base cm. radius of 3 cm and a height
g Tem wants you to make the development of 5 cm. of the box for him.
SOLVING PROBLEMS
1.
I ~ 15.7 cm
I The development of the lateral face of a cyli'ldrical box is represented by the rectangle shown In the dagram. Draw the development of the cylinder and make the box.
2. The disc shown In the diagram represents one of the bases of a cylinder with a height of 5 cm.
Make the development of this cylinder. 2cm
"'-- ./ 3. Some of these drawings are developments of cylinders. Find the ones which are and
say why the others aren't.
0 D
~ I 2 I l 3 J Ua 5 oU 4. Complete this drawing to make It Into the development
t;=J of a cylinder.
page 145
Cytinders: development and corrshucllon
LET'S FIND OUT
1. This Is a cylinder:
c ircular face • What ore the circular faces cailed?
, - What is the distance between the two , ~ body cl the cylinder circular faces coiled? , , , - What Is the body of the cylinder called? , , distance between the two circular foees ,
/'
'IC 2. Try and draw ftJe development of a cylinder. Use anything cylindrical osogufde.
LET'S LEARN ABOUT IT
I. _The circular faces are called the bases. :4- They are parallef and of the some me. , , The distance belween the Iwo bases Is called the height. , It is measured on 0 perpendicular line between the bases. , , , , The body of the cylindre Is called the kltet'aI foe •.
2. To draw the development of 0 cylinder. do the following:
./ ~velopment of a c vllnder 0
, , , ,
/ , , U ,
Roll up the lateral face In paper. 'When urvolled It will form 0 rectangle one of the dimensions of which will be ~ual to the perimeter of the circular bases and the other to the height of the cyll er. Draw the rectangle and complete the development by adding the two cl1cular bases.
page 144
Perimeter - Calculation of one dimension
LET'S PRACTISE
1. Complete the following table:
length 35m 1l5m ? 25dm 95 cm ?
WIdth 26 m 67 m BOcm ? ? 45m
Half perimeter ? ? 180 cm ? 170cm ?
Perimeter ? ? ? 90dm ? 210 m
2. The perimeter of a rectangle is 84 cm. 3. A square measures 30 m In perimeter. • What Is the length of one side? Its width Is 17 cm.
- What's the length?
4. The perimeter of an equilateral triangle Is 18 dm. - Calculate the length of one side
of the tnangle.
5. A farmer has a triangular field which sides measure 125 rn, 175 m and 90 m. He fences it In with a triple row of barbed wire, costing 18 vr per metre and sold In rolls of lOOm. • Calculate the expense.
6. A garden consisting of a rectangle and 0 right-angled triangle Is shown In the figure below. Its perimeter Is 148 m. • Calculate the length of the rectangle.
c
: [L-______________ ~~J~ __ ~_~~ E CD = 12 m CE=20m DE= 16m
o 16m
SOL VING PROBLEM
Steven eXChanges the square piece of land he owns for the rectangular one belonging to John. They both hove the same perimeter of 160 m. John's land Is three times os long as It Is wide.
• Calculate the dimensions of the two areas.
. Has Steven mode 0 good eXChange?
John 's land
Steven's /and
page 33
OMsJon of whole numbers: parlicular cases
LET'S FINO OUT
1. To empty a tonk containing 2 052 litres of water, 9 barrels are used. All have the same capacity, - What is the capacity of a barrel? Will any water be left in the tank? Write down the
corresponding equation.
2. Set out and do the following:
14600 ~ and
- What do you notice? - How do you go from the first division to the second?
3. A merchant hos to dispatch 5 201 kg of cotton in bales of 17 kg. - How many bales does he need to manufacture? - Set out and work out. Explain the different steps.
1460 ~
4. A cooperative shares 370 bottles of liquid fertilizer equally among 12 villagers. _ How many bottles will each villager receive and how many bottles will be left over? - Set down and work out. Explain the different steps.
LET'S LEARN ABOUT IT
1. In the first situation, the remainder of the division of 2 052 by 9 Is O. The quotient is said to be exact. This is written: 2 052 '" 9 x 228.
2. 14 CI:JJ + 60 and 1 460 + 6 give the same whole quotient: 243 but the remainders are not the same.
To Dnd the whole quotient In a division between two numbers both ending In 0, elIminate the same numbel of zeros in the divisor and the dividend.
3. If, after having brought down a digit , a number smaller than the divisor results, write o added to the quotient before bringing down the following digit and continuing the division.
ol 5201 1 7 bl 5201 1 7
-sq 30 ... , because 10 < 17 -5 1 H 305
[IQ] 1 0 1 - 85
16
page 34
MuHiplicalkm
LET'S PRACTISE
1. set out and do the c alculations: 7 h23mlnx4 7 h23min+4 16 h 41 mln x 5 16 h 41 min + 5
2. On 0 race course, a rac ing cor going at constant speed does four laps in 15 min 20 S. Calculate the time tor one lap.
3. It takes Dad 40 mln to get to work. Given that he does this trip there and back twice a day and that he works five days a week. work out how much t ime he spends travelling in a week.
4. A cassette ploys for 60 min and contains 8 songs. How long does a song last on average?
5. Every day Benny watches TV from 6.30 p .m. to 9.15 p .m. How muc h time does he spend viewing per week?
6. On a building site. 6 labourers work from 8 a.m. to 6 p .m. with a break of 1 h 30 min. How many hours will be paid to them for In a day? in a week?
7. Three divers take turns at repairing the hull of a ship . The total duration of the dives Is 2 h 45 min. What is the average t ime spent under water by each diver?
QUICK SUMS
1. Multiply: 2, Multiply: 3. Multiply: 52x25", ... 12 x 50 '" .. . 12 x 75 = ... 72x25", .. . 48x50", ... 40 x 75 '" ...
144x25", ... 92x 50", .. . 32 x 75 ::::: ... 96x25= ... 124x50 = ... 160 x 75 = ...
120x 25= ... 36x50= .. . 72x75= .. . 360x25= ... 184 x 50 = .. , 244 x 75= ... 284x25= ... 250 x 50= .. . 96 x 75 = ." 248x25= ... 318x50= .. . 124x75 = ...
1 240 x 25 = .. , 416x50= .. , 368 x 75 = .. , 1 648x25= ... 1 320 x 50 = .. , 1 200 x 75 = ...
page 143
Mufflp/Icaffon and division of measufements,of duration
LET'S FIND OUT
1, A man-made satellite completes a circuit of the earth In 1 h 52 min. What will be the time taken for 5 circuits? Look what Canner has written:
h mln
1 52 x 5
5 260
+ ------.. +4· (60x4)+20
Explain his calculation. 9 20 In the same way, work out (3 h 8 mln 24 s) x 7
2. A cyclist takes 2 h 35 mln to cover a course, Given that on average a car travels four t imes more quickly, how long will It take a motorist to cover the same distance? look at how Ken does the calculation:
Hours Minutes seconds
2 35 4
*-.2 x60--' + 120 Oh38min45s --
155 - 12 --
35 - 32 --
3
+ ~3x60 -+ 180
-16 --20
- 20 --a Explain Ken's method. In the same way. calculate (4 h 16 min 25 5) + 5
LET'S LEARN ABOUT IT
page 142
Division of whole numbers: particular cases
LET'S PRACTISE
1. 160 cars ore to be parked In a multlstoried car park:. 60 cars to each storey. How many storeys ore there? The cars ore to be parked In rows of ten In each storey. How many rows are there per storey?
2. A merchant sells 23 metres of cloth tor 4 715 VT. What Is the price of one metre?
3. For the same price. Bobby exchanges 42 m of cloth costing 480 vr per metre for a off-cut of lining at 360 VT per metre. What length Is the off-cut of lining?
4. A daily labourer earns 37 500 VT fOf thirty days' work, What would he eam If he only worked 10 days?
5. A transporter has to deliver 38 (XX) socks of millet and 750 socks of rice. He can load 250 socks onto his truck at one go. How many trips will he make fully loaded?
6. A company puts up an electricity line with three cables between two villages 12 km 6CX) m apart. How many metres of cable ore required? The reels used are 200 m long. How many reels are needed?
7. After their victory In the final of the World Cup. the eighteen members of the winning team are given equal shares of a bonus of 551 160 VT. How much will each player receive?
QUICK SUMS
50+ ... '" 100 89 + ... '" 100 100 - ... = 25 54+ ... • 100 18 + ., ... 100 100 - ... ,. 55 .. . + 83 = 100 .. . + 25 = 100 100 - ..... 43
67 + ... = 100 35 + ... = 100 100 - .. . - 19 ... + 63 '" 100 .. . + 87 - 100 100 - ... = 37 , .. + 22 .. 100 11+ ... • 100 100 - 83 - . ..
page 35
ProporIionaltty: graphicrepresentalfOn
LET'S FIND OUT
look at these two graphs carefully:
• ~ I-+-++-+--I--l-H ,~ I--+-++-+-+-l-H .~ 1-+-++-+-+--:l~H .~ I-+-++-+--,f'--l-H .~ I--+-++-:¥-+-l-H ,~ 1-+-+.,.,~t-+-++-1 , ,~ I-+--¥-+-+--I--l---'H
DIspoIch role h Vl
,
, , ,
, , ,
, , ,
• ax! l(oolSOOlllXJ2SOOlllXl Welghlofpockoge
h' 1 2
_ From graph L read the price of 2, 4, 6, 6, 10, 12 m of cloth and from graph 2 the dispatch of a packet of 500, 1 OCX), 1 5CO and 2 OCXJ g.
_ Draw up a table showing the weight of the packet in g and the dispatch price. - Which of these situations shows proportion? _ Look carefully at and compare the arrangement of the points in each graph. _ What feature do you look for in a graph showing proportion? _ From graph 1, find the price of 6 m of cloth, then the number of metres you can buy
with 7 COJ VT.
LET'S LEARN ABOUT IT
Nl.mber of m of cloth 2 4 6 8 10 12 (x 500 )
Prlcek'lVT 1000 2000 3000 4000 SOOO 6000 ..-----Starting with the number of metres, you con find the coefficient needed to work out the price. Situation 1 shows proportion.
Weight of package In g 500 1 000 1500 2000
Dispatch rate In VT 1 000 2000 2500 4000
There is no coefficient between the dispatch rate and the weight of the packet. Situation 2 does not show proportion .
page 36
L-SU_rfac __ e_a_~_ea_of_rec_ta_n_gu_ta_r_prlsm _ _ S_a_n_d_C_U_bes ____ --jtJ LET'S PRACTISE
l. 651 dm The length, width and height of a rectangular prlsm are
I~I C:I~ respectively 651 dm, 40 m and 27.8 m.
Calculate in dm2 the total area of this shape .
2. Complete this table: A'80 3. The exterior surfaces
Lenath Width Height of a cube-shaped
,Ide ""'" total packing case are to 4 dm 3 dm 50 cm be painted. The edge
6.S m 4.1 9m 7 cm measurement Is
S dm 44 cm 3.8dm 82 cm. The PQint costs 165 VT per m2
,
What will the Job cost?
4. The table gives the measurements for the edges of 5 cubes. Complete.
Length of on edge in cm 9 20 8.S li S 4.S
Area of one face In cm2
Total surface areo in cm2
SOL VING PROBLEMS
1. Calculate the total surface area In dm2 for each of the parallelepiped rectangles In the follolN'ing cases:
Height 19 cm 17 m 7.2 m 22dm
Length 69 cm 12.S m 410 dm 7m
Wldlh 9cm 4.5 m 70dm 630 cm Total area
2. Michael has had his bedroom painted. The measurements are: length: 5.5 m. width: 4 m and height: 3.5m. - Calculate the surface area which has been pointed, given that the door and the
window take up 6 m2.
- How much will It cost If 1 m2 of paint costs 480 VT?
3. A cube-shaped box is to be mode wtth a lid and
ITJ ~ edges of 0.90 m. The plonks It Is to be made from are 0.15 m wide. What will be the cost of the wood for this job If
W 1 m2 of plonk costs 650 VT?
page 141
LET'S FIND OUT
1. This is the development of a cube wtth a side of 8 cm: ,---6
@ II 2 3J4J • Calculate the area at one face of the cube.
5 • Calculate the total surface area of the cube. ~
2. this Is the development of a rectangular pl1sm (parallelepiped rectangle):
1:1 1::1 I =20cm w=6cm h =8cm - Which faces hove the some
area?
- Calculate the total surface area 2 of this rectangular prism.
Bcm
I 1 3 5 I 6 I ~ - How do you calculate the total 20 cm surtace area?
Bem 20 cm 4
LET'S LEARN ABOUT IT
/ 1. Thecube / 2. Rectangular ptIsm
Area of one face: The faces wtth the same area 8x8=64cm2 are: 1 and 5; 2 and 4; 3 and 6.
The 6 faces have Areas of: the same area. land5:(8x6)x2=96cml
Total surface area of 2 and 4: (8 x 20) x 2 = 320cm2
3 and 6: (6 x 20) x 2 = 240 cm' the cube: 64x6=384cm2 Total surface area of the
rectangular prism:
/' ---- ,.... 96 + 320 + 240 = 656 cm'
""'" --r>o you know any other war 0 calculatIng the surface aroo of a
rectangular prism? ./' -------page 140
LET'S LEARN ABOUT IT (continued)
Proportion Is shown in a graph by points formIng a straight line. PrIce rl vr
From graph 1 on poge 36 It Is clear that 12 metres of c loth cost 6 0:::0 yr.
With 7 (XX) VT you con buy 14 metres. (starting from 7 0:::0, move across to the right in a hOOzontol direction to the point of Intersection wtth the straight line, then drop vertically to the base of the graph where you' ll find 14).
LET'S PRACTISE
1. 20 VT are worth 1 FRF (1 French franc).
... , .. ... ... ... , .. , .. , .. , V
v V'
IL V ,
~ V
, V
,
, • • • 10 12 W 16 Nu'nber 01 m oIclolh
- How many French francs will you get If you change 1 500 VT. 2 500 VT, 5 OCO VT?
2.
- How much do you have to pay In VT fa receive lOO FRF, 500 FRF? set the situation out In 0 table. Draw the graph on squared paper (taking one square for lOO VT and one square for 10 FRF).
WeIght of cocoa
L Study the graph and answer the
"0 ques1lons, , .. ~
IV IV - How much cocoa is needed tor
~ 1. 2, 3, 4 blocks of chocolate? V c
V - How many blocks of chocolate .. V con be obtained from 1 cm 9
otcocoo? , , , , • • """""' of bIodo 01_. page 37
LET'S FIND OUT
1. e
A B
Using your pl"otractor. measure the angle A. - What kind ot triangle is ABC? Using a ruler or compasses. measure the sides AB and AC and compare them. - What Is a triangle ~ke this colleen - How many degrees ore the angles B and C?
2. Using a set square, draw a right angle XOY. Complete the figure to make a rightangled Isosceles triangle.
3. Draw a line BC 4 cm long. On the same side of BC construct angles XBC and VCB, each of 45 degrees. BX and CV intersect at A. - Check that angle A Is 0 light angle. - What kind at triangle Is ABC?
LET'S LEAIlN ABOUT IT
1. With your protractor, you have checked that angle A measures 90 degrees. SO we may soy that triangle ABC Is 0 IIght-ongled triangle. Also, you have measured sides AB and AC and confirmed that AB = AC. TherefOfe triangle ABC is on isosceles triangle.
Conclusion: tl10ngle ABC Is a rlght-ongkKt Isoscekt. triangle.
Measure the acute angles: angle B and angle C are each 45 degrees.
I In a ffICfr:IngIJIa 1««; ... #t1c::InQAP. ",. acW anpI!t belli trNJCIICR 45 degt8- I 2. Construcflon at 0 IIght-ongled Isosceles triangle. beginning trom 0 light angle,
a ) With your set square draw the right angle XOY.
b) Measure OA on OX and do the same on QY to obtain B.
y
B
O'--+'AC-X
OAB Is a righfangled IsosCeles triangle.
With your set square check that the angle A Is 0 light angle.
The isosceles trtangle ABC Is also a rightangled triangle.
L-C_Ube_ $_:_O_Ihe_ ' _de_V8_/opmen ___ Is_a_n_d_C_onstrucIion _______ ----j~ LET'S PIlACTISE
1. Frank wonted to work out three different developments ta a cube wtth 0 side of 1 cm. Copy his drawings onto squared paper and look (0( the squares which need to be eliminated before making a cube.
2. Copy the developments 1, 2 and 3 to make cubes to r- I the dimensions given:
lJ 0) With development 1. construct 0 box with an edge J of 5 cm. 2 b) With development 2, construct a box with on edge ,-- ~
of 4 cm. c) Wrth development 3, construct 0 box with on edge 1
of 6 cm. ~ I I
3 ~
r- 4. Does the drawing 3. This Is 0 cube L1J and Its shOw a development
development: I of Q cube? Why?
Draw the development of a box with on ~ EEm edge of 4.5 cm.
5. look at the following figures. 'Nhlch ones represent the development of a solid you hove studied?
-
Cf8 ,-
~ I I I ~ ~
Oraw the shape in each case and work out what the solid is,
6, Copy the drawings and complete them to show the development of 0 cube, Cut them out and construct the solids.
db $ page 139
RIght-angled Isosceles triangle - Construction
LET'S FIND OUT LET'S PRACTISE
This Is a cube, BI Are the drawings bek>w developments tor this cube?
1. Construct a rlght..ongled triangle of which you know that one side of the right angle measures 6 cm.
2. Construct a rlght.angled Isosceles triangle of which you know that the side opfX)SIte
0- r--0-
3 the right angle measures 8 cm.
•
2
I I , ~-
I
5 I
• R 3. Construct a square ABeD with the side AB = 6 cm. Draw the
TSJ diagonal AC. then cut out the square and fold it a long AC. Unfold and cut along AC . • What type of triangles have you obtained?
o C
SOL VING PROBLEMS
7 8 I 9 1.
I Here Is a circle with Its centre at O. Drow tt on a piece of paper. Drow two perpendicular diameters AC and BD .
- • Join points A and B. B and C, C and D. then D and A. 0 FM all the rlght-angled Isosceles triangles In the diagram and
C the deve nts. cut them out and construct 0 cube where name them. Be careful! There are more than four.
LET'S LEARN ABOUT IT 2. Starting with two perpendicular straight lines. construct two right-angled Isosceles
• The drawings 1,3,4,5, 7, 9 are developments of the cube. triangles with a common side.
• What kind of shOpe Is formed by these two triangles? - What does the side common to both right-angled Isosceles
triangles represent In the new shope? Justify your answer. I d::OJ R 0 I Cb' I I ~ 110;1 :FtP1 3. Construct 0 quadllateral ty1NOP in which the diagonals ore of the some length and
ore perpendicular at their middle Q. What type of quodi\ateralls MNOP? What type of triangles ore MNQ, QPM, QON, PNO, PNM, MOP and MON?
4. Cut out 0 square ABCD with sides of 4 cm, and draw the
B ~~~r dlagona~ AC and BD.
Try to find the greatest number possible of right-angled isosCeles trlongles. A 0
_ Drawing 2 ~ Is not 0 development of the cube because It has two faces (0
and b) both on the some sk:te of the bond of the four lined up faces.
- Drawings 6 and 8 I R l ore not developments of the I I I I I I I I I
cube because they hove more than four faces In a row.
"'" cJeve/oprnMJI at a cube _ not _two -.. on .. .,.".-01 ~ ItGlIout a.tcx:'8I tJ Q IOW. "'-0IfI_~_ ""_ naldepklWl_.
Ttylo __ at_.
page 138 page 39
Units of length - Ccnverslons
LET'S FIND OUT
25hm • •
Village A Village B
- What unit has been used to show the distance between the two villages? Express this In other units. Express this distance In kilometres using a decimal point. Remember the symbol km.
- The length of the table is 195 cm. Express this in metres.
- Draw up a conversion table and write down the following lengths: 1 075 mm; 007 cm; 25 hm; 75 m
LET'S LEARN ABOUT IT
- The distance between the two villages A and B has been expressed In hectometres. It may also be expressed In other units: 25 hectometres :: 250 dam:: 2 500 m. In kilometres 25 hm is written as 2.5 km (25 hm = 2.5 km).
- In metres the length of the table is: 195 cm = 1.95 m.
Conversion table
km hm dam m dm cm mm
1 0 7 5 1075 mm 5 0 7 007 cm
2 5 25 hm 7 5 75m
'. . . , Examples:
r" ~~"l'4;;".'~" 1 m=O.l dam 1 dam =0.1 hm
i ;~~~'*r~f'I'1II'" ~r: 0.'
1 dam = 10 m 1 hm = 10 dam = 100 m 1 km = 10 hm = 100 dam = 1 o:xJ m
page 40
ProportIonatJfy: average $pHd I
LET'S PRACTISE
1, Copy the table and calculate the overage hourly speed for each of the vehic les Indicated.
AoropIono BIcycle .... _ ...... Distance covered In km 5 700 76 744 657
T1rne kIk.o In h 6 3 12 9
$pHd In km/h
2. Following the same model. calculate the overage speed in rnls for the sprinters.
Abo
Dtstance covered In m 576
D,,,aIIon 66s
Speed In m/I
SOL VING PROBLEMS
1. Kenneth covered 1.4 km In a quarter of an hour. What's his hourly speed?
3. What distance does a motorist cover who drtves at an average speed of 120 km/h for 2 h 15 mln?
Davld JOMPh J""'" 3 o:xJ 3540 1 763
6mln4Ds 9m1nSOs 3mln25s
2. The Concorde does a three-hour flight at an average speed of 2 400 km/h. What distance has It covered?
4. A cyclist does a ride of 30 km In 1 hour 15 min. What was his average speed?
5. A motortst does the 420 km between two cities at ' 05 km/h. How long will It take him?
6. A plane flies for two hours. During the first hour It covers 780 km. In the second hour because of an unfavourable wind It con only cover 624 km. What was tts average speed during the flight?
7. A passenger in a train watches the telegraph poles going post. He counts 7 poles in 30 seconds. The poles are 100 m apart. - What is the average speed of the train? - How long after his departure station will the traveller arrive In the first station, given
that the dstance between the two stations is 15 km?
page 137
LET'S FIND OUT
A bus does a Journey in four hours. During the first hour. it covers SO km, during the second hour 100 km, during the third hour 60 km and during the fourth hour 90 km. 1. At what moment was it going at top speed and at minimum speed? 2. What is the distance covered In at the end of four hours? 3. If It hod covered this d istance at the same speed, what distance would it have
covered In one hour? What is the distance covered in one hour called? 4. Calculate the distance covered at this speed at the end of 2 hours, 3 hours, 7 hours.
Draw up a table for this situation. Is It a proportion table?
5. How long would it take this bus to cover 825 km if it kept up the same speed?
LET'S LEARN ABOUT IT
1. The bus was travelting most quickly during the second hour and most slowly during the third hour. Its speed was not constant during the whole Joumey.
2. In four hours, the bus covered in km: 80 + 100 + 60 + 90 = 330.
3. To convert the situation to a situation of proportionality, you calculate the average distance covered in one hour. This average distance would have been in km: 330 + 4 = 82.5. This is the average speed. It is written as 82.5 km/hour and is read 82.5 km per hour. Speed can also be expressed In metre/second, in km/mln, etc. The speed therefore represents the distance covered in one unit of the given duration.
4. The number of kilometres covered at the end of 2 h Is 82.5 x 2 = 165; 3 h is 82.5 x 3 = 247.5; 7 h is 82.5 x 7 = 577.5.
The table below shows that, in this situation, the distance covered Is proportional to the duration. The coefficient of proportionality is 82.5 (In km/h).
Duration I 2 3 7 ~ in h ,82.5 +82·0
Distance 82.5 165 247.5 577.5 .,;r :..--.r-In km
5. To find the duration taken to cover 825 km, use the coefficient of proportionality or one of the properties of proportionality. It is easy to calculate the speed, the distance covered and the duration taken for the distance by applying the properties of proportionality.
page 136
Utjils:of length - C· ohversio:1fis
LET'S PRACTISE
1. Complete: 4km= ...... hm
0.9 km = ...... m 0.05 dam = ...... mm
400 m = ...... dam
2. Complete: 4kmSdam= .. .... km
17mScm = ...... m Shm= ...... km
Skm624m= .. ... . hm
3. The dimensions of a rectangular sheet of paper are 3.1 dm x 2.1 dm. Calculate its perimeter. using the metre as the unit.
SOL VING PROBLEMS
1.8km= ... ... m 6.27 m = ...... dam
90.4 dam = ...... hm 1 OCOcm = ...... km
820 m = ...... dam 400hm = ...... km lCXXlm= ...... km
S.2 dam = ...... cm
4. Give in decimetres the width of a cover with a perimeter of 88 dm and a length of 230 cm.
1: Students are to run a distance of 5.5 km In a race. Michael gives up after 2.348 m. What distance did he still need to run?
2. Kylle has to go 1.252 km to school. What is the distance there and back in metres?
3. A field measures 6.5 dam long and 450 dm wide. The farmer piants shrubs all around it at 1 m intervals, How many shrubs does he plant?
4. A SQuare garden measures 7.5 m by 7.5 m. A roll containing 10 dam of wire is bought to make a fence. How many t imes will this go round the garden and how much will be left over?
5. Jane's footstep measures 75 cm. What distance in metres wiiJ she cover when she takes 50 steps?
6. A cyclist covers a distance of 89.5 km in two stages. In stage 1 he covers 46.254 km. What distance does he cover in stage 2? Express this in hm.
page 41
Decimal numbers
LET'S FIND OUT
1. Four apprentice Joiners are measuring some planks of wood with a tape measure.
• Toby measures his plonk and writes I 2 m I down: «Measurement of plank = 2 m. ~~~~~~~~~~~~==:::z~
• Sean measures his plank and writes r 1 m ond 5 dm I down: «Measurement of plank - 1 m ~ 2 5dm. - -
• Bruce measures his plank and writes down: .Measurement of plank = 1 m 54 cm.
• Gerord takes the measurement and writes: .Measurement of plonk = 1 m 342 mm.
I 1 m and 54 cm
1I m cnd 342 mm
- Using decimal numbers express all the measurements In metres. Expklin what you hove written.
2. Read the number 5.327. - What Is Its whole port and its decimal part? - Write this number in a table of numeration. - Write the digit for the tenths. hundredths and thousandths. - Write the number 5.327 In the form of an addltkln of two numbers, clearly showing
Its whole and its decimal ports.
LET'S LEAIlN ABOUT IT
1. In addition to metres: - Sean uses the dm os unit of measurement. The length at the plank Is 15 tenths of 0
metre or 15 dm or 1.5 m. - Bruce uses the c m os unit of measurement and so he writes: the length of the plonk
Is 154 hUnQredths of 0 metre or 154 cm or 1.54 m. - Gerard uses the mm os unit and so he writes: the length of the plonk Is
1 342 thousandths of 0 metre or , .342 m.
2. • Every decimal number contains a whole part and a decimal part. In 5.327:
. 5 Is the whole port: 327 Is the decimal port: 3 Is the digit for the tenths, 2 for the hundredths and 7 Is the digit for the thousandths.
- Any whole number can be written In the form of 0 decimal: 11 = 11 .IXL - Any decimal number may be broken down In the form of on addition of Its whole
and decImal ports: 5.327 = 5 + 0.327 0.327 = 0.3 + 0.02 + 0.007 (3 tenths, 2 hundredths, 7 thousandths) Therefore: 5.327 = 5 + 0.3 + 0.02 + 0.007
page 42
Area: agricultural measurement l1li11 111-1 '------------111/-J LET'S PIlACTISE
1. Express In ores: 2, Express in ha: 3. - How many centlares are there In 0
200 ca 75400 a square decametre? 16 6(X) m2 80 CXX) m2 - How many square metres ore there 7ha 10 hm2 in a hectare? 10 (XX) dam2 45 dam2
4. Convert Into: a) ares: 1 800 co; 4 300 m2; 24 dam2; 0.58 ha; b) hectares: 3500 ca; 470 m2
; 7.2 a; 3 hm2; 5 km2
;
c) centiares: 0.5 ha: 0.a25 hm2; 52 dam2
; 348 m2;
d) square metres: 3.4 ha; 0.C06 hm2; 8 ca; 0.24 a .
5. A piece of land Is made up of a portion of 68.50 hectares and a plot of 195.75 ores. What In ores Is the total surface area of this piece of land?
SOL VING ProBLEMS
6. A field of 67 a 5 ca Is divided Into three equal ports. What in square metres Is the area of each of these ports?
1. A farmer wonts to spread manure on a square field with a perimeter of 360 m. What will be his expenditure at the rote of 12 (0) VT the qulntol, if he plans to spread 275 kg per hectare?
2. A former sells 8 0 45 ca of 0 field wtth 3. A cooperative has 945.8 ha of land. on area of 7 500 m2• What orea wll still There are 247 ha of rice plantations and be left? How much has he made at the 62 500 ores of miDet fields. Calculate the . rate of 8 (XX) vr per are? area not cultivated.
4. A rectangular pIece of land measures 560 m long and 216 m wide. What Is the area In ha? It Is p lanted In the following way: 2 ha 5 a of moize. 45 a 50 co of beans, 520.4 a of maniac. Calculate in ores then In hectares: - the total area planted;
- the area remalnng fQ( other crops.
5. Alon buys a rectangular plantation 150 m long and 120 m wide, - Calculate Its perimeter; - Calculate Its orea; - Calculate the purchase price of the
land, if it costs 150 tlXl vr per hectore,
6. A square field has a side of 81 m. A rectangular field has the some perimeter and Its length Is twlce Its width. - What are the dimensions of the
rectangular field? - This field has been bought for
280 o:xJ VI per ha. What Is the purchase price?
page 135
LET'S FIND OUT
Often special measurement units are used to measure the surfaces of p ieces of land. These are agricultural measurement units. let's see what we can find out about them. Compare the square A and the square 8.
1 dam
~ Suppose that the side of square A measures 1 dam. What is the area of the square?
In agricultural measurements, this area Is:
1 are (1 a)
. The side of B ~ ten limes that of A.
- What is the Qrea of 8? - Convert It Into ares. - What would you call thls
ur/!?
B
r!J 1I1esqu",.C. compared to A,
I m would not be 'cl bigg'" than ~ o dot.
The square C has a side of 1 m. - What is its area? • Compare It to the area of
A. - Convert It Into are. - What would you coli this
unit?
- What do you coli these new units of measurement? 2 2 - Draw up a table of correspondence fOf the units of measurement of area (hm , dam .
m2) and these new units of measurement. 2
- Write these measurements In your table: 100 m2, 10 CO) m2, 250 dam .
LET'S LEARN ABOUT IT
The area 01 square B with a The area of the square C is side of 1 hm Is 1 hm2 = 1 m2. 1t represents 0 hundredth 100 dom2
, this unit (1 hm2 = part of the area of A 100 a) is coiled the hectare 0/100 a). this unit Is called the ( I ha). centime (I ca),
If ..,.
!lie _ (QI, _,..",. ..... ..., .. .,.. ... (cq) 7
. 1. ••. __ '" .........-•
Table of correspondence and hm' dam' rn' converslon: ha a ca
100 m2 1 0 0
10 (XX) m2 1 0 0 0 0
250 dam2 2 5 0
11 AlIn"'c:c-."'~~:
page 134
DecImal numbers
LET'S PRACTISE
1. Write the following decimals in figures: Ten units and three tenths; twenty-nine units and four hundredths; three hundred and twenty-five units and eight thousandths; twelve units and two hundredths.
2. Write the following in figures: 72 tenths; 415 hundredths; 302 tenths; 1 813 thousandths; 12 thousandths; 8 hundredths.
3. Write the decimal numbers shown in the following additions: 20 + 6 + 0.4 + 0.05; 13 + 0.05; 19 + 0.6 + 0.08; 40 + 7 + 0.3 + 0.028; 215 + 0.12 + 0.0CI9; 59 + 0.22 + 0.004
4. This Is what Roger wrote down during a dictation of decimal numbers. Correct it If necessary: - twenty-three tenths . 0.23 - fifty units and ninety-two hundredths . .. ................. . 50.92 - one thousand and twenty-eight units and eight hundredths 1 028.8 - nine hundred and four thousandths . . . . . . . . . . . . . . . . 0.904 • six hundred and five hundredths . . ..... .. 0.605 - three thousandths ......... ... 0.03 - twenty-nine units and nine hundredths . . . 29.09 - three hundred and nineteen units and five thousandths . '-'" 319.05
5. Write the following numbers in the form of an addition of the whole part and the decimal part:
105.210 = 228. 15 = 3 005.625 = 5 713.043 = 9.307 = 4.008 =
6. Delete the unnecessary zeros in the following decimal numbers: 80.03; 12.90; 47.302; 56.00 125.690; 348.900; 108.050; 500.006
QUICK SUMS
10+10= .. . 15 + 25= .. . 10+20= .. . 25+25= .. . 15+15= .. .
30 + 10 = ... 35 +15= ... 30+20= ... 40+50= ... 60+40= ...
200+300= .. . 250 + 100 = .. . 300 + 400= .. . 350 +425= .. . 550+450= .. .
804.119 = 13.05 =
1 OCO+ 1 OCO= ... 2000+ 1 000= .. . 2500+1500 = .. . 1250+1750= .. . 55OJ+45CO= .. .
page 43
SOLVING PROBLEMS
1. look carefully at and draw the graph shown belOw (1 square for 3 sweets and 1 square for 25 VT),
Is this Q case of proportion? Explain your answer.
PI1ce In VI
175 - -------------------- - - .
150
125 -------------- -- 0 100
75
50 -----4 25 - - •
• •
o 3 6 9 12 15 18 21 Number of sweets
Look at the graph and without doing any calculations. answer the following:
- What Is the price of 15 sweets?
- What Is the price of 3 sweets? 9 sweets? 12 sweets? 18 sweets?
- How many sweets con you buy with 150 VT? 50 VT? 75 VT?
2. A magazine announces the following subscription rotes:
Period (In months) 3 6 9 12
Price In VT 650 1250 1 &Xl 2300
_ Show the situation In 0 graph. put the period on the hOrtzontal axis and the price on
the vertical (l square for three months, 1 square for 100 VD.
- Are the prices in proportion 'WIth the subsCrlpHon periods?
page 44
I· Parallelepiped rectclngles: development and construct/orl
LET'S PRACTISE
1. Here are two developments of 0 parallelepiped rectangle. Copy them on squared paper. Ale they for the some parallelepiped rectangle?
2. Among these tour figures find the developments of a box In the shope of a parallelepiped. Copy one of these developments, cut It out and construct the box.
3. Copy the shape onto squared paper. Cut n out. Is It the development of 0 parallelepiped?
4. M
N
Ea
~ab
b a c
Is the drawing M the development of the parallelepiped N?
Using the letters A. B. C mark the surfaces on the drawing coo9sponding to those on N.
5. Only one of these shapes is the development for a parallelepiped. Which one?
3
page 133
POraIIelep/ped rectangles: c/eve/opnlflnt and
LET'S FIND OUT
These drawings have been done by Kenneth to show the development of 0 box In the shape of a parallelepiped.
-- I a • a a
01$ •
I a + 1 0
0 • 0 r,;-4 1 f--':- , . r-~ 3 • 0
- ~
r-a
0 ·1 ». ~ • a a 7 5
-=-- o • • 0
Point out the drawings which are a development (or pattern) for the box. Copy them, cut them out and make the box.
LET'S LEARN ABOUT IT
Drawings 1, 4, 5 and 6 are models for the box.
- I a • a I a
cbe I a
1 !-~
o I-
4 ~ o
• c....
'a 0 ·1 a
5 ~ r£t To check this, use the letters to compare the faces two by two. Opposite faces have the some letter. For example In drawing 4, by folding the band marked b, C , b, C , you will form the sides of the box. The faces marked a are the top and bottom.
You can also see this from the fact that there are not more than four surfaces lined up. You can also observe that two faces are placed on either sides of the .lateral surface.
page 132
Proporlionolity
SOL VING PROBLEMS (continued)
3. Complete this proportion table:
Weight of mushrooms in kg 5 10 20 30 40
PrIce in VT 1500 300l
• Draw 0 graph with the amounts on the horizontal axis and the price on the vertical (1 square for 5 kg, 1 square for 1 500 VD. What Is the price of 15 kg of mushrooms?
- How much mushrooms can you buy for 15 (XX) VT?
4. l ook at the table below. Give the price of the toilet soap:
Quantity of soap 8 10
Price in VT 680 850
• Does this table show proportion? Explain your answer. - How do you obtain the numbers from the first Una if you begin from the second? - Draw the graph (1 square for 170 VT and 1 square for 2 soaps).
QUICK SUMS
1. Calcula te: 50+ 40= 60+ 20= 70 + = 100
800 + 2(X) = 1 200+ ... .. 20cx)
+ 300 = 1 (XX) 9CX) + 700 =
800+ ... :::1(XX) + 100::: 1 (XX)
500+ ... =10lJ 600+ ... = 1 0c0
1 (XX) + 200= + 9CX) = 2 (XX)
+ 500= 1 300
2. Place the following numbers between the nearest tens, hundreds or thousands: Examples:
10 < 16 < 20 32 ; 129
400 < 472 < 5(X) ;
347 ; 769 ; 7 639 ; 2 (XX) < 2 835 < 3 (XX)
1652 ; 3299 ; 6782
pogo 45
LET'S PRACTISE
1. Draw a rectangle ABCD and its diagonal BD. - What types are the triangles ABO and BOC?
2. This Is a right-angled triangle ABC: B
30" A L-='-----_ _ __ ---l C
The angle SAC measures 30 degrees. On the extension of BC, choose a point D so that C is the half way point on BD.
- Measure the angles BAD, ABO and BOA - What type of triangle Is the triangle ABD? - What does AC represent for the mangle ABO?
3. This polygon has equal sides. With 0 protractor, measure the angles and fill In the table.
B C
A
F
4. Draw these polygons. Check if they ore concave.
pogo 46
D
E
convex or
- What do you notice? - What Is a polygon like this called?
? - ~ DivisIon of decimal numbers: eX9rclces and pi'obIems ~--~
LET'S PRACTISE
1. Fill in the table:
Quotient Quotient QuoIlent Quotient Dividend Divider to the nearest to the nearest to the nearest to the nearest
un. tenth hundredth thousandth
4.55 52
0.85 0.75
1 058.6 56
2. Work out an approximate value for the following quotients to the nearest unit, hundredth, thousandth.
Quotient Quotient Quotient to the neaest to the nearest to the nearest
un. hundredth thousandth
510. 75 divided by 6.25
637 divided by 6.75
0.2076 divided by 5.63
SOL VING PROBLEMS
1. A roll of cable weighs 41.31 kg and measures 10.8 m. What Is the weight of a metre of cable?
2. A farmer has harvested 16.48 t of clusters from his palms. He delivers this harvest to the 011 works with his pickup truck which takes a maximum load of 0.85 t. How many trips will he have to make to deliver the whole harvest to the works?
3. Mlrlam and Shella have bought together a remnant of 12.5 m of material for 2250 VT. Mlrlom takes 5.2 metres and Sheila the rest. How much does each girl pay?
4. A pharmacist fills his stock of '29.377 litres of alcohol Into 235 identical bottles. What is the contents of a bottle? What does a bottle of alcohol cost If the price per litre is 2000 VI?
5. Nell wants to paint a rectangular wall measuring 4.50 m long x 2.60 m high. He chooses a brand of point advertised os ((Covering 9 rrf /kgll. How many tins of 0.750 kg will he hove to buy if he wants to give the wall three coats of point?
pogo 131
DMsIon of a decimal by a decimal
LET'S FIND OUT
A barrel contains 15.66 kg of 011. 1 litre of oil weighs 0.9 kg. Calculate the number ot litres of 011 in the barrel. • Jack gives the following explanation: .To find the answer, I hove to dlvfde 15.66 by
0. 9. As I don 'f know how to divide a decimal by 0 decimal, I convert the divider Into a whole number by putting kg fnto hg.»
0.9 kg = 9 hg and 15.66 kg = 156.6 hg
1 5 6 . 6 }-'9-::--:
-9 117 . 4 66
The number of litres is therefore: 1 7.4
-63 3 6
-3 6 o
• Thomas says: "Converllng the kg into hg is the same thing as multiplying the dlvfder and the dMdend by 10 .• He sets this out and does the calculation. • Compare the two results and state the rule for dividing 0
decimal by 0 decimal. • Write down the equation for this division. • Apply this rule to the following calculations:
940.5 divided by 4.75; 14.625 divided by 3.25 and 19.6 divided by 0.14.
LET'S LEARN ABOUT IT
n 15,6.6 0,9 -9 1 7 • 4
6 6 -6 3
3 6 -3 6
0
To divide a decimal by another decimal, begin by making the divider as a whole number. by multiplying it by 10, 100, 1 (((), etc. Then muttipty the dividend by the some number before doing the calculation.
this means moving the decimal point 1, 2, 3 ... digits towards the right in both the dividet' AND the dividend.
Examples:
LET'S PRACTISE
1. Work out to the nearest thousandth: 9.78 + 3.2 = ... 70.43 + 0.35 = ... 12.29 + 0.68 '" ... 63.4 + 7.1 = .. . 5.73 + 4.5 = ... 89.7 + 0.27"" ... 35.9 + 0,92:::: .. . 13.4 + 9.5 "' ...
2. Set out and work out to the nearest 0.01: 32.41 + 2.7 '" ... 235.4 + 9.35 '" .. . 47.034 + 1.94 = ...
139.37 + 8.12 = ... 470.39 + 4.232 = ... 933 + 12.25 = ...
page 130
Revision: geometry
SOL VING PROBLEMS
1. Draw five trklngles and soy jf they ore convex or concave. What do you think of the results?
Try to construct a concave triangle. Is this possible? Compare your attempts with those of your classmates,
2. Draw an angle AOS and cut it out. Fold It In two with the sides meeting exactly. Unfold. • What con you soy about the line along ~f~d? 0
Cut along this line. • What con you say about the two new
angles?
3. Taking the results from question 3 In LET'S PRACTISE. construct a pIoygon with slx equal sides. using a protractor and compasses. • join A and O. What does AD represent for
the angle A, for the angle O?
• Join A and C. Measure the angle C In the triangle ACD. What type of triangle Is ACD?
• Join C and E. What type of triangle Is CED?
- Complete the triangle EAC and measure the angle A
- Measure the angles EAD and OAC.
- What does AD represent for the triangle EAC?
- Measure the sldes of the triangle ACE. What type of triangle is It?
•
A
A
B
c
o
E
F
page 47
LET'S FIND OUT
1. At the end of the school year, tour students obtain the following averages;
Mlchael: 13.25 Trisha: 9.07
Julie: 12.19 Koran: 13.75
- Compare these averages In twos. - Which Is the best average? the poorest? - Put them in decreasing order. - How do you go about comparing decimal numbers? _ Use the some method to arrange these numbers in increasing order: 115.7; 172.83;
206.04; 118.52.
2. Here are the heights (In metre) of four children:
Oono: 1.26
Seamus; 1.4
Peter: 1.08 Kathy: 1.153
- Compare the heights of these children two by two. Who Is the tallesn the shortest?
- Arrange these heights In Increasing order. _ Use the same method to compare and arrange in sequence the following
numbers: 32.4; 32.08; 32.47; 32.479; 32.48.
3. Write down all the decimal numbers: - between 1 and 2 to one d igit after the decimal point; - between 1.1 and 1.2 to two digits after the point; - between 1.15 and 1.16 to three digits after the point.
4. Take the number 1.153. Place It between two numbers: - to the nearest units; - to the nearest hundredths;
page 48
- to the nearest tenths; - to the nearest th~sandths.
Perlrnilfet 01 a circl8
LET'S PRACTISE
1. Complete the following table: 2. Calculate the perimeter of
Perimeter (P) D_or (0) 'odl"'~1I) a circle with a radius equal to 3.03 dam.
62Bcm ... dm ... m
... dam 7.5m ... cm Compare the radius of this circle with the radius of a
... m 0.75 dam ... m c ircle whose perimeter is
... cm ... cm 3.5 cm 20 dam.
3. If the radius of a circle Is 4. Complete the table: doubled, what happens to
I Radius
I 20
I 40
I 60
I 80
I its perimeter?
Perimeter
What do you notice?
SOL VING PROBLEMS
1. A round table has a diameter of 1.30 m. How many people con sit around this table if each one takes up approximately 80 cm space?
2. The surface of a small table is to be covered with A B protective material. The figure In the diagram shO\NS
Cl I) the shape and the measurements.
AB = 15 dm, BC = the half of AB.
Calculate the perimeter of the table . D C
3. The figure in the diagram represents a stadium.
~!( ) - What's the perimeter of the stadium? - What's the distance covered by a runner who completes
5 laps of the stadium running on the outside edge? • • l40m
4. The geometric figure shows a circular park. of 4 dam diameter
6i R ~ in which alleys have been laid out. To go from A to B always keeping In the direction of B, one has y / '1 to walk e ither along the arc ARB or the arcs AMC and eNB,
A • C ___ B
- Find 011 the possible routes for gOing from A to B.
~ s'-P - Are 011 these routes from A to B the some length? Justify your answer.
page 129
Perimeter of a circle
LET'S FIND OUT
Matthew Is roiling a tyre from position 1 to posItion 2 to position 3 so that point A
reaches the ground again, Le. the tyre has made a complete revolut ion.
,
i'\ 2
- Measure the distance from 1 to 3. What Is it?
- Measure the diameter D of the circle. - Compare the distance 1 to 3 with the diameter of the circle.
LET'S LEARN ABOUT IT
9.4 cm
• , ,
3
• , , ,
A'
3
- The distance between 1 and 3 corresponds to perimeter P of the circle.
- The distance between 1 and 3 is approximately 9.4 cm.
- The radius R of the circle Is 1.5 cm. Its diameter 0 is 3 cm. - The distance between 1 and 3 is almost equal to 3.14 x 3 or 9.4 cm. - The number 3.14 is called 1[ (pronounced: pi).
I The perimeter of the circle Is exprB$$tSd by th8 foIrnu/a: P - Tt X 2 x R 01 P ~ 1t X D I
page 126
Order of decimal numbers t/J L---~~------j
LET'S LEARN ABOUT IT
To compore two decimal numbers, begin by comparing the whole ports.
1. If they are different the larger number is the one with the greater whole pat.
Example: Compare 13.25 and 9.07.-The whole parts ore 13 and 9. As 13 > 9 we may write 13.25> 9.07 .
2. It the whole ports are equal , then we compare the decimal parts, beginning with the tenths; If the number of tenths is equal. compare the number of hundredths, thousandths ... The larger of the two numbers Is the one with the greater number of tenths. hundredths, thousandths ...
Examples: • Compare 1.26 and 1.4. As the whole ports are equal, 1 = 1, compare the number of tenths, 4
and 2. As 4 > 2, we can write 1.4 > 1.26. • Compare 32.48 and 32.479.
The whole ports and the tenths are equal, so we compare the hundredths, 8 and 7. As 8 > 7, we can write 32.48 > 32.479.
3. Between two numbers, we can Insert other numbers with decimol$ as follows:
Example: Between 1 and 2, we can insert: • numbers with one digit after the decimal point:
1; 1.1 ; 1.2; 1.3; ... ; 1.9; 2 • numbelS with two digits ofter the decimal point:
1; 1.01 ; 1.02; ... ; 1.1 0; 1.11 ; ... ; 1.19; 1.20; ... ; 1.85; ... ; 2 • numbers with three digits after the decimal point:
1; 1.001 ; UXl2; .. . ; 1.100; 1.101 ; ... ; 1.900; ... ; 1.999; 2
4. In this way we con insert all decimal numbers between:
• the nearest units • the nearest tenths
• the nearest hundredths • the nearest thousandths
Thus:
1 <1. 153<2 1.1 < 1.153 < 1.2
1.15< 1.153< 1.16 1. 152 < 1.153 < 1.154
1.152 Is the nearest number below 1.153 to the nearest thousandth: 1.154 Is the nearest number above 1.153 to the nearest thousandth.
page 49
Order of c1ecimQI numbers
LET'S PRACTISE
1. Compare two by two then arrange In Increasing order: 18.26; 9. 175; 14.43; 3.24; 18.3; 2.018
2. Insert the following numbers between: • the nearest tenths: 12.183; 327.06; 8.714 - the nearest thousandths: 62.529; 76.012; 7.435
3. Give the approximate value: - to the nearest unit below: 1.926; 743.98 • to the nearest tenth above: 704.92; 5.06 - to the nearest hundredth below: 4.53; 166.85 - to the nearest thousandth above: 105.318; 25.007
4. Write 011 the numbers between: - 113.47 and 113.48 wtth three digits after the decimal point; - 26.2 and 26.3 with two digits after the decimal point; - 11 .05 and 11.06 with three digits after the decimal point.
QUICK SUMS
Arrange in sequence according to tens, hundreds and thousands. look at this carefully: - Number 26 comes between 20 and 30;
so we can wrne 20 < 26 < 30. - Number 318 comes between 300 and 400;
so we can wrtte 300 < 318 < 400. - Number 4 672 comes between 4 CD) and 5 COJ;
so we can write 4 COJ < 4 672 < 5 (0),
Arrange the following numbers in the same way: 1. between two consecutive tens:
18; 31; 49; 58; 71; 92; 87; 26 2. between two consecutive hundreds:
924; 318; 539; 492; 768; 827; 245 3. between two consecutive thousands:
3 476; 8 504; 6 246; 9 328; 5 992; 4 106; 7 658.
poge SO
Proportionality: percentage
LET'S FIND OUT
The price ot 0 bulldozer is 8 750 COO VT. The dealer offers DarTen a price of 8250 oc:o VT If he pays in cosh. lJVhot Is the discount offered? What percentage Is the discount of the ticket price? Give your answer to the nearest tenth of a percent.
LET'S LEAllN ABOUT IT
- The amount of the discount In VT Is 8 750 (XX) - 6 250 oc:o z: 500 OCO. - The porportion for this situation will look like this: x 87 500
Price 100 8750 IXXl " VI
[2J = 500 OCO + 87 500 '" 5.7 Dlscoonl [2J 500 IXXl In VI The suggested deduction is 5.7 VT for 100 VT or
i I 5.7 % of the purchase price. x 87 500
+ 87 500
SOL VING PROBLEMS
1. To enlarge his house, Jonathan borrows 5 (XX) (XX) VT from the bonk. At the end of flve years of repayment, he does his accounts and finds that he has repaid 6 (XX) (XX) 'VT. What was the annual rote of interest on this loon?
2. A new car was bought for 12 125 (XX) VT. At the end of one year's use, Its value is estimated at 10525 (XX) VT. At the end of 5 years, this has decreased to 5 775 IXXl VI. What Is the percentage of devaluation at the end of 1 year and of 5 years?
3. A sewing machine costs 35 CO) VT. It is suggested that Kattw buys it on hire purchase, paying 6 monthly Instalments of 7 (XX) VT. What percentage of the price will she hove to pay In credit? If Kathy hod 30 OCO VT savings In the bank at 6 % Interest, how long would It hove taken her to save enough to pay for the sewing machine In cosh? N.B. A loan is never a free gift and It Is often preferable to save up ond pay In cash.
page 127
Axial symmetry: conslnJc/ion of symmetrfcal figures
LET'S PRACTISE
1. Look at the __ ---A---.d 2. Point B Is symmetrical to point A with diagram. regard to d. Drow d.
A • Find the point symmetrical to point A A with regard to straight line d .
• B
3. Construct the line symmetrical to line AB with regard to the straight line d.
a) b) B
L c)
B
A~ d A d d
4. Construct the figures symmetrical to the figures shown in the diagrams. with regard to the straight line d .
a) d
b ) c) d
F B
d
5. Copy these figures and draw In their 6. Copy these figures and draw In their axes of symmetry. If there ore any. axes of symmetry. If there ore any.
2
•
page 126
MuHlp/es and divisors - Divisibility by 2, 5 and 10
LET'S FIND OUT
1. - Write down all the breakdowns of 30 os the products of two factors. - Write down 011 the divisofs of 30. What Is the largest, the smallest?
How do you knOw if a number will divide 30? - Write down the mutfl:>les of 6. Can you find them all? What's the smallest?
How do you know If 0 number Is Q multiple of 67 - Indicate the phrases which have the same meaning.
a) 30 ~ dlvlsible by 15 d) 30 Is a mul11ple of 6 b) 6 Is 0 divisol' of 30 e) 15 is a divisor of 30 c) 30 Is 0 multiple of 15 0 6 divides of 30
2. Look at the numbers: 12; 15; 27; 48; 59; 73; 104; 230; 375; 420; 673; 828; 902; 1 006; 1 065; 1 250
- Make a list of the numbers divisible by 2. - Make a list of the even numbers. What do you notice? How can you recognize If a
number is divisible by 2? - Make a list of the odd numbers. How do you recognize these? - Make 0 list of the numbers divisible by 5. How do you know they ore? - Make 0 list of the numbers divisible by 10. How did you find them? - What con you say about a number that Is divisible by both 2 and 5?
LET'S LEARN ABOUT IT
30 = 5 x 6; 30 = 2 xiS; 30 '" 3 x 10; 30 '" 1 )( 30 The divisors of 30 ore 1; 2; 3; 5; 6; 10; 15; 30
- To soy that 15 is 0 dMsor of 30 is the same as saying that 30 is a multiple of 15. - An even number Is a number divisible by 2. It ends In 0, 2, 4. 6 Of 8. - AA' odd number ends in 1, 3, 5, 7 or 9. - A number Is dMslble by 5 if it ends In 0 or 5. - A number Is divisible by 10 If It ends In O. It Is also dMslble by 2 and by 5. Inversely, 0
number divisible by 2 and by 5 Is divisible by 10. Example : 20 Is divisible by 2 and by 5; n ~ dMslble by 10.
LET'S PRACTISE
1. lNhot Is the whole number that divides Into 011 the others? 2. What Is the least common multiple (LCM) fOf 3 and 47 3. Make a list of the divisofs of 50 and of 75.
What Is the greatest common divisor of 50 and 75? 4. Using each of the digits O. 2. 3 and 5 only once, write down 01 the four-digit
numbers: - d;v1siblle by 5; - dMslblle by 2.
page 51
,.
LET'S FIND OUT
This year W1l11om. Nlgel and Bob have produced 2 tons, 4 tons and 6 tons respectively, of premium grade cocoa. At the end of the season, the cooperative decides to divide among them a bonus of 48 IXO VT in proportion to the quantity of premium grade cocoa produced.
- What is the total quantity of cocoa produced by these farmers? What Is meant by paying out the bonus in proportion to the quantity of cocoa?
. Draw up the proportion table for this situation and determine the coefficient of the proportion.
- By means of this table work out the amount of the bonus received by each one.
LET'S LEARN ABOUT IT
1. In tons, the total quantity of cocoa produced is: 2 + 4 + 6 = 12. This problem involves proportion.
2. This Is the table representing this proportion situation.
Quanflty of cocoa 12 2 4 6 I--in IoN (x4CXXJ
Amount of conesponclng bonus in VI 480c0 8 000 160c0 240c0 ~
From the entries In the table, we can confirm that the coefficient of proportion Is 4 OCO.
3. We can work out the amount of the bonus paid out to each farmer in several ways:
a) By using the proportion coefficient: In VT, the bonus for Wllllam Is 2 x 4 (XX) = 8 !XX), for Nigel 4 x 4 (XX) :::: 16 (XX) and for Bob 6 x 4 (XX) :::: 24 CO)'
b) We can also use the properties of proportion. It is c lear that Bob has produced halt of the total tonnage (12 tons), so he will receive half of the total bonus; 48 (XX) + 2 = 24 CO) VT. William has produced 2 tons (12 + 6 = 2).so he will receive: 480c0 + 6 = 8 OCO VT. Nlgel has produced 4 tons, twice William's yield (2 x 2 :::: 4), so he will receive 16 OCO VT.
c) There are other ways of calculating the bonus paid: find them.
page 52
Axial symmetry: construction of symmetrical ffgures
LET'S FIND OUT
1. ABeD is an isosceles trapezium with an axis of symmetry.
Can you find it?
A B
D L---------------~C
2. Draw the figures shown below ond mark in the axes of symmetry if there are any.
LET'S LEARN ABOUT IT
1. To find out if the trapezium ABCD has on axis of symmetry, draw the perpendicular line d at H, midpoint on AB.
- B is symmetrical to A with regard to d. - d meets DC at K.
A r--id:7,--, B H
, DL-______ ~K~ ____ ~C
As AB is parallel to DC, d is also perpendicular to DC. Measuring KD and KC you see that KD = KC. The points D and C are symmetrical with regard to d. d Is the axis of symmetry of the trapezium ABCD.
2. ~~~n~ft~~ ~~ ~~u;~r~~~ ~:th~d~ymmehy and to draw these axes, if they exist, you
- When you fold the figure in diagram 1 at Z,' ",N, ,"""""" ~ .-o,:,'/~ _ -~ -- 0 the opposite corners or through the " middle of the opposite sides, the two parts are not superlmposable. M I P 1 The parallelogram does not have an axis of symmetry.
- When you fold the figure in diagram 2 at the opposite corners Q and S, the triangle QRS is superimposed on the triangle QTS.
R
, And the same applies for the triangles TQR and TSR. ' 0
Q ----------~- - ----- - -- S
The diamond-shape (rhombus) has two axes of symmetry: Its two diagonals.
T 2
page 125
Axial symmetry: construction of symmetrical figures
LET'S FIND OUT
B
A<-----.JC
How can you construct mirror
images?
d
LET'S LEARN ABOUT IT
B
A
, ,C M;' ~ ~
E 0
F
- Draw the pOints D, E and F which are symmetrical to points A, Band C with regard to line d.
- Draw the segment which is symmetrical to AB with regard to line d.
- Draw the figure which Is symmetrical to the triangle ABC with regard to line d.
- What is d for the triangles ABC and OEF?
d
- To construct 0 symmetrical to A draw the perpendicular line AM from A to meet d . Extend AM so that MD = AM. In the same way, construct E and F symmetrical to C and B in relation to d.
- Segment OF is symmetrical to segment AB. OF and AB are superlmposable by rotation (folding).
• Triangle OFE is symmetrical to triangle ABC. The triangles ABC and OFE are superimposable by rotation (folding).
· d Is the axis of symmetry of the triangles ABC and DFE.
• We can soy that ABC and OFE are symmetrical with regard to d.
In an axial symmetry, the symmetrical equivalent of a triangle is a triangle,
page 124
We can soy that the division of the pay-out Is proportional to the quantity of cocoa produced .
• so, if farmer A produces 2, 3, 4 Hmes more than farmer B, A will receive 2, 3, 4 times more pay-out than farmer B.
• in other words, if farmer B produces 2, 3, 4 times less than farmer A, he will receive 2. 3, 4 times less pay-out.
LET'S PRACTISE
1. Medication is administered in proportion to the body weight of the patient: a dose of 5 drops fO( every 25 kg.
- Work out how many drops wilt be given to a patient weighing 50 kg, 75 kg. - What is the body weight of a patient receiving 6 drops, 11 drops?
2. A cooperative distributes 90J kg of fertilizer among four farmers In proportion to the respective a reas cultivated.
If James has a total area of 3 hectares, and John and Michael have respect ively 6 and 18 hectares, work out the quantity of fertilizer each receives. Seamus, who has a field of 9 hectares, decides to join the cooperative. What Is the
total quantity of fertilizer now required so that the first three farmers will still receive as much as before?
3. A boss divides 115 CXXI VT among three labourers, In proportion to the number of days worked. Edward has worked 10 days, Kenneth 15 and Selim 21. What are their wages?
QUICK SUMS
Do these additions In your head:
3500 2700 1 250
+ + +
1400 = = 3100
300 =
2500 3600
+ =3CXX1 + 300 = + 5(() = 1 600
page 53
LET'S FIND OUT
o c) Give the number of the shapes having at least one pair of parollel sides. What
family do these Quadrilaterals belong to? b) Name the shapes In this family with only one pair of parallel sides. c) Among these last shapes name those with 0 single pair of opposite sides equal.
Whot o re these Quadllaterols called? d) Among these now name the shapes with a single pair of perpendicular sides.
What ore these quadrilaterols c oiled?
LET'S LEARN ABOUT IT
c) These shapes have at least one pair of parallel sides:
Cj, '----'----'1 ' 1 /r-:---". \ B 8 (> h ~ <2> These shapes belong to the trapezium family.
b) These are the members of the trapezium family with 0 single pair of parallel sides.
/. h These shapes are generally ca11ed 0 trapezium and this Is what Is meant when the trapezium Is mentioned.
C) From the trapeziums above. these are the shapes with a single pair of opposite equal sides:
/ .\ These are isosceles trapeziums.
d) No. 7 Is on example of a trapezium with a pair of perpendicular sides. It Is a right-angled trapezium.
I A "-'*"" 1$ a ~ having 01_ """ P<* "' __ -
page 54
Addition and subtraction of tractions
LET'S PRACTISE
1. Calculate and simplify If necessary:
~+~ . L+~ . 2+~ · ~+~ . 2+~ . 8 8 ' 12 12 ' 10 10' 9 9 ' 17 17
5 1 . 23 11. 53 37. 11 7 47 32 3" - "3 ' 30 - 30 ' 100 - 100 ' 25 - 15 ; 65 - 65 .
2. Calculate and simplify if necessary:
2.+1.. 2.+1.. . ..L+~. J3+.i . 2+2. , 9 3' 6 12' 15 13 ' 17 3' 12 7
.§. _ .l. 2. _ .§.. Q _ ..!Q . ~ _ 7 . 21 16 9 3' 9 8' 22 13' 15 12' 100 - 25 .
QUICK SUMS
1. Do these in your head: 0.25 + 10 = .. .
0.25 x 100 = .. . 0.25 + I (XX) = .. .
0.125 x 10 = .. .
2. Order of size of a product:
... )( 0.25 = 2.5 . .. xO.2=2OO
10XlxO.2= ... 100 x 0.125 = ...
0.4 x 10= ... ... x 100 = 0.4 0.4 + ... = 0.004
0.125 + ... = 0.00125
By rounding one of the terms to the nearest ten , hundred or thousand, give the order of magnitude tor the following products. Example: the prOduc t of 18 x 8 is approximately 18 x 10 = 180.
21 ~ 9 = .. . 35x 8= .. .
196 )(15= .. . 403)( 8= .. .
3. Order of magnitude of quotients:
5.39x 9= .. . 14.1.8 x 8 = .. . 25.75 x 11 = ". 59.9 x 12 = ...
6.37 x 9 = . I 596 x 9.3 = .. .
39.18x8= .. . 1 993x9= .. .
By rounding the dividend or the divisor to the nearest ten, hundred or thousand, give the order of magnitude for the quotient in the following divisions. Example: the quotient of the division of 395 by 9 Is nearty the some as that for the
division of 395 by 10, thot Is 39.5.
425+ 9= .. . 1993 + 9= ... 1475+5= ... 638+11= .. . 296+11 = .. . 3618+ 5 = .. .
47 + 5 = .. . 900+11= .. . 749 + 9 = .. . 150 + 8 = .. . 1 (XX) + 12 = .. . 8380+99 = .. .
DOge 123
Add/ffon and subtracffon of tractions
LET'S FIND OUT
1. A merchant sells ~ of a piece of c loth to Des and then : of the some cloth to Ben.
Show each of these troctions on a band.
_ What fraction of the cloth has the merchond sold? • What fraction of the c loth is left? _ State the rule fO( adding or subtracting two fractions with the some denominator.
2. During a P.E. lesson, ; of the students in the c lass play football and ; play basket
ball.
- What traction of the class are taking part In sport? . What fraction do not participate? _ How do you do the calculation in each case?
LET'S LEARN ABOUT IT
1. Let's look at the situation 1. - The traction of c loth sold Is the sum
of the two fractions 1.. +~ = 3 +4 =2. 8 8 8 8
3 8
4 8 ?
_ The traction of cloth remaining is the difference between the fraction for the whole cloth and the fraction for the cloth sold' .!!. _ !.... = 8 - 7 =..!. . 8 8 8 8 .
I To add 01_ two fractions .."" /he same denom/naIor, add 01_1 /he numeralon and ""'" write /he denominok>I /0 /he ,..uf/.
2. _ The fraction of the c lass taking part in sport is the sum of the two fractions: 3 2 5+"7= ?
When these two fracHons do not have the same denominator, reduce them to the same denominator: 35 (35 = 7 x 5).
1.",,3 )( 7=~ . 1.",,2x5=JQ 5 5x7 35 ' 7 ?>C S 35
1. + 1. = ~ + JQ = 21 + 10 = ~ 573535 35 35
_ The fraction of the class not taking part in sport is shown by: 35_l!.= 35-31 =~ 35 35 35 35
To add or sub/racI two frocIIom .."" _ denomIna/oIS. I8ducfl /he Iracllonl to the same dfInomInaIor. do the addJIlon Of subIroctlon 01 ItNf num«a1on and write /he common __ or /0 /he result.
page 122
Trapeziums
LET'S PRACTISE
1. Indicate the trapezium shapes. What kind are they?
0~~[0GJ~~b 2. Draw 0 square ABeD with sides of .4 cm. on the extension of BC, mark a point E so
that CE = 2 cm then join 0 and E. - What Is ABED?
3. ABC Is an Isosceles triangle with a right angle at A. From the midpoint of AB (I), extend a perpendicular line Intersecting BC ot J. B~: - What type of shape is AIJC?
I
4. Draw the Isosceles trapeoom ABCD.
- Draw the diagonals AC and BD. I' - Measure the angles ABD and ACD wtth a protractor. A
SOL VING PROBLEMS
1. Construct a triangle ABC. On the side AB mark the midpoint M and on AC the mid-point N. Draw the height line AH of the triangle ABC.
- Check that AH Is perpendicular to MN at P. - What are:
MPHB. PNCH. MNCB. BPNC and MHCN?
2. Draw 0 line AB 8 cm long. Wrth 0 protractor. construct two angles of 60 degrees on the same side of AB at A and of B and can them XAB and YBA - On XA and YB, measure the lines AD = BC = 4cm. - Complete the quadrilateral ABCD. - What kind of quadrilateral is It?
A
M/~ N j)\ P .~
B H C
x y
A 11:l.J' I:I.J' )., B 8em
page 55
Units of weight - Conversions t/J '----------"-----,
LET'S FIND OUT LET'S PRACTISE
,..,..,
7! I ~~.t\ What equipment do you use Whot ore the other units
~...:for_measuring weight? of weight?
What is the principal unit for measuring weight?
~ ~
W· . . . .
- Draw up 0 table of conversion for units of weight. Use it to do the following conversions: , 548 9 into kg, into hg, into dg; 25 quintals into kg and Into tons.
- What Is the appropriate unit for expressing the load of a ship? or the weight of an aspirin tablet?
LET'S LEARN ABOUT IT
A pair of scales or weighbridge is used to measure weights.
I The pdnc/pa/ unlloI w~t /sJhe gram (g) .' . . . I Table 01 conversion The other units of weight ......... milligram (mg)
ton 01 •• Ieno '. hg dog • do c. mg centigram (cg)
'" 01 •• decigram (dg) quintal decagram (dog)
1 5 , • hectogram (hg) kilogram (kg)
2 5 qulntol (q) ton (t)
1 548 g = 1.548 kg = 15.48 hg = 15480 dg 25 q = 2 500 kg = 2.5 t
I 1he un/l$ of welgh~·Jtic.reose aIKJ dfIC~ by tens. I
e;elM:j B~ The load of a boat The weight of an aspirin tablet
is expressed in tons (I). Is expressed In mllligroms (mg).
page 56
1. ~ A plot of ground ABCED is represented by the figure right on which ABCD is a parallelogram with AB = 24 m; MH = 12 m and ME = 4m.
- Calculate In dam2 the area of the plot of land.
2. Calculate the area of the D figures shown: ,------------ ----- --, 5cm , ,
B E C u - 0 0 -0 3
G 20 cm , ,
5cm ,
A ______ __ ___________ J
H
Polygon ABCDEFGH
SOL VING PROBLEMS
1. A plot of land ABCDEFGH is represented A by the figure in the diagram.
Calculate its area.
H
2. On a piece of land shown In the figure. the field ABDE is in the shape of a polygon.
- Calculate the area of the field (in dam2).
- The crop is maniac and the yield is 15 t per hm2
. What will be the amount of the harvest?
D H C
~ A M B
E F 20 cm E G
u L1.£!rl '" ,H
E D - , 0 , 0 Bern ' 3 I
_______ .J
• ' E
F B ,....<::. ' u
'",
'" J 0
A 3
K
Polygon A8CDEFGHIJK
"~ , :~ D , ,3 ~ , :K C J 3
51 m r~ ~- ------------, 50m 22m G w
'" 3
F E
p A B Q
IY '2m 25m
E~ 32m
S D R
page 121
Area of complex figures
LET'S FIND OUT
Look carefully at the two polygons ABCDE a nd PQRS. P A 42 cm B 25cm Q
E 4 u ~
E
E u
~ - Compare the surface of the polygon ABCDE with that of the rectangle PQRS.
~ - Wtich Is larger?
c
- Calculate the area of rectangle PQRS. - Calculate the orea of polygon ABCDE from
the area of the rectangle and the areas of the triangles 1. 2, 3 and 4.
0'0 _ Explain how you did it. 2 0
L-______ ~~~ __ _73 ~ 3 N
S 50 cm D 32cm R
LET'S LEARN ABOUT IT
_ I see that the area of the rectangle PQRS is larger than the or90 of the polygon ABCDE.
82 cm
Area: 4 428 cm2
Dimensions of the rectangle: Length: 50 cm + 32 cm '" 82 cm W1d"fh: 15 cm + 39 cm '" 54 cm
Area of the rectangle: 82x54=4428cm2
15cm .42 cm 25 cm ; ----;---=='--,- ---- --; ,
E : 4 u, - , ~,
, E: u , f3 : 3 ,
, , , ~ ,,0 '0 :3
,-------- ------- - ----------so cm 32 cm
Area of triangle 1: 25; 39 ::: 487.50 cm2
Area of triangle 2: 15; 32 ::: 240 cm2
Area of triangle 3: 50; 23 ::: 575 cm2
The small side of the right angle in triangle 4 : 82 - (42 + 25)::: 15 cm
Area of triangle 4: 15; 31 ::: 232.50 cm2
Sum of the areas of the 4 triangles: 1 535 cm2
Area of the polygon ABCDE: .4 428 - 1 535 = 2 893 cm2 I 11 Area of the polygon: area of the rectangle minus the sum of the areas of the triangles. 1
page 120
Units of weight· COnvelSiohs . ::'.
LET'S PRACTISE
1. a) Write in grams: 2. Complete: 3 kg: 25 hg: 45 dog: 0.8 kg = ... dog 140g= ... dg 55.7 dog: 1.75 hg: 8.6 hg 24.8 dog = .. . g 1 q = ... kg
b) Write in kilograms: 10.77 kg = ... hg 200c0dg= .. . kg 1 500 g: 400 dog: 57 hg: 750 g: 4q= ... kg 3.141 = ... kg 5 dog: 2.6 hg: 0.9 I: 1.27 t: 24 hg 18 1 = ... hg 1348dg= ... dog
3. Complete the following: 5.75 9 = 575 .. . : 4.21 = 42 ... : 0.57 g = 57 ... : 0.361 = 36 ... : 7.4g=74 ...
4. An empty lorry weighs 1 250 kg. It is loaded with 45 sacks of maize, each weighing 50 kg. What is the weight of the lorry when loaded?
SOL VING PROBLEMS
1. From a block of butter weighing 5 kg Tony cuts off 1 kg 5 hg. What is the weight of the remaining butter in grams?
2. Annette receives 3 kg of carrot seeds. She pocks them in sachets of 15 grams. How many sachets can she fill?
3. A 1 kg packet of sugar contains 225 lumps. A family uses on average 30 lumps of sugar per day. What weight of sugar do they use in a month of thirty days?
4. A shop assistant places twenty-five 200 gram sachets of sugar in an empty packing case weighing 2 kg. - What Is the weight of the packets of
sugar in dog? - What is the total weight of the case
when full?
5. 62 SOJ kg of cement was stored In a hangar before being loaded equally onto five lorries. What Is the weight loaded onto each lorry? Give your result In tons, qulntals, kilograms.
6. A carboy of oil weighs 4 kg 9 hg 5 dog. Empty, it weighs 95 dog. What is the weight of the o il in kg?
7. To store peanuts. 0 buyer has 8 silos, of which:
~~ 2 hove a capacity of 540 quintals, 5 of 240, and 1 of 200 quintals. This season. the first seven ore completely full and the eighth
~ (200 quintals) half full. - Calculate the weight of his stock in quintals and then tons.
~~~~~ - What weight of peanuts con he sell and still have 1 400 kg for his own consumption?
page 57
SImple fractions
LET'S FIND OUT
1. Read the fo llowing fractions: I; 1: -i: i: l~'
- Write the fractions corresponding to the shaded areas:
11 ~ 11
2. Look carefully at A. B, C, D: • How many sections are there In each band? · For each band, write the fraction corresponding to
the shaded area. · In each case Indicate the numerator, denominator
and their meaning.
~ I1 1 lA I ~ I B
~ Ic
~~D - Now you show each of the following fractions In a simple figure:
4 .5 . 7.2.3.4,7.8.7,9 5' 6 ' 9 ' 2' 5' 4' 10' 9']' 9 '"
3. Look a t band D. - What fraction of the band Is shaded? - Can you find a 1Nf"l0le number corresponding to the fraction shown In band D?
Write out the fraction and show ......not It equals,
4. Look carefully:
Unit: 'I ' I-rl 'I-rl -lrTl' E~I F~ G~
- What is the fraction:
- In each case, write down the fraction of the unit corresponding to the shaded area.
equal to the unit? smaller than the unit? greater than the unit? - What can you learn from this to help you recognize a fraction equal to, smaller
than or greater than the unit?
page 58
Axial symmetry
LET'S PRACTISE
1. Two lines AB and CD ore perpendicular at their midpoints. - What is the relation of the line CD for the points A and 81 - What is the relation of AB for the points D and C?
2. Are the triangles ABe and EFG symmetrical In relation to the A~C straight line d?
Justify your answer. G~E d
3. look at the isosceles trapezium In the diagram.
~ Are some of the straight lines axes of symmetry? If so, say which ones and justify your answer.
o J C
4. M A 8
Q 0 This Is 0 regular hexagon.
F C - What are AD, BE. CF In this polygon?
P R - Is It so for MN, OP. QR if M. N, 0, P. Q, R are the midpoints of these sides?
E N 0
5.
A~ ~ - O f these two straight lines d 1 and d 2
M which Is the axis at symmetry for the two triangles?
~ d , \(
cl, - Find the points symmetrical to B. M, P in
relation to this axis.
SOL VING PROBLEMS
1. Construct a square ABCD.
- What are the axes of symmetry in this square? Mark each with two letters. - Draw all the axes of symmetry on the figure.
2. look carefullly at this polygon. F C
- Does the figure ABCOEF have an axis of symmetry? 8 E
- If so, draw this axis on the figure. A
0
page 119
Axial symmetry
LET'S FIND OUT
A' B D' •
C
d
• E • F H •
1. look at the points and the strolght Une d. Draw the figure In your workbook . • Which point Is symmetrical to A In relation to d?
Which point Is symmetrical to B In relation to d? Which point Is symmetrical to 0 In relation to d ? Justify your a nswers.
- What can you soy about AB and EF? 2. - What Is d In relation to the figure ABFE?
- What con you say about 0 point near the axis of symmetry? - Which point Is symmetrical to C In relation to d?
LET'S LEARN ABOUT IT
1. The diagram above shows 0 straight line d which we wll call on axis of symmetry.
E Is symmetrical to A In relation to d, F Is symmetrical to B In relotk>n to d . H Is symmetrical to 0 In relation to d.
o • , ,
~ , , c , , , , H •
d
JustWIcatlon: If the lines AE, BF, OH, or9 drawn. you see that the straight line d Is perpendicular to these segments at their midpoints.
AB and EF are symmetrical In relation to d.
2. The straight line d Is the axis of symmetry in the figure ABFE.
The nearer 0 point Is to the axis of symmetry. the nearer the symmetrical point (mirror image) Is to the axis of symmetry.
Because the point C is on the straight ~ne d , It merges with its symmetrical equivalent.
page 118
~~_m_~_e __ ~_o_cti_.on __ s ____________ ~ ______________ ~~ LET'S LEARN ABOUT IT
1. Each of the bands A, B, C, 0 have been divided into 6 equal parts: 2, 3, 4 and
6 parts hove been shaded respectively, giving the fractions: i ; %; ~ ; % . - A fraction Is made up of two numbers. Example: ~ .
9 Is the denominator: it shows the number 01 equal parts Into which the unit has been divided. 4 is the numerator: It shows the number 01 porf5 5hoded .
2. The shaded part of band D represents % of the band. It is a complete band. We can
write % = 1 band or % = 1.
3. In the case of the bands E, F, G the unit has been divided Into 7 equal parts; then 6, 7 and 10 ports respectively have been shaded resulting In the following fractions for the shaded areas:
2. for band E' !... for band F· .!Q for bond G. 7 ' 7 ' 7 In a fraction if the numerator Is:
• tess than the denominator, the fraction Is smaller than the unit. Exampie: ~ . • equal to the denominator, the fraction is equal to the unit. Example: t . • greater than the denominator, the fraction is greater than the unit. Example: 1~ .
LET'S PRACTISE
1 Sh th f 11 ' ht" , le d ' 3 , 4 , 9 , 7 , 12 . ow e 0 OWIng ac Ions In Slmp IOgrams: 3" 15' 25 ' '9 ' 12 .
2. Look at this shaded band: I I I I I I I ~ What Is the fraction repre-sented by the shaded area? the non-shaded area?
3. Draw this diagram and place the 3
fractions on It: 0 "5 1 12 . 8 , 5 . 4 . 1 . 14.9 I , , , , I 5 ' 5'5 ' 5 ' 5'5 ' 5 '
A B 4. Write the fractions for the
shaded and non-shaded areas In the figures A. B, C and O. RE
7 5 2 3 , , , , I , , , , I
c D
~ page 59
Addition and subtraction of decimal numbers
LET'S FIND OUT
1. Ben wants to buy gold framing fOf three pictures. The perimeter of the first is 1.50 m, of the second 2.85 m and of the third 1.30 m. How much framing does he need? Knowing that the framing is only sold In lengths of 3 rn, how many lengths must he buy? Explain your reasoning and the stages of your calculation .
2. At the beginning of the week the adorneter in Mlchoel 's taxi shows 89 745.7 km. At the end of the week, the reading Is 91 015.9 km. How many kilometres has the taxi covered in a week? Show how you work this out.
LET'S LEARN ABOUT IT
_ The length of the framing required is In m: 1.50 + 2.85 + 1.30
_ The distance covered by the toxi in one week is in km: 91 015.9 - 89745.7
1. Generalisation
l.oo + 2.85 + 1.30
5.65
91015.9 - 89 745.7
1270.2
The addition and subtraction of decimal numbers is the some os for whole numbers. Be careful to set them out correctly with the numbers of the same value and decimal points exactly above or below one another.
2. Specific case: 800 - 796.25
Remembering that 800 = 800.00 this Is set out and calculated as shown: 800.00 - 796.25
3.75
LET'S PRACTISE
Set out and calculate: 986.483 + 356.69 . ...
726.3 + 914.35 + 67.8:: ... 5759.75 - 32<16.968· ...
29 274 - 257.25· ...
pag060
0.0145 + 1.0608 = .. . 0.0665 + 0.09507 . .. . 66.045 - 0.532 . .. .
19 645.07 - 4 120.75 . .. .
Percentage - Interest rates _ Capital 1IfIl '--------------J1I1-J LET'S LEARN ABOUT IT (continued)
2. CaplIaIInVT 100 121 Inter.st In VI 6 25122
This proportion table lets us write:
100 x 25 122 = 6 x G and G· 100 x 25 122 = (2 512 200) + 6 = 418 700 6
In order to have 25 122 VT Interest at the end of the first year, the planter needs to bank 418 700 VI.
SOL VING PROBLEMS
1. A bookseller offers a discount of 10 % for 1 250 725 VT worth of books sold. Calculate the amount of the rebate and the price paid.
2. To fit out her restaurant Kathy decides to use all her savings and take out a loon. She withdraws the 750 0Xl VT she has hod in the bank for two years at 6.5 % Interest per annum and borrows 9(() ())J VI repayable In one year at 19 % Interest. Calculate the cost at fitting out the restaurant and the amount of Interest she will have to pay.
3. Naoml wants to buy a television set 01 which the retail price Is tIJ (XX) VT. The company suggests two payment options: 1°) a cash payment with a 8 % d iscount; 2°) a down payment of 8 600 VT with payment of 10 monthly instalments of 5 B:OO VT. Which Is the more favourable option for Naoml? How much will It allow her to save?
4. A school cooperative is planning to buy a computer of which the retail pf1ce is 560 ())J VT. They solicit the help of the community and the parents' association who agree to pay 20 % and 40 % respectively of the total amount to be paid. The shop offers 12 % discount on the purchase. How much Will each of the parties have to pay?
pogo 117
Percentage - Interest rates - Capital
LET'S FIND OUT
1. A planter bonks 395 (XX) VT at the bonk. They suggest to him that he opens a savings account with 6 % annual Interest on the capital Investment. - What Is meant by _6 % annual interest on the capital Investment-? _ Calculate the amount of interest at the end of a year. _ What is his new capital at the end of the year? _ Calculate the amount 01 interest at the end of two years? _ Calculate his new capital at the end of two years. _ Compare the Interest accrued at the end of one year with the Interest accrued at
the end of two years. What can you conclude?
2. How much would he need to bank at 6 % per annum, to accrue 25 122 VT In interest at the end of one year?
LET'S LEARN ABOUT IT
1. 6 % reads «six per cent.. The rate of interest suggested to the planter Increose~ every year by 6 VT for every 100 VT invested. Another way of saying this Is that the Interest
represents ....!!.- of the amount Invested. 100
6 % is l~ or 0.06.
x 3950
CopItoI in VT 100
Intarest In VT 6
x 3950
+ 395 OOJ
[2] ~
therefore [2] = 6 x 3 950
·1237001
The amount of Interest at the end of one year Is 23 700 VT, so the capital In VT at the end of one year Is: 395 COJ + 23 700 = 418 700. The interest of the secund year Is added up on basis of the new capital (418 700 VD.
x4 187
CopIIcO In VI 100
Interest In VT 6
x4 1B7
~
41B 700
[2] ~
therefore m = 6 x 4 187 • 25122
The interest accrued during the second year increases to I 25 122 VT I At the end of the second year the capital in VT has Increased to: 41B 700 + 25 122.443 B22 " We can say that the longer an Investment is left , the greater the Interest accrues.
page 116
L-A_dd_'ItIon __ " _and __ SU_b_"'_OC_~tIon _ _ of_ dec_ Itria_· _f _nU_m_bers ___ ---'_~6J SOL VING PROBLEMS
1. When a traveller with a 20 kg baggage allowance, goes to check-out, his two suitc ases are found to weigh 18.5 kg and 9.8 kg.
Is he within his allowance?
If not , calculate the excess baggage for which he must pay.
2. Arthur and Kathyare 1.62 m and 1.84 m tall respectively.
Jack Is 25 cm shorter than Kathy.
Is he taller than Arthur?
3. A lumber truck Is loaded with three logs weighing 3.5 t , 4.8 t and 2.9 t .
Give an apprOximate value of the load.
How much more timber may be loaded If the maximum lOad Is 20 toos?
4. A wholesaler makes delivery to a retailer of three barrels containing respectively
2.25 hQ. 3.10 hQ and 0.80 hQ of oil.
Give the quantity of 011 in hectolitres received by the retailer.
How much oil has he now if he already had 10.35 hQ?
5. To dry some washing, Jason attaches a line to two posts 6.45 m apart. He needs
50 cm more for attaching it. The shopkeeper sells him a ball wtth 10 metres.
How much wilt he have left?
6. An athlete jumps 6.97 m In the long jump. Another athlete Jumps 39 cm further.
What distance has the second athlete covered In metres?
page 61
Trapeziums - Construction
LET'S FIND OUT
Frank has constructed three quadrilaterals:
L\ 1:\ A 8 A~8
o~
A~8 • In each Instance, what Is the relationship of AB to DC? - In each Instance compare AD and BC. then compare angle A and angle 0 in shope
3. What can you soy about them? - Determine the type of each shape. - Explain how you would construct each shope.
LET'S LEARN ABOUT IT
In each 01 the quadilaterals 1. 2 and 3 side DC Is parallel to side AB. Therefore Frank has constructed three trapeziums. In 0 comparison of the length of AD with that of BC, in each of the shapes, we see that,
a) In l : AD~BC.
b) In 2, AD = BC.
c) In 3: AD < BC.
Steps In construction:
Shapes
I
t1.
2
A '
3
A'
page 62
The angles ore not equal. Therefore: 1 is on ordinary trapezium.
Angle A Is equal to angle B; angle C Is equal to angle D. So: 2 Is on isosceles trapezium.
Angles A and D are each 90". They are right angles. SO: 3 is a right-angled trapezium.
Step I Step 2 step 3
, 0 e C 0 \ 7
/ " Ii '8 A
", 0
" C~ 7
C
\ •
'. A' '. Z:·:S A B
' 8 1=8 t=S: A 8
DIvIsIon of Q whote number by Q decimal number
LET'S PRACTISE
1. Set out and calculate: 513 divided by 3.4 (to the nearest 0.1) 749 dMded by 5.8 (to the nearest 0.01) I 827 divided by 3.25 (to the nearest 0,001)
2. WIthout working out the divisions, Indicate 'Nhlch ones will have the some quotient: 47 divided by 7.3 85 divided by 2.8 85 divided by 28 748 divided by 8.1 847 divided by 1.08 470 divided by 73 84 700 divided by 108 74.8 divided by 81
3. Work out the divisions: 176r 425~ 5829F
379 r 417~ 281 r Calculate the quotients: - to the nearest unit: - to the nearest tenth (0.1); - to the nearest hundredth (0.01).
SOL VING PROBLEMS
I. Dad has bought 3.5 kg of meat. He paid 1 330 VI. What's the price of one kg of meat?
2. As the 11 th September 1997. 1 Papuan kino was worth 64.25 VT and a New Zealand dollar 77.44 VT. How many Papuan klnos and New Zealand dollars could one receive on that day for 24 900 VT?
I 3 How many 094 m ties con be mode out of 0 8 m silk band?
4. Uno bought 2.65 m of material tor 640 VT. How much does 1 m of that material cost? Give the resuH to the nearest votu.
5. A 2.5 kg box of washing powder costs 1 155 VT. What's the price of one kg of that washing powder? Another box containing 0.9 kg of that same washing powder Is sold 450 VT. How much does 1 kg of this washing powder cost? Which packaging Is the best value for money, the 2.5 kg box or the 0.9 kg box? Which saving will be realised If the most attrocttve packaging Is chosen?
page 115
Division of a whole number by a decimal number .
LET'S FIND OUT
1. A shopkeeper has 25 litres of 011. She wants to sell it in O.7S Q bottles. How many 0.75 Q bottles will she obtain? To find the answer to this question you have to divide 25 by 0.75. - Moria observes and says: «/ cannot divide a whole number by a 25 0 0 75
decimal number. But, by converting the litres fnto centilitres it comes -225 + --
to a dIvision of two whole numbers.» 33 She writes: 0]5 Q = 75 cQ and 25 Q = 2 500 d. Then she divides: 25 0 2 500 + 75 = 33.3 cQ = 0.33 Q. -225
25
- Carl has not ice that the conversion done by Moria comes to multiplying the dividend and the divisor by 100, 2500 0 ... ..75 Carry on that division to the 0.001. - 225 + 33
250 -225
2. State the method of division of a whole number by a decimal 25 number. App ly that method to divide 52 1 by 3.4 to the 0.001. Using that result. write the numbers which frame the quotient to determine its approximate value: - to the nearest unit below and above; - to the nearest tenth below and above; - to the nearest hundredth below and above; - to the nearest thousandth below and above.
LET'S LEARN ABOUT IT
- Dividing a whole number by 0 decimal number with 1, 2, 3 ... digits after the point comes to d ividing by a whole number after having multiplied the d ividend and the d ivisor by 10, 100, 1 eXXL.
5210
-~: tl - 1 70
1 1 0 - 102
BO -6 8
1 20 - 102
page 114
1 B 0 - 1 7 0
10
153 . 2 3 5 153 < the quot ient < 154
153,2 < the quotient < 153,3 153,23 < the quotlent < 153.24
153,235 < the quotient < 153.236
153, 153.2, 153,23, 153.235 are approximate values of the rounded down quotient . The quotient has been rounded down to the nearest unit, to the nearest tenth (0.1), hundredth (0.01), thousandth (0.001).
154, 153.3, 153.24, 153.236 aTe approximate values of the rounded up quotient. The quotient has been rounded up to the nearest unit, to the nearest tenth (0.1), hundredth (0.01), thousandth (0.001).
L-~_a_pen ___ u_m_s_-_c_o_ns __ u_U_C_"_on ____ ~ ____ ~ ________ ~~~ LET'S PRACTISE
1. On a piece 01 paper folded In fwo, construct a right angle trapezium ABCD with the right angles at A and at D. Unfold and separate the two right-angled trapeziums, then arrange them in such 0 way as to form one single trapezium. What type Is It?
2. Construct a trapezium ABCD with AB = 4 cm, BC = 3 cm and CD = 2 c m and with AB perpendic ular to AD.
3. Construc t a trapezium MNOP, with MP perpendicular to OP and ON perpendicular to OP. Also, MP = OP = 3 cm and ON = 5 cm.
4. Draw a triangle ABC with AB = 4 cm, BC = 3 c m and AC = 2 cm. I and J ore midpoints of BC and AC. Join I and J. Check that IJ Is parallel to AB. What kind of quadrilateral is UAB?
5. Can you construct a trapezium so that AB = BC= CD=AD? If It's possible, what type is it?
6. Name 011 the trapeziums In this figure and soy whether they are isosceles or not.
M A \ZSZD
F E
SOL VING PROBLEMS
1. Draw two parallel straight lines X and Y and man.: the points A and B on X. Draw c ircles with centres A and B and with radii AD and BC (AD ~ BC). The circle centred on A should intersect Y at D and F. The circle centred on B will Intersect Y at C and E. Draw and write down all possible trapeziums.
2. Construct a rectangle ABCD which Is 8 cm long (AB) and 3 cm wide (BC). Inside the rectangle construc t triangles BCE and ADF right-angled at C and D, with CE = 2 c m and DF = 3 cm. What types ore ABEF, ABED and ABCF?
3. Karl wants to cut a piece in the shope of a trapezium ABCD from a cardboard box. Angle B is 65", AB = 8 cm, BC = 7 cm and angle C = 90". Construc t the outline of this trapezium for Kart.
4. Construc t a convex quadrilateral ABCD in such a way that AB = 8 cm and BC = CD = DA = 4 c m. What can you soy about lines AB and CD? Give this quadrilateral the most exact name you are able to.
5. Construct a triangle ABC in such a way that angle B Is t:J:r, AB = 4 cm and BC = 8 cm. Complete this tr10ngle so as to obtain on Isosceles trapezium.
page 63
Units 01 duration - Converrlons
LET'S FIND OUT
/1 02:35// WALL CLOCK
(ALARM)
What time Is shown on these two clocks?
How many hours are there in a day? How many minutes ore there In an hour? How many seconds in 0 minute? in an hour? Write In minutes: 2 h 45 min; 1 h 15 min;
In hours: 80 mln; 125 min; in seconds: 1 min 20 s; 1 h.
LET'S LEARN ABOUT IT
In I day. there ore 24 hours. 1 d '" 24 h The time on the olarm is In 1 hots, there ore 60 minutes. 1 h - 60 mil
2 h35m1n In 1 mi1ute, there ore 60 seconds. Imln -60s On the wall clock, In 1 tlOu', there ore 3 tOO seconds. 1 h ",,3(;00$
~ls
2h30mln
ConverskMll: hours minutes l8Conds
in minutes: 2h45 min (2 x lA)) + 45 "" 165 min
1 h 15mln (1 x lA)) + 15::: 75 mln
In hours: BOm" BO+60 '" 1. with 20 left over
-lh20mln
125mln 125 + 60 ·2, wI1h 5 left 0\I8f
:::2h5m1n
., seconds, l m1n20s (1 xlA)) +2O=a)s
1 h 1 x60=60mln 60x60". 3(;OOs
This Is how you convert 8 h 45 min Into minutes: 8 h represent 8 x ro - .480 min 480 + 45 '" 525 mln
This Is how you convert 125 min Into hours: 125 mln represent 125 + 60 ::: 2. with 519ft over
=2h5mln
page 64
Comparison of fractions
LET'S LEARN ABOUT IT (continued)
3. To compare the tractions -+ and ; .
• Calculate the decimal value of each fraction: 1- = 0.33 .. ,; ~ = 0.60.
Written as 0.60 > 0.33 or ; > -+ . - Reduce the fractions -} and ; to the some denominator 15 = 3 x 5.
3 _ 3 x 3 _ 9 . 1 _ 1><5 _ 5 5-5 )( 3-15 ' "'3-3 )( 5-15 '
- To compare two fractions with dttferent denomlnatot's. you con compare the decimal values of these fractions, or reduce these fractions to the some denominator and then compare the numerators.
- In the case of two froctions with the some numerator, the larger Is the one with the
smane, denominator. Example: ; > 132 .
LET'S PRACTISE
1. And the decimal value for each of the froct1ons and then compare: _I and..l. · ..1.. and 2 · ~ and ~ . .it and 2 32 ' 45'85 ' 43 '
2. Reduce to the some denominator and then compare:
..1. and ..1. . 3 2 '
.it and ~ . 8 7 '
~ and ~ . 4 9 '
~and~ . 6 5
3. Compare two by two and then arrange In Increasing order: ; , ; and 1~ .
4. Simplify these frac tions and compore: .it and 2 . ..§.. and ~ . .!!. and 25 , 6 7 12 8 . 20 36 ' 21 35 ' 12 and 28 .
5. Find a common denominator and then compare: ..1.. and ..1. . ~ and 2. . 2 and .l. 2. and ~ 3 6 ' 7 21' IS 5' 5 ~ .
QUICK SUMS
3.2 + 10 = .. . 173.5 + 100 . .. '
... + 100 = 3.03 3.75 + ... . 0.0375
1.5 + ... ·0.015 75 + ... . 0.075
page 113
LET'S FIND OUT
The fractions ;; . 1; , + and ; are shown In the diagrams A. B. C, 0:
A~
B~ff~
C~
D~
3 12 5
12 1
"3 3 5
1. Compare the fractions I; and I; . How will you start? State the rule for comparing
two frac tions with the same denominator.
2. Compare the fractions -} and 1; . l ook at the denominators, What do you notice?
How wlII you set al:xJut comparing t'NO fractions when one of the denominators Is a multIple of the other?
3. Compare the fractions -} and ~ ,
• How will you set about comparing two fractions when the denominators are different?
- Compare the fractions 132 and ~ . How will you set about?
- State the rule fOf comparing two fractions with the some numerator. Compare ~
and ; in the same way.
LET'S LEARN ABOUT IT
1. The figures A and B have superimposable graduations. So we can write 5 > 3 or
~>~. 12 12
- To compare two fractions with the same denomInatOf. compare the numerators; the larger fraction Is the one with the larger numerator.
2. In the case of the fractions -+ and 1; • 12 Is the muttlple of 3: 12 = 3 x 4 .
- Find the fraction equal to -} wtth 12 os the denominator:
1_ 1 )( 4 _4 "3- 3)(4 -12'
54 ·5 4 51 • Compare 12 and 12 . ~ 5 > 4. we can write 12 > 12 or 12 > '"3 . When comparing two fractions 'Wt18fe one of the denominators Is a multiple of the o ther, start \NIth the smaller denominator and find a fraction equal to it with the same denominator os the other fraction. Then use the same methOd as In 1.
page 112
UnHs of duration - Conversions
LET'S PRACTISE
1. Convert Into seconds: 2. Complete, 3h8min 25s 425 mln : .. . h 2 h 13 mln 65 - ... s 2 h 17 min 37 s 3h57 mln = ... mln 3h30 min = ... min 4 h 12 mln
3. Change to days, hours, minutes: 4. Alison goes to bed at 20 h and gets up 138 min: 648 mln: 2 565 rnln; 5 272 min. at 6 h. How long does she spend In bed?
5. How many minutes 6. How many seconds ore 7. How many minutes are there ore there in the there In the following? in the foilowing? foilowing?
1 mln t ; 2 mtn: • 0 Quarter of on hour (* h); 24 h; 2 h;
- a ha~ hour <t h); 1 h t ; 4 h;
3min j.; 5mln; - one and a ha~ hour
2 h t; 5 min; 10 min. (1 t h); 15 h.
- one and three Quarters of
on hour (1 t h).
SOL VING PROBLEMS
1. A student leaves tOf school a t 7 h 40 min. He spent 45 mln getting washed and dressed and 20 min having his breakfast. At what tJme did he get up?
2. A minibus has orrlved ot Its destination at 21 h after 0 trip kJsting a 15 h 35 min. At what time did it set out?
3. A fop fills t of 0 CE3 4. A heart beats at least 85 times per minute. How many times does It beat In on hour?
tank in 1 h 25 min. How many times does It beat In 0 day (24 h)? How long would It fake to fill It compietely?
5. A watch Iooses 2 seconds per hour. .. " ,
It Is set correctty at 8 o'clock on Monday morning. .. J ,
How much time wIU It hove lost by 8 o'clock the following • • SOturday morning? • , , , •
6.
--The light at the sun takes 8 mln 18 5 to reach the • earth. The speed at light is 300 COO km per second. Work out the distance from the earth to the sun.
page 65
DIvIsIbIIJfy by 3 and by 9
LET'S FIND OUT
- Complete the toble:
Number 0 ., 72 73 81 .S •• 107 117 121 99 399
RelT'lCinder when 0 1 0 1 dtvldlng by 3
Remainder when 0 7 0 1 dividing by 9
SUm 0 7 9 10 of the diglt5
_ Make lists of the numbers divisible by 3 and 9: compare these two lists. What do you notice?
- l ook at the row showing the sum of the digits contained In the numbers. - Pick out the numbers which sum of the digits con be divided by 3. - Pick out the numbers which sum of the digits con be divided by 9. - What do you notice? _ How con you recognize a number which can be divided by 3, by 9?
LET'S LEARN ABOUT IT
- A number is divisible by 3 If the «sum of its digits- Is divisible by 3. • A number Is divisible by 9 if the «sum of its digits- is divisible by 9. - Any number divisible by 9 Is also divisible by 3, but not every number divisible by 3 is
also divisible by 9. Examples: 1 B is divisible by 3 and by 9;
24 is divisible by 3 but not by 9.
LET'S PRACTISE
1. Pick out the numbers divisible by 3 and those divisible by 9: 3 107; 2 123; 1 149; 9 999; 10 703; 111 111.
2. lOB and 36 are divisible by 9. Are (lOB + 36), (108 - 36) and (lOB x 36) also divisible by 9?
3. A contractor wants to pave an area of 12 m x 15 m with square paving stones 30 cm x 30 cm. The paving stone touch each other. Will he need to cut any of the pavers? How many paving stones will he need to finish the job?
page ..
L-A_rea ___ O_f_c_o_m_p#_e_x __ ~_u_~_es ____ ~~ ______ ~~~~ __ ,~ LET'S PRACTISE
1. Calculate the areas of the figures ABCDE and ABCDEF: A E Ir---~
p0'!l,- 2om', JOm B -- ',to
, , 3 , , ~', '- 0 3', ,_-'0
Mm' , .,,. 51m3
C
2. Calculate the value of this field.
• The angles at A C and E are right angles.
- The slde AB Is parallel to ED. - 1 m2 of land costs 350 VT.
27 m E ~ 3
c
A\::r 3 E B
'" oll 0
C
3. Calculate the area of this figure in 4. Calculate the area dm2
. of His shape. J7m
E N L_--'63""""""m--...J
SOL VING PROBLEMS
~ 3
Scale: ~
Idm
1 < 2 A pIece of land has been divided up among five people.
- Calculate the area received by each.
$~,~ < {» 3
- The plot In the shape of a trapezium h~S been planted with vegetables and yields 102 kg per dam . Calculate the total production of this plot. 4
< "", ~ 5 'I",
2. The rectangular ground in the diagram Is mode up of a grassland and a plantation. The dimensions are: AB=95m; M =20m; AD=68m; AE = 40 m; GH = 45 m. - Calculate the area of the plantation and that of the gra~and.
- 1 m of plantation is <xX) VT and 1 m2 at grassland Is 600 VT. Colculate the price of the ground ABCD.
r-!<-----, B
" " G E --------',-,------- H "
~~~~
o , C
page 111
Area 01 complex figures
LET'S FIND OUT
B The polygon ABCDE has been broken down Into polygons the area of W'hlch we may eoslty calculate.
H A ...... +-~------'rr--"' .. C - What kind of dIfferent figures now make up the G polygon ABCDE?
E The dimensions are os follo\oVS: AC = 7.5 cm AF = 1 cm
o AG=2cm AH=6crn He", 1.5cm FC=6.5cm FH=5cm EF= 1.5 cm
BG = 2 cm HO = 2.5 cm - Calculate the area of the potygon from the areas of the different figures.
LET'S LEARN ABOUT IT
The polygon ABCDE was broken down into three triangles and a trapezium.
1. ABC Is a triangle with the height BG. 2. AFE Is a triangle with the height EF. 3. DHC Is a right-angled triangle with
the height HD. 4. EFHD Is a right-angled trapezium with
the height FH. Its long base Is HD. The short base is EF.
Calculate the areas o f the four ABC
B
~ , , , 1 A~C
F rG _____ --,H H C
E ____ 4 ___ ID DV AC x BG 7,5 x 2 7.5 cm' -,-
figures fOfmlng I-----+---"----I----,--+-------i 2
1 x 1.5 the polygon AFE ABCDE,
AF If Ef 0.75 cm' --2 2
OHC HC x HO 1.5 x 2.5 1.875 cm'
2 2
EFHO (Ef. HO) x Hf (1.5.2.52 If 5 10 cm'
2 2
The area of the polygon ABCDE In cm2: 7.5 + 0.75 + 1.875 + 10:: 20.125
page 110
Perimeters - Revision
LET'S PRACTISE
1. A rectangle Is 61 m long and 37 m wide. Calculate the side of 0 square with the same perimeter as the rectangle.
2. An area In the shope of a parallelogram has a perimeter of 522 m. Its length is 200 m. Calculate Its width.
3. A rectangle with a width of 7 cm has the same perimeter as a square with a side of 9 cm. Calculate Its length.
4. A rectangular field 120 m long and 80 m wide Is enclosed by two rows of barbed wire. A gate 2 m wide is InstaJled. Given that the wire costs 85 vr per metre, calculate the cost of the fencing.
5, A student measures the dimensions of a sports pitch with the aid of a rod 90 cm long which he has marked off In ten sections of equal lengths, The length of the pitch is 8 times the rod plus 5 tenths, and the width 6 times the rod. Write out the dimensions of this pitch, first using centimetres, then metres as a unit. Calcula te the perimeter of the pitch.
6. A covered area 3.50 m x 61 m Is built along the length of a rec tangular school yard. a) What Is the length of the shetter?
E shelter b) The perlmeter of the yard Is 90 m. What is
~ rts width? n
7, This piece of ground is In the shape of 0 rectangle. It Is decided to build a street running past It, 20 m wide, os illustrated In the
1 drawing. The remaining areo becomes a square with a perimeter of 100 m. " Calculate the dimensions of the piece of ground before construc ting the street. 20m
B. railing A sports field Is 120 m x 60 m. It Is surrounded by a railing situated 20 m from the
+- sports oreo -0 edge of the field, os shown in the drawing. 0) Calculate the dimensions of the railing .
• b) What Is the perimeter of the roiling?
page 67
LET'S FIND OUT
1. Look at the following numbers: 0; 1; 3; 8: 11 ; 25; 36; 41 ; 51 Find the divisors for each of these numbers. Indicate the numbers which have only two divisors.
What are they? What are numbers called which have only 1 os a divisor besides themselves?
2. Break down 210, showing It as the produc t of factors. Give a breakdown containing only prime factors.
LET'S LEARN ABOUT IT
1. Any number other than 0 can divide by O. The quotient Is O. Number 3 has exactly two divisors: 1 and 3.
3 is said to be a prime number.
In the same way, 11 and 41 ore prime numbers.
In contrast. L 8, 25, 36 and 51 are not prime numbers because:
1 has only one divisor: 1 8 has the divisors: 1; 2; 4; 8
25 has the divisors: 1; 5; 25 36 has the divlsoo, 1; 2; 3; 4; 6; 9; 12; 18; 36 51 has the divlsoo, 1; 3; 17; 51
A number" a ptIme __ '"-. hyo_ namely , and -.
2. Any number can be reduced to a product of prime factors:
210:: 21 x la = 3 x 7 x 2 )( 5
page 68
- ~
product of prime foctors
Polygons: regular hexagons
LET'S PRACTISE
1. Which of these are regular hexagons?
(0 Q 0 [ __ 4_J
2. Draw a circle with 0 radius of 4 cm and mark a point A on the circle. Construct a chord AB = 4 cm long, then chords BC, CD, DE, EF all 4 cm long. What figure is formed by the points A, B, C, D, E, F? Mark the points midway on the sides of the polygon, then join them two by two to form a different polygon. What kind of figure is this?
3. Construct a regular hexagon ABCDEF. Join A and D then Band D. Check that the angle ABO Is a right angle.
4. Construct a regular hexagon. ABCDEF. Join B and 0 then D and F. Measure and compare the sides BD, OF, FB. What kind of triangle is BDF?
SOL VING PROBLEMS
1. Construct a diamond-shope ABCO so that angle A = angle C = 60 degrees, and angle B equals angle 0 :: 120 degrees. Extend AO from OD = OA, then CO from OF = QC. Construct the diamond-shape ODEF. Measure the sides then the angles of this polygon, What can you soy about the polygon ABCDEF?
2. Construct a regular hexagon ABCDEF. Join A and D. Check that BC is parallel to AD. What kind of quadrilateral is ABCD? Write down all the trapeziums that con be formed using tour points of the regular hexagon.
3. Construct an equilateral triangle AOB 4. Draw a circle with centre 0 and radius with sides of 5 cm. Continue to R = 5 cm. construct other equilateral triangles BOC. COD. DOE. EOF. Check that · FOA Is an equilateral triangle. What kind of polygon is ABCDEF?
On this circle construct a regular hexagon ABCDEF. Using the sides of the hexagon and diameters AD, BE and CF, name all the diamond-shapes and triangles you can form.
page 109
LET'S FIND OUT
1. This Is a convex polygon. B - How many sides and angles does tt have? - What's this p<*ygon coiled? - Measure ond compare the sides of the polygon. A c - Measure and compare the angles. · Draw the polygon ABCDEF. - Draw the lines AD, BE. CF. • AD and BE Intersect at poIntO. • Measure OA, OS, CC, OD, OE, OF,
What do you nottce? F - Compare the sides of the polygon with the distance
OA.
o
· Drow a circ le with the centre 0 and radius OA. What do you not1ce? E
2. How would you construct a pIoygon like this?
LET'S LEAllN ABOUT IT
1 .• The polygon ABCDEF has 6 sides and 60ngles.
- This ploygon Is coiled a MXagon. AS = BC = CD = DE = EF · FA
- The angles A, B, C. D. E, F are equal and each measures 120 degrees.
A B
E o
OA =OS= CC = OD=OE =OF The sides of the polygon are equal to OA. Tha circle with centre 0 and the radius OA touches aU the vertices of the polygon.
2. To construct a regular hexagon with side AB - 3 cm (tOf example), draw 0 circle with 0 radius OA = 3 cm. Using compasses divide the circle In 6 equal arcs which lengths equal the radius. The dMsJon points on the circle are the vertices of the regular hexagon.
o
A polygon wilt! 6 equal _ and 6 equa/. "0 __ hflxOflOll. ·
page loa
LET'S PRACTISE
1. Copy the table and tick off the following: - the multiples of 2 greater than 2 - the multiples of 3 greater than 3 - the multiples of 5 greater than 5 - the mult1ples of 7 greater than 7 - the muttiples of 11 greater than 11 - the mult1p1es of 13 greater than 13 . the multiples of 17 greater than 17 - the mult1p1es of 19 greater than 19 - the multiples of 23 greater than 23 - the multiples of 29 greater than 29
0 1 2 3
10 11 12 13
20 21 22 23
30 31 32 33
40 41 42 43
50 51 52 53
4
14
24
34
44
54
5 6
15 16
25 26
35 36
45 46
55 56
- Make a list of the numbers you haven' t tlcked off. - What do you coU these?
7 6 9
17 IS 19
27 26 29
37 3B 39
47 4B 49
57 56 59
- Why Is there no point in deleting the multlples of 4. 6. 6, 9, la, 12, 14, 15, 16. 16. 20. 21. 22, 27 and 26?
- Using the some method. find all the prime numbers less than 100.
2. Breok down into products of the prime factors: 36: 49: 54: 62: 95: 120.
3. Look at the follOwing products: 3 x 21; 15 x 19; 13 x 17; 26 x 12; 3 x 5 x 11 x 17; 23 x 51. What are the products of prime factors? Write the other products In the form of products of prime factors.
page 69
LET'S FIND OUT
How can f corutnJct the bJsecffng Ins of the angle XOY. ..
1 2 3 ... using a protractor? '" using compasses
and a ruler? ... using only a ruler?
y X y
o L----X o o'-----x
LET'S LEARN ABOUT IT
1
y
c
l..!::::;~l'--X o XOY measures 72".
I mark half of it 36 degrees at point C.
I draw QC.
OC Is the bisecting line of the angle XQY.
page 70
2
y c X
o From 0 I draw a circle which Intersects OX at A and OY at B.
Then, taking A and B as . centres, I draw arcs of circles having the some radius and intersecting at C.
I draw OC.
OC Is the bisecting line of the angle XOY.
3
y
O~-'oD---=""Et-X
On OX and QY I measure segments OD and OA of same length.
Then I measure OE and OB of same length.
I join A and E, than Band D. Segments AE and BD bisect In C.
I draw QC.
QC is the bisecting line of the angle XOY.
DMs/on of a decimal by a whole number
LET'S PRACTISE
Set out and do these divisions:
51.8 + 14 : '2f:N.76 + 23 : 7248.5 + 7 109.25 + 23 339.5 + 97
SOL VING PROBLEMS
1. A truck transports a load of 21 .2 tonnes, mode up of 53 Identical iron girders. What's
the weight in tons of one g irder?
2. 41.4 kg of cake dough tills 18 identical moulds. What overage weight of dough does each mould contoln?
3. In a roosting plant. 1837.75 kg of coffee hove been packed Into 7351 identical pockets. How much coffee does each pocket contain?
4. At the butcher's, mum buys 6.5 kg of meat for 3 120 VT. What's the price per kilo?
5. A roll of Iron wire weighs 27.75 kg. A metre of the wile weighs 150 g. A rectangular
piece of land 26 m long x 15 m wide Is to be fenced in. How many times YAII the roll
go round, given that there Is a gate 1 m wide?
QUICK SUMS
1((()+1O:::: .. .
230 + 10 ", .. .
3250 + 100 •... 375.8 + 10 :::: .. .
... + 10 '" 500 125 + .. . '" 1.25
3.54 + 100 • ...
4.17 + ... • 0.0417 ... + 1 ((() - 1.02
. .. + 100 • 0.08
page 107
DIvIsIon 01 a decimal by a whole number
LET'S FIND OUT
Mk:ky the motorist does 599.2 km In 7 hours. What is his average speed per hour? To calculate the overage distance covered In on hour. Mane says: «599.2 km = 5 992 hm and I on dMde 5 992 by 7_
5 9 9 2 1-1;-7.....,-_ 5 6 856 therefore. In 1 h. Mk:ky does 856 hm = 85.6 km.
39 -35
42 -42
o Mlrlam sets out In this way and works It out:
599 . 2 ~7~~ -56 ! 85 . 6
39 -35
• Compare the two results. 4 2 -4 2
o - Study the way Mlrlam worked It out and expkJln how she dIvided
o decimal number by a ~ number. - State what the rule is.
LET'S LEARN ABOUT IT
To dMde a decimal number by a whole number. you do the some os when dividing 0 whole number by a whole number. You put the decimal point In the quotient after bringing down the first digit of the decimal port of the dividend and contnue with the division as before wtth the whole part of the dividend.
Examples:
39.9 ~ 228 . 35 24 15 . 90 1 7 -38 2 . 1 216 9.51 - 0 0 . 93 --19 1 2 3 15 9
- 1 9 - 1 2 0 -15 3 -0 35 60
-24 -51 - -I 1 9
pogo 106
LET'S PRACTISE
1. Draw on angle ADa = (jJ degrees. Using compasses and a ruler. draw the bisecting tine QC of the angle AOS. Check that the two angles formed by the bisecting line are equal.
2. Use 0 protractor and 0 graduated ruler.
0< 1 Draw the angle ROT shown In figure. Draw the blsecfing line OA. Measure each of the angles ROA and AOT. What can you say about these angles?
3. Draw a line OX and a line OY so that 4. Do exercise 3 again. this time wfth the XOV = 64 degrees. angle XOV = 164 degrees. Bisect the angle XOY with the line OZ.
5. On a piece of paper, draw the angle shown using your protractor:
o<.X
• Measure this angle. • Construct the bisecting line 0 1 of the angle XOY. • Measure the angles IOX and 10Y. • What do you noHce?
6. Draw a line ox. then construct the . 7. Draw four angles measuring respectively 60, angle XOY = 45 degrees. 90, 120 and 180 degrees. Draw the bisecting line OZ of the • Construct the bisecting line for each of angle XOY. these angles.
In each case. you will obtain two angles . • Check If they are acute. right Of obtuse.
B. On a line x:r mark the point O. then draw 0 9. Construct the fonowing four line OT. angles. Draw the blsect1ng lines OM and ON fOf each 7[1>. 130".30". 45". of the angles XOT and TOY. Draw the bisecting line for each Measure the angle MON with your protractor. of these angles.
10. Construct the angle XOT given that OY 11 . Draw two lines )N and MN Intersecting is the bisect1ng line of XOT. at O.
X X N 0 0 V
M Y • Construct the bisecting lines OA and
OB of the angles XOM and YON . . What do you notice?
page 71
Units of capacHy - Conversions
LET'S FIND OUT
t" ~ CWho1ismyC~ '-- ./
/
~ 200 1
? •
What is the p rinclpal unit for the measurement of capacity'? 5 hI = ... dol Give other units of measurement of capacity.
Draw up the table of conversion fOf the units of capacity. 345 dl = ... I Using the table, do the following conversions: 907 c l = ... dol
27 dol = ... mQ
LET'S LEARN ABOUT IT
I !he p!tncipal unit "" measuring capacity Is the 1IIre.
The other units to( measuring capacity ore:
I hl = 10 dol; I dol = lOt 11= IOdl; I dl = 10 cl; 1 c Q '" 10 mQ
5 hI
345 dl
907 c l
27 dol
pagen
= 50 dol
= 34.5 I
= 0.907 dol
= 270000 rnl
hI
5
2
dol
0
3
7
Converlion table
I ., 4 5
9 0
0 0
cl
7
0
hectolitre (h#) decalitre (do#) decilitre (df) centNltre (cD) millilitre (me)
ml
0
Declmal.value of fractions - Decimal fracttons
LET'S PRACTISE
1. Calculate the decimal value of these fractions: 3 , ~ . .2... . ]! . 36 , 9 ~' 12 ' ~ . 8 ' ~. m'
2. Write these fractions os decimal numbers: 9 25 42 57. 125 . 7
10 ; 100 ; 10 ; 1 cm ' 100' 1 cm .
3. To the nearest thousandth, calculate the decimal value of these fractions: 2 . 22. 12 . 11 . 11 . 32 3 ' ]' 32' 8' 18 ' 42 ' Which ones are equal to decimal fractions?
4. Divide 4 by 8. I by 5. 7 by 60nd 4 by 3. When possible. write the quotient In the form of 0 decimal fraction.
5. And the froctlons wfth the smallest denomlnotOf equal to the following fractions: 5 25 125 . 144. 75 . 375
15 ; 100 ; 1 coo ' 100 ' 100' 100 '
6. Write down the fractions equal to these decimal numbers: 0.4; 0.8; 12.5; 0.16; 0.35; 0.6.
7. There are eight portions of cheese In a box. Nlcky eats 3 and Millie 1. What fraction of the cheese has each eaten?
QUICK SUMS
Calculate:
7.2 + 100 = .. . 51.3 + 1 (XX) ::: .. .
367.2 + ... :::: 3.672 ... + 10 ::: 0.8
62.8 + ... = 6.28 12.63 + 1 (XX) .... .
117 + I 000 = .. . 0.53 + 100 = .. .
page 105
Decimal value of fractions - Decimal fracttons
LET'S FIND OUT
1. Three pieces of cloth each measure 1 m. The first Is divided Into two pieces of the same length, the second Into four and the third Into eight. - Write down for each the simple fraction corresponding to a port.
- In each case, what Is the length In metre of each piece?
2. Read the numbers 0.5; 0.25; 0.125. Write each of them In the form of fractions with the denominators 10, 100 and 1 (XXl
3. Find the decimal value of the fractions ~ . j and ~ .
Can you write these fractions as a decimal fract ion?
LET'S LEARN ABOUT IT
1 . • The length In metre of each piece Is:
~ = 0.5; ~ = 0.25; ~ = 0.125 .
· The numbers 0.5, 0.25 and 0.125 are the exact decimal value for the fractions ~ ,
1 1 4 ' 8 '
• To find the decimal value of a fraction. divide the numerator by the denominator.
2. The number 0.5 Is fead os -five tenths- .
We can write this os: 0.5 = l~ ; 0.25 = 1~ ; 0.125 E 11~ .
- The fractions l~ , 1~ and ll~ are decimal fracHons because they hove the
denominators 10, lOO, 1 (XXl
3. When the quotient from the division of the numerator by the denominator Is not exact. you find the approximate decimal value of the froctlon.
Examples: ; = 0.571. .. : ~ '" 0.333 ... ; ~ "" 0.777 ...
The numbers 0.571. 0.333.. . are the approximate declmot values for the fractions
~ and J. 7 3 .
• The tractions ~ • ~ and ~ do not hove equivalent decimal fractkins.
page 104
UnHs of capacity - Conversions
LET'S PRACTISE
l. Complete: 2. Complete: 2 dot • ... I 7 I· .. . dl 1 I= ... dl · .. . cf 1 dof - .. . f· ... dl 6 hf = ... I 41= ... c l tl= ... dl •... e l tdol= ... I = ... dl 1 cl :lE ••• rn' 9 Q = ... rnQ
~ Q .. ••• dQ = ". et t dof= ... f= ... df 25 dl = ... rn' 901= ... dol SOdl = ... I SOOI= ... hl t 1- ... dl = ... cl ~dol= ... I= ... dl
400 c l = ... I 50 1= ... dol
3. Calculate In f: 8daQ+4daf+5f= ... f 6hf+9daQ+6f= .. . I 3h1+5da~+QQ= ... f 9hl-4hl ,. ... f 5hl-SOI = ... 1 5dal- 71 s . . ' t
SOL VING PROBLEMS
1. A farmer has harvested &Xl sheaves of corn. Eoch sheaf yields 1.4 Q o f groin. What Is the total yield in doQ?
2. A pond when full contains 32 hI of water. A gardener fills his watering can 96 times; Its capacity Is 10.5 I. How muc h water Is left In the pond?
3. A barrel has a capacity of 68 Q. Arst 4.5 daQ of coconut 011 Is filled Into it. then 1.7 doQ. How many litres of coconut 011 has been filled Into it? How much more Is needed to fill It completely?
4. A pump delivers 75 c Q of water when the handle Is activated once. How many litres of water are delivered If It Is used 225 times? What quantity of water must stili be pumped to fill 0 drinklng trough of 250 I capacity?
5. A shepherd feeds his sheep 18 do' of millet per month. He has harvested 85 hI of millet. How much millet must he buy In to be able to feed his herd over the whole year?
6. A breeder has 15 milking COINS In his herd . Each cow yields 3 Q of milk per doy. Whot is his weekly milk yield? It he sells his milk for 180 VT per Q. how much will he have mode by the end o f the week?
page 73
Equal fractions
LET'S FIND OUT
1. Look at these marked lines A, B, C, D, E:
A I I 1
B~~~®~~~%~®~%~%~@~~~%~%~~~----------I+
C IWf.$!6if«~:~ I :
D 1«i.W'~ I :
E 1(if;W~~ I :~
Give other fractions which are equal to 1- .
- Write down the fraction which corresponds to the shaded section of each line. What conclusion do you draw? Wri t e down th e corresponding equation.
- How do you get from one to the next In the following _1 t0 2 J...to~ 2to 2 4'~ 8'4 ~ ~ to-? 16 '16 4'
2. - Write down three fractions equal to -+ . - Find the fraction of the smallest denominator equal to i; . _ State the rule which allows you to find a fraction equal to a given fraction.
. . 1 . .4 28. 12 '. 16 _ .4 - Complete the following equations: 2 = 12; T = - , 24 = 2 ' 12 - - .
LET'S LEARN ABOUT IT
1. For diagrams B, C, D, E, the fractions 1- , ~ , : ' 1~ represent the shaded parts which are superimposable.
_ You can write' J... = 2 = ~ = ..i. = ... They are equal fractions. . 2 4 8 16
You can write' 2- = .!!..- = Q = ... They are equal fractions. . 4 8 16
2. You can write: 1><2_2 . lx3=~. 3 x2- 6 ' 3x3 9'
1 2 3 I ~ " _ - - ore aqua "acllons. 3 ' 6 ' 9
12 12 + 4 3 1£ and ~ are equal fractions. -You can wrtte:16= 16-+4 ="4 ' 16 4
~ Is the fraction with the smallest denominatot' equal to ~ ~ ; it is said to be
Irreducible.
. A fraction· egIJaI to .a ~ _ /$ touf,d by ffIf!/IfpIyIng 01 dividing both ,."",. bY'''''' IIQI7HI num_.
- You can write: _1 = 1 x6=.!!..- . 2 2 x 6 12'
page 74
Q = 12 + 12 :::_1 24 24 + 12 2
I
Circle - Disc
LET'S PRACTISE
1. Draw 0 circle with centre 0 and a 2. Draw a line AB 4 cm long. then draw 0 radius of 3 cm. then draw two circle with the diameter AB. Draw a perpendicular diameters AB and CD. second circle with the same centre and Join the points A, C, Band D. with the radius AB. What kind of shape Is it? Colour in the annulus.
3. ON Draw the points 0 , M and N.
0° Draw a circle with centre 0 and the radius OM. Is N inside or outside the circle?
OM Explain your answer.
4. Complete the table then dravr CircMts C, C, c, c, C, the circ les C2 and Cd' Radii On cm) 4 ? 16 ? 30
Diameters (In cm) ? 10 ? 3 ?
5. Draw a circle with a radius of 5 cm. then inside this two circles. each with a radius of 3 cm. which Intersect at two points. Colour in the common area to all three discs.
SOL VING PROBLEMS
~Q:) To construct the figure shown In the diagram first 0 circle was drawn with centre 0 and a radius of 3 cm, then an arc with the same radius and centre A on the circle. This arc intersects the circle at Band F . Draw an arc with centre B and the same radius intersecting the c irc le at
F C. What type of polygon Is ABCF?
2. Draw two circles with the same centre 0 and radII of 5 cm and 3 cm respectively. Mark two points A and B on the circle with the radius of 5 cm and join them (the line AB Intersects the circle with a radius of 3 cm at C and 0, C being nearer to A than to B). Compare the measurements of AC and DB. Let M be the middle of line CD. What is point M in relation to the line AB?
3. Draw two circles with the same centre 0 as shown in the ;,~ ~ diagram. wtth radius RI = 4 cm and radius ~. = 3 cm. Mark
B the points A. C, 0 and B os shown In the diagram. From 0 A draw the perpendicular to CD which Intersects the circle .~, with radius RI at Hand K and the circle with radius R2 at M 0 and N. Cut out and fold the figure along HK. -:/ What do H and K represent in relation to the arcs AB?
page 103
Circle - Disc
LET'S FIND OUT
1. Draw a circle with centre 0:
A
" 0
c
B
_ What do we call the surface formed by the circle and the Interior points?
_ On the circle, mark the points A, B, C. Compare the distances OA, OB, OC. What does OA represent In relation to the circ le?
_ Extend AO to intersect the circle at D. What Is AD for the c ircle? Compare AO and AD.
- Join A and C by a line. What do you coli this Une?
_ The two points A and C mark two curves on the circle. What are they caned?
2. Draw two circles with the same centre and having different radii. Colour In the area between the circles. What Is this area coiled?
LET'S LEARN ABOUT IT
1.
" , ' , ' , " , , , , , , , , , , ,
, ", 0 , , , , , , ,
D
- The surface formed by the c ircle and its Interlor points is called the d isc.
B _ Comparing OA with OB ond QC, we can say that the three lines are equal.
OA = OB=OC
OA Is a radius of ",., cJrc/e.
If AO Is extended to D, you get the line AD which is twice the radius OA.
AD = 2 OA
!he line AD Is a _ /er of ",., ckc/e.
The line AC Joining two points on the circle Is called a ChOfd.
The two CUNed lines marked on the circle between A and C are arcs. N.B, The diameter AD is the largest chord: it divides the
c ircle Into two arcs of the same length.
page 102
2.
The space between the two circles wtth the radii RI and R:z and with the same centre 0 , (shaded area) Is coiled rtng Of
annulus or c irclet.
Do you remember what these ore:
radius arc
chord annulUl?
Equal fractions
LET'S PRACTISE
1. Find two equal fractions to the following:
~. ..!...... 13. ~ . ..L. 11 . 17 5 ' 12 ' 15 ' 8 ' 9' 8"' 5" '
2. Write fractions equal to ~ which have
- os numerator: 21 ; 63 ; 93 ; 54 ; 42;
- as denominator: 14; 21 ; 42 ; 112 ; 84.
3. Find the fraction with the smallest denominator equal to: _4 . 7 6 . 16 . 48 . 105 . 75 . 126 168 12 ' 7 ; "9 ' 24 ' 90' 180 ' 48 ' 189 ; 196 '
4. Complete so that each equation Is correct;
120 = ~; 1~ '" 192; ~ = -i- ; ~=- ' 13 91'
QUICK SUMS
25x 10= ...
158 x 100 '" ... 25 x = 75
7 x ... = 63 B x .. . = 400
... x 9= 72
230 x 100 = ... ... x 100 '" 2 50J
11 x .. . = 121
11 x .. . = 55 .. . x 9= 63 ... x 10=1270
7x ... = 56
... x8= 48 22x ... =2200
... x8= 720
... x 7= 490
34x ... = 3400
page 75
PropottIonaIity and rule 01 three
LET'S FIND OUT
1. Look at this table of proportionality:
Number 4 2:jf(~ l1 26 of pencils
PrIce in VT 200 550 1300
Work out: 4)(250 and 5 x 200 5><1300 and 26 x 250 4 x 450 and 9 x 200 l1x 250 and 5 x 550 4 x 550 and 11 x 200
What do you notice?
2. look at this table of proportionality.
Number l1 19 ? of 8 1"0$81'$
PrIce In vr 440 ? 3200
Alison does the following calculation: the price of one eraser is 440 ... 1 1 or ~ .
The price of 19 erasers is therefore: (440 + 11) x 19, so she writes down:
~ = 40 a nd ~ x 19 = 40 x 19 = 760 .
Jane uses the property discovered in 1: 11 x [2J = 19 )( 440
therefore [2J = (440)( 19) + 11 = 4401 ~ 19 = 760.
Compare the results obtained by AJison and Jane.
Complete the table.
LET'S LEARN ABOUT IT
Look at these numbers from the table in situation 1:
1. ~ []J1J [illiJ CSJEJ ~~~~ In each of these tables, you do the equation with the diagonal numbers. Example: 4 x 250 = 5 x 200 = 1 OCXJ . In this way you con find a missing number In a proportion toble when you are unnable to make use of the Information given. According to A1ison and Jane's
calculations, you can write down: 440 x 19 = 440 x 19 :lE 7f:IJ 11 11 .
page 76
1, 0.01, 0.001)
LET'S FIND OUT (continued)
3. When the remainder of a division Is not 0, we say that the result Is 0 quotient to the nearest unit, to the nearest tenth, to the nearest hundredth, to the nearest thousandth , It there are 0, 1, 2, 3 ... digits after the decimal point in the quotient. In the division of 23 by 3: A. 7 Is the quotient to the nearest unit. B. 7.6 Is the quotient to the nearest tenth. Written: to the nearest 0.1. C. 7.66 is the quotient to the nearest hundredth. Written: to the nearest 0.01. D. 7.666 is the quotient to the nearest thousandth. Written: to the nearest O'(X)l.
LET'S PRACTISE
1. Calculate the exact decimal quotient of: 1 025 divided by 82. 2 320 d ivided by 64 and 8 456 dMded by 64.
2. Calculate the quotient to the nearest:
a lO. 1: 756r 643~ 4 596~
bl 0.01 : 569~ 967 f1- 6579~
cl 0.001 : 479f 215 ~ 8957 f
3. Complete the following divisions:
747491%hs 2 26 5
43791~ 810 . 9
627281 46 1363 . 65
89264173 1222 . 794
SOL VlNG PROBLEMS
1. T~rPrice of a 14 carat gold bracelet Is 76 CO) VT. 1 gram of gold Is 4 020 VT. What Is In rams the weight of gold in the bracelet to the nearest 0.001 ?
2. The magnetic bond of a VCR unwinds at the rate of 320 cm per min. Calculate in cm to the nearest 0.1 and the nearest 0.01 the length of bond unwound each second .
3. Antonic buys cheese costing 880 VT per kg. She gives the cashier a 5 CXXl '-;g~ote and receives 3 9IXl VT change. How much cheese has she bou ht?
4. The annual subscription to Vanuatu Magazine (52 Issues) costs 5 500 VT. Calculate to the nearest tenth the price of each Issue. The ne'NStand price is 125 VT per Issue. What saVingS -ore to be mode by being "a subscriber?
pago 101
file nearest O.
LET'S FIND OUT
1. An iron girder of uniform cross section measures 5 m long and weighs 237 kg. What is the weight of 1 metre of this girder? Here's how Kenneth, Cameron and Neil solved the problem:
Kenneth
237
1 - 20.47 37
-35 2
What is the remainder from this division? Wrfte It down. At this stage, the dividend has not yet been used up as 2 units remain. We can say that 47 Is the quotient to the nearest unit of the division of 237 by 5.
Cameran
2370~
- 20+1474
37 -35
20 -20
o Cameran wants to obtain a more precise result . He knows that: 237 kg = 2 370 hg. So a metre of girder weighs in hg: 2 370 + 5 = 474 hg or 47.4 kg.
Nail
2 3 7. 0 f05'=-7 - 2 O. : 47 . 4
37 : -35 ;.
2 0 -2 0
o Nail does a direct calculation by setting down: 237 = 237.0 and working out the dMsion as shown. What's the remainder from this division? Write It down. Look closely ot the steps and say what method he has used.
2. Use Nail's method to work out the quotient of 23 d ivided by 3 to 1, 2. 3 d igits after the decimal point. Will you obtain an exact quotient?
LET'S LEAIlN ABOUT IT
1. In the division at 237 by 5, the remainder Is 0 and 47.4 Is the exact quotient. because 47.4 x 5 = 237.
2. In order to calculate the decimal quotient of a whole number divided by another whole number: 1°) do the division to the last given digit of the dividend; 2°) lower the digit denoting the tenths of the dMclend (0), place the decimal point
after the quotient and continue dMding the tenths; 3°) then lower the digit for the hundredths and continue your division, then lower the
digit fo r the thousandths, and so on ...
A. 23~ - 217
2
page 100
B. 23. - 21
2 - 1
Olt-z*1
76
8 2
c. 23 .00 h -2~ * 117.66
-1 8 --20
-18 2
D. -~t · *1010
I~ .666
::.U. 20 1 8
20 - 1 8
2
2. To complete the table in situation 2, let's write down:
[llI2J ~
11 x3200=440x [2J .
Then: G = 3~1l = 80 .
With 3 200 VT, we can buy 80 erasers.
Calculating a fourth number when you already know the three other numbers, is called "the rule of three". Any problem to do with the rule of three can be solved by using the properties of proportionality.
LET'S PIlACTISE
1. Complete these tables of proportionality, using the property of «crossed .. diagonal products.
[Hj[ili]m~ 18 35 20 183.6 25 45 70 ffitj[lli 18 5
2. SOy if these tables are proportionality tables.
~ ~ [ffiJ 70 90 110
22 44 5.2 16.9 22 18 84 lOB 132
SOL VING PIlOBLEMS
Show each of the following situations In a proportionality table and find the answers with the «cross» products method.
1. 4 kg of cocoa beans yield 1 800 g of cocoa powder. How much could you obtain from 15 kg of beans?
2. A car consumes 6 litres of fuel per 100 km. What will the fuel consumption be over 325 km?
3. 50 Q of milk make 9 Q of c ream and 1 Q of c ream makes 350 9 of butter. How many litres of milk will It take to make 2 450 g of butter? How much butte r can be produced from 75 Q of milk?
4. A piece of land 40 CXXl m2 has an estimated value of 5 CO) CO) yr. It is divided into two plots of 15 CO) m2 and 25 CO) m2. Calculate the price of each plot.
page 77
Multfplication of a decimal by a whole number iJ{Il ~----------------------------------~~ LET'S FIND OUT
1 litre of oil weighs about 0.912 kg. Calculate the weight of oil in: • 10, 100, 1 (XX) bottles of 1 litre; • 1 carton of 12 bottles; • 1 pallet of 64 cartons?
Explain how you work out these calculations. Work out the rules for mutiplylng a decimal by:
• 10; 100; 1 CXXJ; • any whole number.
LET'S LEARN ABOUT IT
- The weight In kg • of 10 Q is: 0 .912 x 10 = 9. 12; • of 100 Q Is: 0.912 x 100 '" 9 1.2; • of 1 OCO Q is: 0.912 x 1 CXXl = 912.
To muttiply a decimal by 10, 100 or 1 CXXl ... move the decimal point 1, 2, or 3 ... places towards the right.
• The weight In kg
1<,) of 1 carton of 12 litres is: 2<') of a pallet of 64 cartons each of 12 litres:
[1l1J 3 digits after x O. 9 ~~ ---+ the decl1mal point
18 24 9 I 2
r:::-:-:l 3 digits atter 1 0 .~ ---+ the decimal point
x 64 1 43776
65664
~ 3 digits atter 1 O~ +--the decimal point
~ 3 digits otter 7 0 O~ +-- the decimal point
When multiplying a decimal by a whole number. forget about the decimal point until you come to the end of your calculation, then place the decimal point so that there are the same number of digits atter It in your answer, as there were in the original decimal number.
LET'S PRACTISE
1. Give an approximate value of the following products. then do the calculations: 145.06 x 26 307.95 x 203 38.4 x 143
1 050.25 x 105 2 030 x 35.08 0.47 x 259
2. Do these in your head: 13.25 x 10 0.12 x 1 CO)
147.4 x 100 4.8 x 100
page 78
17.342 x I 000 10.6 x 100
1 050.04 x 1 COJ
LET'S PRACTISE
1.
:'11 :3
A piece of land Is In the shope of a diamond-shope. The long diagonal Is 4.5 dam and the short one 39 m. Find the area.
--- -----'--- -----4.5dam : ,
2. A field h the shape of a diamond-shope has an orea of 372.9 dam2,
Find the length of the long diagonal If the short one measures 165 m.
SOL VING PROBLEMS
1. The short diagonal of a dlamonct.shape shaped field measures 245 m. Given that the long diagonal measures twice the short one. calculate the area of the fleld.
2. The long diagonal of a diamond-shape with an area of 1 850 cm2 measures 92.5 cm. How much longer Is the long diagonal than the short one?
---- - ----~ ---- --- --, , , ,
3. If I odd both diagonals of a diamond-shope, the total Is 560 m. If I divide the long diagonal by the short. the result Is 7. Cak:ulate the area of this dkunond-shope.
4. One of the diagonals of a diamond-shope measures 200 cm. Its area Is 400 m2.
What's the length of the other diagonal? What other name con you give to this diamond-shape?
page 99
LIANG ____ d __ dO __ '_Knd ___ -m_~ ___ ' ____________________ ~~ LET'S FIND OUT
B
COpy ABCD onto paper. A ----- ------ ---t -------------- C oK o
o
- Cut out ABCO Into four triangles. along the diagonals. - Rearrange them to fOfm a rectangle. - What can you soy about the length and width of the rectangle In relation to the dlagona~ of the diomond-shape?
- Calculate the orea of the diamond-shape ABCD based on the lengths of the diagonals.
LET'S LEARN ABOUT IT
B
2
Cuffing along the diagonals, we obtain four right-angled triangles:
A ---- ----------,--- ----------- C :K
The mangles ore rearranged to form a rectangle. The area of the rectangle Is equal to the area of the diomond-shape ABCD. The side BD of the rectangle Is also the short diagonal d of the diomond-shape. The side MS of the rectangle Is equal to AK 'Wtllch Is the half of the long diagonal D,
page 98
d : short diagonal D: long diogonal
4 o o o o o o o
o
3
M ,----""71 B
3
A IE--------j K
4
2
o
Multlpllcaffon of a decimal by a whole number A{Il L-______________________________ -,~
SOL VING PROBLEMS
1. Cala decides to put some screens on his windows. His living room has three windows each measuring 1.60 m long and 0.9 m wide. His bedroom has a window 1.25 m long and 0.9 wide. The netting for the screens Is sold In rolls 0.9 m wide. Rnd the approximate length of the screening required.
The netting costs 525 VT per metre. What will be the total cost of Colo's project? He has saved 15850 VT. How much wlR he have left att9f buying the screens?
2. At the Pay Less supermar1<et, 0 kg of potatoes costs 160 VT. 1 kg at Imported rice 130 VT and 1 kg of apples 380 VT. Mum buys 3.7 kg of potatoes. 4.8 kg of rice and 5 apples weighing 0.3 kg each. She has 3 COl VT in her purse. Do a quick sum In your head to see If she has enough money in her purse to pay for these purchases. If so, work out how much she wiU have left.
3. A retailer buys a demijohn of perfume. WIth the contents, he fills 24 bottles of
0.125 R, 14 bottles of 0.050 , and 20 bottles of 0.65 R. What was the quantity of
perfume in the demijohn?
4. A workshop has an order for 24 shirts, Each shirt takes 1.65 m of mater1al. How much cloth will be needed? The workshop has 75 m of cloth, so how much wl,lI be left?
5. Frlda buys a drum of 011. costlng 250 VT/litre. WIth the contents from the drum, she fills
60 bottles of 0.60 Q. What Is the capacity of the drum bought by Frlda? What does a bottle of 011 cost? What profit does Frlda make If she sells a bottle for 270 VT?
6. A grocer packs 142 tins Into a crate weighing 8.750 kg when empty. Each tin has a gross weight of 0.575 kg. What's the weight of the crate when full of tins?
7. A blacksmith needs 9 bars 1.40 m long to make a screen. What length of Iron does he need?
page 79
Parallelograms and diamond-shapes
LET'S FIND OUT
Look carefully at the quadrilaterals and then fill out the diagram at the right. 1 2 3
/ \ L><7 ~ I 1. 2. 3. 4. 5. l 1 6, 7, 8,Q, 10
4 5 6 ,- ~ "",-
~ L==J 1><1 I 11
~ """ -7
$ 9
~ C><J Iv l I I 1 111
~ - "'" - Iv 10 $
_ Measure and compore the angles linked by the same diagonal in the figures 2. 3, 4, 7, 8. 9 and 10. What do you notice?
_ In the figures 2, 3, 4, 7, 8, 9 and 10, check that the diagonal Hnes Intersect exac tly In the middle.
LET'S LEARN ABOUT IT
I All these shapes are quadllaterals. In shapes 2. 3. 4. 7. 8. 9 and 10 the angles linked by the some diagonal are equal.
11 111e shapes 2. 3. 4. 7. 8, 9 and 10 ore parallelograms beCause they have two pairs of parallel sldes. 111elr diagonals also Intersect exactly in their middle.
III The parallelograms with equal diagonals (3. 4. 8 and 9) are rectangles.
IV 111e parallelograms with perpendicular diagonals are diamond-shapes or rhombi (4 . 8 . 10).
V The diamond-shapes wtth equal diagonals or the rectangles wtth equal perpendicular diagonals ore squares (4. 8).
Area of parallelograms and trapeziums
LET'S PRA eTISE
1. A piece of land Is In the shope of a 2. A parallelogram Is 25 hm high with 0 parallelogra m. with height 118 m and base 45.3 dam. base 750 drn. Ca lculate Its area. Calculate Its area.
3. A piece of land In the shape of a right-angled trapezium has bases of 128 m and 72 m and 0 height of 64 m. What's Its area ?
4. What's the short base of 0 trapezium with a n orea of 4.4 dam2, it the long base Is 57 m and the height 11 m?
5. Calculate the height of 0 trapezium with an area of 24 cm2, If the long base is 7 cm and the short base 5 cm.
SOL VING PROBLEMS
1. A square plot of land 75 m of sides is exchanged for a flekj 'With the some area but In the shope of 0 parallelogram having lED a base of 45 m. lAND Calculate the height of the parallelogram.
76 m
451"11
2. The long base of a trapezium measures 64 m. 111e short baSe Is two times shorter and the height four times shorter than the short base. Calculate the area.
3. These two carpets have the same area 36 m2• The one Is a rectangle 4 m wide and the other a square. Both .
ore to be trimmed with sotin ribbon. Will the rectangle .; need more ribbon than the square or will the same
~, »rd amount do to( both?
...;" -. .;. .~.,
Area of parallelograms and trapeziums
LET'S FIND OUT
1. Copy ABeD on paper.
Al ze H 8 K
2. Copy ABeD on paper. o e
A 8
LET'S LEARN ABOUT IT
Now draw the perpendiculars DH and CK from 0 and C. Cut out the figures AHD and BKe. Superimpose these two triangles. What do you notice? Compare the area of the rectangle HKCD and that o f the parallelogram ABeD. How can you calculate the area of ABCD?
Extend AB from A so that AM "" CD. Extend CD from 0 so that ON = AB. What kind of quadrilateral is BCNM? Compare the area of quadrilateral BCNM with that of trapezium ABeD. Calculate the area of ABCD. knowing AB, CD and the height of the trapezium.
1. AHD and BKe cover each other exactly and have the same area. The area of the rectangle HKCD Is the same as that of the parallelogram ABeD. BKe replaces AHD. The ared of the parallelogram is therefore equal to DC x OH. I
A H
As OH (or CK) is the height of the parallelogram and DC its base, we can soy that the area of a parallelogram is equal to its base multiplied by its height.
B: base h: height
2. AB is the long base of the trapezium, called B, CD is the short base of the trapezium, called b . BCNM Is a parallelogram. Its length CN Is equal to CD + ON, w ith ON = AB. We can say that the length (or base) of the parallelogram Is:
N 0 e
<'------\--7-/-------;\ I=AB+CD
The area of the parallelogram is A = J x h
or A=(AB+CD)xh A = (B + b) x h
Because the area of this parallelogram is double that of the trapezium ABCD, divide the area of the parallelogram by two to find the a rea of the trapezium.
page 96
M A H
B: long base b: short base h' height
8
Parallelograms and diamond-shapes A{Il L-________________________ ~ ____ ~~
LET'S PRACTISE
1. In a c ircle with centre 0, draw two perpendicular diameters AC and BD. Mark the midpoints I of OB and J of OD. Draw AICJ. What kind of quadrilateral .is It?
2. Draw a line AC = 6 cm. Through its midpoint 0, extend a straight line XY. Mark OB = 3 c m on OX and OD = 3 cm on OD. What type at quadrilateral Is ABCD?
3. Construct a square 5 cm x 5 cm. Is it a rhombus? a rectangle? a parallelogram? State your reqsons.
SOL VING PROBLEMS
1. Draw a line MN '" 6 cm. Construct an 2. Construct a parallelogram on a piece ot angle NMX = 60 degrees at M and at N, paper. Draw the smaller diagonal and an angle MNY of 120 degrees on the cut along it. Place the two triangles side some sid e os MN. Draw MP '" NO = by side In such a wcry as to obtain two 8 cm on MX and NY. Soy what MNOP is. new parallelograms.
3. Draw and cut out four triangles each with the sides 6 cm, B cm, and 10 cm. Arrange the triangles to make a quadrilateral. Is It a parallelogram?
4. Cut out four right-angled isosceles triangles the sides of which at the right angle are 3 cm. Arrange them to form: a) a square:
b) a rectangle: c) a triangle.
5. Draw and cut out two triangles AOB then two triangles AOD so that you have four triangles. Arrange these triangles to form a parallelogram.
L~ol 8 Can you make different parallelograms by rearranging the four triangles?
0
6. Draw the figure shown here twice and cut out the triangles AOB and AOD.
A~B - Arrange these triangles to form a quadrilateral. What type Is n?
- What other figures can you form by rearranging the four 0 triangles?
page 81
Units of area · ConverJlons • L-______________________ ~ ______ ~~
LET'S FIND OUT
This figure represents a square with 1 m sides. In real lite. the side would be 10 times longer thon this.
lm
- How many err? are there In 1 dni? - How many dnT are there In 1 m'? - How many rr? ara there In 1 knT? - How many dam' are there In 1 hm'? - How many mrrr are there In 1 rrf?
To the same scale here Is 0 square with sides of 1 dm:
D - What Is the area of a
square with a side of: - 1 m? - 1 d m? - 1 cm? - 1 hm?
- Do you know other units of area? Wrfte these units in a correspondence table.
How many times greater Is a unit than the unit ImmedIately below It?
LET'S LEARN ABOUT IT
A square with 1 m sides has an area of 1 m2,
A square with sides of 1 dm has an area of 1 dm2,
At the r1ght. study the correspondence table showing the relationship between length and area:
... 1m 1 dm
1= 1 hm 1 km 1 <km
km'
I ~ , page 82
hm'
1 0 100 0 0 0 0
1 0 0
o
1mm
= ' o o
10000 0
, '~ -' j , ,.. ""'_ oI .... ~ai_",.,,. , ,
mm'
... ~ 1 m' 1 dm'
1=' 1 hm' 1 km' 1 <km' 1 mm'
o
" n A
.
Enlargement and reduction of shapes
LET'S PRACTISE
1, Copy the triangle ABC, 2, Copy the square ABCD,
A ~ Draw on enlargement DEF of the triangle ABC In such a way
a C thatDE=3xAB,
ADD a C
Then construct another larger square to the scale 4 ,.
3. Copy the parallelogram ABCD.
a) On squared paper. construct a paralelogram EFGH so that:
EF"'3xAB and EH =3xAD Dl l a b) likewise, construct the parallelogram IJKl so that U = ~ )( AB and Il = ~ x AD.
2 2
4. Andy has drown figure A and Jonah has drawn figure B which Is an enlarged version of A. Measure the different dimen-sions of these two A shapes then calculo- 7 te the scale of enlargement.
SOL VING PROBLEMS
a
'------~/
1. On the plan of his town Michael measures the d istance from his home to school
which Is 12.5 cm. GIven that the scale of reduction is l~' calculate the real
distance Mlchael has to go to school.
2. A rectangular classroom measures 6.40 m x 6.50 m. On the longer side there Is 0 door ·0.00 m wide and situated 1 m from one of the corners of the room. In your
exercise book draw the plan of this room to the scale of reduction 1~ .
3. A designer wants to represent the head of a pin 2 mm In diameter by a circ le 6 mm in diameter, What ",,11 be the length of the pin in the deslgn ~ Its real length ~ 13 mm?
page 95
Enldrgement and reduction of shapes
LET'S FIND OUT
1. This rectangle represents the reduction of 0 piece of land:
I~ 4cm
What are the actual measurements If the ratio
is -'-? 500
How many times larger are the actual measurements than those of the rectangle shown?
LET'S LEARN ABOUT IT
1. As the scale of reduction
is ~. each cm on the
drawing represents 500 cm In the actual neId. The measurements of the piece of land are 500 times greater than than those of the rec tangle In figure 1.
l ength: 4 x 500 = 2 (0) (cm)
Width, 1.5 x 500 = 750 (cm)
The measurements of the piece of land ore: 2 (0) cm x 750 cm or 20m x 7.5Om.
page 94
2. Joson has to draw a 3. scale model of a rectongukJr field 10 km x "~ 8 6 km. Help him calculate the measurements given that the scale Is 1 cm to the km. Draw this field in your notebook.
2. Joson did the following calculations: 1 km is represented by 1 cm, therefore 10 km are represented by 10 c m and 6 km are represented by 6 cm. The fleld is represented by a r ec tang le measuring 10 cm x 6 cm.
6cm
A
Two towns A and Bore 32 km apart. On the map of the district this distance Is represented by 8 cm. What Is the scale of the map?
3. To be able to calculate the scale of reduction . you must flrst have the same un it s of measurement. First of all change 32 km into centimetres. 32 km = 3 200 (0) cm. SO the scale of reduction Is 8 cm for 3 200 (0) cm. ThIS Is written as:
13 ~ (XX) '" 4OO1(XX) I
In other words. 1 cm on pap e r rep r esen ts 400 (0) cm or 4 (0) m or 4 km on the actual piece of land, 4 km of road are represented by 1 cm on the mop of the district.
UnHs 01 Olea - Conversions
LET'S PRACTISE
1. Express In dm2: 3 m' 0,06 m2
2. Express In rr?-, 40 dam2
327.5 dm2
3. Express In cm2;
8 dm2
0,02 dm2
4. Complete, 4.27 c m2 = ... mm2
23 700 cm2 ::: ••• m2
11 km2 = , .. dam2
17.5 dam2 = ... km2
10 hm2
9 km'
0.04 dm2 = ... c m2
0.06 m2 :: ••• mm2
O.W hm2 = , .. dm2
24 hm2 = '" m2
0.38 dm2 = .. . mm2
4 100 mm2 = ... dam2
1 2(X) km2 = ... m2
O.l cm2 ... ... m2
5. Complete , 7 dm2 3Omm2 + 8 m2 5 dm2
'" .. . m2
2684 dm2 + 8 750 cm2 '" •• • m2
00 (0) cm2 + 75 (0) cm2 = ... hm2
24 m2 90 dm2 + 32 dm2 50 cm2 = ... dm2
8 m2 6 dm2 + 5 B()) dm2 - ... dam2
SOL VE THIS PROBLEM
•
•
L.....lf"rW." A" 2hm'37-..' '"5km' 85 C- 386m2
0- 2 kmt lOtmt
"""'-M. Coley COnan -...-.a.c. Ph: 66-<>IH>6 •
This poster makes the announcement that during the month of July some sect ions of land are being put up for sale. Each section will sell for 600 VT per m' .
Work out how much the mayor's office expects to realise If 011 sections of IOnd are sold.
For the accounts. the records must express 011 areas In m2 and rank them from the smallest to the largest. Help the records to be kept correctly.
page 83
MuHlpllcaflon of a decimal by a"decimal . .
LET'S FIND OUT
An experimental agricultural p lot measures 11.9 m long and 2.2 m wide. The c rop Is
peanuts and the yield is 8.2 kg per m2,
_ Give an approximate value of the orea of the plot. - Calculate the exact area. • Calculate the projected horvest. _ Explain how you carried out the calculations.
LET'S LEARN ABOUT IT
11 .9 Is approximately 12 and 2.2 Is approximately 2. _ The area of the plot In m2 is therefore approximately 12 x 2 = 24. _ You can calculate the exact area in m2 in the following way:
1 1 12] ) :i~·1·~2·dlglt;;;n~r:
x ~~~ :.~.~:",r-~~~~n! .: 23 8 ,..- - --------------;
: 2 d igits after ' 26.(!]] .....- , the decimal point:
: --- - - - - - ------ --'
- In kg, the projected harvest is:
page 84
26.[]]
x 8.~
5236 20944 214[ijJ
) ~---------- -----
: 2 + 1 = 3 digits atter :
~~~~~~~~~~~~~~ , : 3 digits otter '
<4- : the decimal point : . - - - ----- - - - - ----'
_mu/llplyingQdeclmd,",",,-~_~_. foIpf>/
about /he decimal point ... """ __ /he - " ""'" ",.",. fIN ~ In y.,... ..,. ..... so ft)CJt ~ .. CIf _r dIQ//S _It CIf ",.,. w ... 1n /he oIIgInaI twO decimal _ . "
The rule of three
SOL VING PROBLEMS (continued)
5. A roll of wire Is 24 m long and weighs 2 kg. What's the weight of 0 roll of 48 m?
12 m? 6 m? 144 m? Draw both the table of proportlonattty and a graph W'lth one SQuare fa 6 m and one square fOf 1 kg .
What's the length of a roll of wire weighing 10 kg?
QUICK SUMS
1. Beginning wtth the number 1 033, a student has made a list of the numbers: 1 133; 1 233; 1 333; 1 433; 1 533.
a) What system did he use? b) Continue the list orolly to 2 033.
2. Complete ,
3. Do these sums quickly:
... x 10 = 400 3.5x ... :35OO
7.82 x 100 : ... 1 !XX) x ... = 25 (X))
10 1 x ... = 1 010
0.403 x 1 !XX) '" ...
... x 1 050 : 10 500
100 x ... = 293 500
13.3 x 10 : .. . 72.91 x 10 = .. .
78.324 x 100 :: .. . 21 x 36 :: .. .
5.3 x 1 (X)):: . . .
37.2 x 20 = .. . 1.703 x 1 (X)) :: .. .
page 93
The rule of three Ml L----------illl-J LET'S PRACTISE Complete the following proportion tables using the properties given.
1.
Length 01 cable 0.5 1 3 7.5 13 Inm
Weight In kg 1 2
2.
LHr.s of honey 2 1.5 7 5 3 18 23
Price In VT 1 000
I~I~
SOL VING PROBLEMS
In order to solve a problem. draw up the table of proportion and use the diagonal method to obtain your answer.
1. The Ingredients for a special kind of coke are 4 eggs. 240 9 sugar, 260 g flour and 80 g c ream. What quantities of c ream, ftour and sugar will be needed to make a cake with 7 eggs? 1 egg? 21 eggs?
2. A worker receives 1 500 VT for three hours overtime. How much will he be paid for 9 h , for 51 h overtime? How much overtime will he have to do to eam 70 CO) VT?
3. 125 kg of husked rice are obtained from 250 kg of paddy nce. How much husked rice would hopefully be obtained from 800 kg paddy? How much paddy would you need to obtain 15 kg of husked rice? Draw a graph, with 1 kg of husked nee represented by one square on the vertical and 2 kg of paddy by one square on the hortzontal line.
4. 72 g of ground coffee are enough for a cups. How much ground coffee Is needed for 16, 24 and 40 cups? Draw the graph with one square for 4 cups and one square for 50 9 coffee.
page 92
Multiplication of 0 decimal by 0 decimal
LET'S PRACTISE
Give apprOximate values for the follolN1ng and then work out exactty: 272.25 x 3.04 1 207.5 x 0.35 7.343 x 12.24
SOL VING PROBLEMS
1. Miriam decides to paint her sitting room measuring 3.9 m long. 2.8 m wide and 2.9 m high. Given that there ore three windows 1.2 m high and 0.9 m wide. state the approximate area to be painted. If 1 kg of the paint covers 15 m2
, w111 10 kg be enough for the job?
2. GMng an approximate value, compare the oreas of two buildings \NIth the following dlmensk>ns: one In the shape of 0 parallelepiped 4.8 m long, 2.3 m Vv1de and 3.5 m high, the second In the form of a cube wtth sides of 3.8 m.
3. To make a door. 0 blacksmith uses 8.5 bars of iron. Given that one bar weighs 2.125 kg , calculate the total weight of the door.
4. A rectangular field measures 47.25 m long and 32.04 m wide. Given that 0.25 kg of seed are required for 1 m2
• how much seed Is needed to sow the 'Whole fleld?
5. A merchant sells 7&J 9 bars of soap in cartons of 24. Each bar of soap Is 0 cube wtth sides of 12 cm. What In kg Is the net weight of a carton of soap? The merchant Is planning to display the contents of a carton. either by arranging the soaps end to end or In 6 rows. What in metres would be the length of the soap In either case?
QUICK SUMS
32 x 15· .. . 19 x 18 . .. . 11 x 21 . .. . 25 x 12 = .. . 25 x 9 = .. . 11x11· .. . 45x11= .. . 29 x 12 = .. . 10 x 11 = .. . 21x15= .. . 58x 9= .. . 25x11 · .. . 37 x 19 . .. . 39x27= .. . 11 x 5O= .. .
page as
Patalletogroms and dlamond-Ihapes - ConsIrucIion
LET'S FIND OUT
1. Draw DAB.
Construct 0 parallelogram ABCD using a rulef and compasses.
2. BD", 3 cm Is the diagonal line of a parallelogram.
Draw It and construct the parallelogram ABCD with the second diagonal measuring 2 cm.
3. Construct a diamond-shape with the diagonals AD= 6cmand BC '" 4 cm.
LET'S LEARN ABOUT IT
Shapes --, stage,
Draw DAB. Draw the o rc from
A
o
B~
--, Drow the arc from
centre B with radius AD . centre 0 with radius AB.
A B A B
oL<7 B ,
0 0 , , , , R-AD
, , : C~--_ , '-
Drow the diagonal BD. MorI< 0 midpoint of BD. Draw AC In such 0 way thatAO ... OC.
2 B~ B~ B~O "cm o l.5em D
0
Draw the diagonal AD. Mor'< 0 midpoint of Draw BC perpendicular AD. to AD in such 0 way
thotOB: QC.
3
A~D A~ A~ 3cm ~.§: ~ o 3cm D B 0
B
o
LA_ ddItIon ___ and __ subIractton_bl_",_-_-_tton_ot_~ ___ _ _ ot_ durl __ 08_IOI'_n _ ___j~ LET'S PRACTISE
I . Calculate: 1 d 23 h 45 mln 16 s + 2 d 3 h 30 mln 40 s 7 d 20 h 15 mln 10 s - 4 d 19 h 6 mln 20 5
2. Calculate the sum then the difference:
12 h 35 min and 5 h 18 mkl 13 h 53 mln and 7 h 24 mln 22 h 00 min and 15 h 59 mln 23 h 55 min and 0 h 05 mln
SOL VING PROBLEMS
1. Kate decides to colour her hair. She p repares the hair colorant. The Instructions advise leaving the colour in the hair for 20 minutes. By Kate's watch, it Is 2.20 p .m. when this time Is up. A what t ime did she finishing putting the colour In her hair?
2. Paul arrives a t the post office at 4.20 p .m. Nick has been waiting for him fOf 45 minutes. Carl arrived 10 minutes after Nick. Give the order of a rrival of the three friends. When did Nick arrive? Carl arrive?
3. Zak has on appointment at the denttst's at 4.20 p .m. He leaves home at 3.45 p .m . and It will take him 30 minutes to get there. Will he be late or early? By how much?
4. At the saw mill, wood Is being dried in the ovens. Drying a batch of meranti takes 15 days and 17 haurs. The same quantity of teak dries in 14 days and 15 hours. The same quantity of oak dries in 16 days and 9 hours. Which wood dries most quickly? Calculate how long the oven will be In operation for the successive drying of the three batches.
page .,
Addition and sublroclton of measulements of dutaffon
LET'S FIND OUT
During a cycle roee In two stages, the cyclists A and B do the times shown In the table bek>w,
• What Is the t1me taken by Stage Stage 1 stage 2 each cyclist to finish both
Cyclist A 3 hOT mln 155 2h59mln55s stages of the race?
- Which of them wiN win the Cyclist B 2h58mln50s 3h04min 155 race?
- What Is the gap between the two cycMsts at the end of the roee? - Explain how you dO the calculation.
LET'S LEAlm ABOUT IT
1. - TIme token by cyclist A: h min • 70s:: 1 min+ 105
6Omin::1h 3 I 15 therefore 5 h 60 mln 70 s '" 6 h 1 min 10 s + 2 59 55
W/!eo acIding unit> of duration, add ft!em 5 I:IJ 70 up column by column, then do any necessary convfHSions.
• In the same way, calculate the time token by cycist B.
2 .• Gap between the two cyclists at the end of the h mln • race:
In the seconds column, you can't subtract 10 6 3 5 from 5, so you have to borrow one minute (60 s) - 6 I 10 from the 3 min (k1 minutes Column) and odd 5 s to have 65 s. The calculatlDn then k:x>ks like this:
h mln • ro "'- units of -. caIc_ /he
6 2 65 dll'enlnces column by column beginning wiItI /he
- 6 I 10 NCOnCIa. ",., mQ/c/ng ""y adjwttnen/$ """""""Y to mQ/ce ""bIrac/kin possible. '
I 55
. In the some way. find the difference between 3 h 03 min 10 s and 2 h 59 mln 55.
page 90
LET'S PRACTISE
1. Draw a diamond-shope so that the larger diagonal is twice the length of the smaller.
3. Draw 0 Quadrilateral with diagono~ of the same length, 6 cm.
2. Oraw this figure in your exercise book and complete It to make a parallelogram ABCD.
4. Draw 0 porollek>grom
ff B 4cm
ABCD wtth diagonals 8 cm. Explain how you did Measure the angles and compare them.
c
the th~.
5. Draw a parallelogram ABeD with the sides AB = 4 cm and CD = 3 cm . Construct a second parallelogram with dimenslons double that of the first.
SOL VING PROBLEMS
1. Draw a rectangle ABCD with AB '" 6 cm and BC '" 3 cm. On segment AB. 1 cm from A mark the point M and 1 cm from B. point N, Join M and D. then construct the parallelogram MNTD. What kind of figures a re AMD and NBCT?
3. On a piece of squared paper. draw three sets of ox right-angled triangles so that the sides of the right angle measure 4 cm and 3 cm. Cut out these triangles. With each set construct a quadrflate ral so that the three obtained convex quadrilaterals are not superimposable.
2. Construct a dlamond·shape. wlth diagonals measuring respectively 8 c m and 6 cm. GIve the exact measurements of each of the sides.
4. M
T
x y ECA IB OF
Draw two perpendicular lines 'IN and UT as shown. Beginning INfth I. mark B. D, F, A. C and E so that 18 = BD = Df = lA = AC = CE = I cm and IM = IN = 2 cm. Join M and N to each of the points marked on 'IY.
N U - What kind of figures are the convex quadrilaterals
AMBN, CMON and EMFN? • What can you say about the quadrflateral MBNF?
page 87
Area of squarBs, rectang les
LET'S FIND OUT
Situation 1 Situation 2 SltuaHon 3
s=4cm I:: 6 cm
,A , SQUARE E RECTANGLE E u u
of ., of ., 4 cm of side " 4cm x 6cm " • • Draw the height line
AH then construct the rectangles HAMC and HANB. - Compare the area
Divide this square up Into - How many squares with sides 1 cm of HAMC with that squares with sides of can you fit along the length, along of HAC. and the 1 cm. the width? area of HANB with - How many small - How many small squares with 1 cm that of HAS.
squares will you have sides will there be in the rectangle? - Compare the area on each side? - How do you calculate the area of of the triangle ABC
- How many small! a rectangle from the measurements with that of the squares wilt you have in I and w? rectangle BCMN. all? - H ow do you
- How do you calculate calculate the area the area of 0 square of a triangle, given when you know the one side and the side s? height line relative
to this side?
LET'S LEARN ABOUT IT
Situation 1 Situation 2 Situation 3
- On each side there will - Number of squares on N A M be 4 squares with sides length /, 6
BVl\c 1 cm. width w, 4 -The number of small - Total number of
squares Is 16 = 4 x 4. squares; - The area of the square 24 = 6 x 4
given side s: - Area of the rectangle H
I A ~sx s ,' 1 from the dimensions I and w: - The area of HAMC Is 2 x the
The area of a square Is I A. 'XW,] area of HAC. equal to the side - The area of HANB is 2 x the area multiplied by itself. The area of a rectangle of HAB,
Is equal to the length - The area of BCMN Is 2 x the multiplied by the width. area of ABC.
_ The area of ABC = BC x AH 2
lA {. S~,h' l A: orea
a-- " ,'de h' height line
page 88
-Area of squares, rectangles and trtahgles
LET'S PRACTISE
1. Find the area of squares with the fo llowing measurements: 14 cm; 18 cm; 95 mm; 2 hm; 25 m.
2. Calculate the areas of Rectmgle Length WJdth Area rectang les with the
B m 15dm measurements shown In A
the table . 8 35 dm 20 cm
C 5 hm 30 m
0 lOOm 1hm
3. A c loth has been made from 24 squares measuring 3 dm x 3 dm. What Is the area of one square? What Is the area of the cloth in m2?
4. Calculate the area of these triangles and complete the table. The sides and height lines are given.
Side 6 cm 9 cm 4dm 5.5 dm 4.5 mm 19 m
Height line 7 cm 13 cm 5 dm 4.2 m 34.5 mm 6.3 m
Areo
SOL VING PROBLEMS
1. A rec tangular piece of land has a perimeter of 212 m. The length exceeds the width by 32 m. What are the measurements of the ground? What Is its area?
3, 320 square boards 25 cm x 25 cm are used to cover a rec tangular floor 4 m wide. How many boards will there be along the width? What Is the length of the floor?
5. How many tiles 25 cm x 25 cm are needed to pave a rectangular room 12 m x 8 m?
2. A smith cuts a square of Iron from a sheet weighing 125 9 per square decimetre. What is the area of the p laque and what Is Its weight?
4. The perimeter of a rectangular piece of ground is 120 m. One of Its sides measures 35 m. Calculate the second measurement and the area of the ground.
6. The ground In a square room 4.50 m x 4.50 m Is to be concreted. What w ill be the total cost If 1 m2 costs 980 VT?
page B9
