Mathematics Decimal Notes
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Transcript of Mathematics Decimal Notes
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MATHEMATICS - FORM 1
DECIMALS
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DECIMAL AND FRACTION
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1. A decimal is a fraction in !ic! t!e denominator
is 1" or a #oer of 1".
For e$am#le %-
&. T!e decimal #oint ' t!e dot ( se#arates
t!e !ole n)m*er from its fractional #art.
A n)m*er ritten it! a decimal #oint is
+non as a decimal.
For e$am#le%-
,. T!e n)m*er of diits after t!e decimal #oint
is t!e same as t!e n)m*er of eros in t!e
denominator.
For e$am#le%-
A( Re#resentin Decimals it! Diarams
Decimals can *e re#resented it! diarams.
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For e$am#le %-
1 , / 1" 1"" 1 """
'1 out of 10 ( ' 3 out of 100 ( ' 7 out of 1 000 (
0( ritin Decimals *ased on i2en Diarams
or+ed E$am#le
Write a decimal to represent the shaded
area in the diagram above.
Solution
The shaded area is 3 .
10
Therefore, the decimal is 0. 3.
C) Conversion of Fractions into ecimals
and vice versa
1. ! fraction in "hich the denominator is
10 or a po"er if 10 can be converted to a
decimal mentall#.
Wor$ed %&le
%&press the follo"ing as decimals.
'a) ( 'c) 13
10 1 000
'b) *+
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100
Solution
Wor$ed %&le
Change the follo"ing into fractions "ith
denominators of 10 or po"ers of 10.
'a) 0. 'b) 0.** 'c) 0.3(-
Solution
+. ! fraction can also be converted to a
decimal b# dividing the numerator b#
the denominator.
3. i&ed numbers can also be e&pressed
as decimals.
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Wor$ed %&le
%&press the follo"ing fractions as decimals.
'a) 3 'b) ( + - *
Solution
or+ed E$am#le
%&press the follo"ing as decimals.
'a) 1 1- 'b) - * 100 +0
Solution
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or+ed E$am#le
%&press the follo"ing decimals as mi&ed
numbers.
'a) 3. 'b) 11. 0+*
Solution
'a) 3. / 3 0.
/ 3
10
/ 3 10
3LACE 4AL5E AND DI6IT 4AL5E
IN DECIMALS
A( 3lace 4al)e and Diit 4al)e
1. %ach digit in a decimal has its o"n
place value and digit value.
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oo$ at the decimal .+3* belo".
Wor$ed %&le
State the place value and digit value
of the underlined digit in each of the
follo" decimals.
'a) 13.- 'b) (.02*
Solution
'a) Tenths 0.-
'b) Thousandths 0.00*
+. ecimal places are the number of
places occupied b# the digits after
the decimal point.
For e&le 45
'a) 0.3
1 decimal place
'b) 0.0
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2 decimal places
'c) 3.1*+
3 decimal places
'd) 2.(-+0
4 decimal places
6) 7epresenting ecimals on 8umber ines
! decimal can be represented on a number line.
For e&le 45
Wor$ed %&le
State the decimal represented b# each
of the letters on the number line above.
Solution
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9ne part / -.2* 5 -.0
/ 0.1*
F / -.2* 0.1* 0.1*
/ *.+*
G / *.+* 0.1*
/ *.-0
Therefore, F / *.+* and G / *.-0.
C) Comparing T"o ecimals
When comparing t"o decimals, arrange the
digits of the decimals according to their place
values and then compare their digit values
from left to right.
Wor$ed %&le
Which is greater, +-.*3 or +-.3+
Solution
!rrange the decimals as sho"n belo".
!t the place value of hundredths, * is larger than 3.
Therefore, +-.*3 is greater than +-.3+.
Wor$ed %&le
Fill in each bo& "ith : ;: or :
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Solution
) 9rder of ecimals
We can arrange decimals in increasing or
decreasing order according to their values.
Wor$ed %&le
!rrange (.1, .-, .3, .1 in decreasing order.
Solution
Wor$ed %&le
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!rrange 11 , +.(3 and +.(3+ in increasing order.
-
Solution
%) 7ounding 9ff ecimals.
! ecimal can be rounded off to a "hole number
or certain decimal places. The rules for rounding
off decimals are as follo"s.
'a) oo$ at the digit to the right of the digit that
is to be rounded off.
'b) =f the digit is more than or e>ual to *, add 1
to the digit that is to be rounded off and ignore
the rest of the digits after it.
'c) =f the digit is less than *, maintain the digit
that is to be rounded off and ignore the rest of
the digits after it.
Wor$ed %&le
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7ound off .3(*+ correct to
'a) 1 decimal place,
'b) + decimal places,
'c) 3 decimal places.
Solution
Wor$ed %&le
7ound off the follo"ing decimals to the
nearest "hole numbers.
'a) .1 'c) 3*+.-
'b) 30.3+-
Solution
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ADDITION AND S50TRACTIONOF DECIMALS
1. ecimals are added and subtracted in
the same "a# as "hole numbers.
+. When t"o decimals are added together
or subtracted from each other, the deci5
mal points must be placed directl# one
belo" the other and the digits "ritten in
the correct place value columns.
!) !ddition of ecimals
Wor$ed %&le
Calculate
'a) 0.3+ -.+- 'b) * .
Solution
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Wor$ed %&le
Calculate
'a) 3-.+ +.(3 2.2*
'b) 2.+* *- 1(.31
Solution
6) ?roblem Solving involving
!ddition off ecimal
Wor$ed %&le
Salman is 1.(* m tall and !r@un is 0.12 m
taller than Salman. Ao" tall is !r@un.
Solution
1. Bnderstand the problem
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iven information 4
Salman:s height / 1.(* m
!r@un is taller than Salman b# 0.12 m
Find 4 Aeight of !r@un
+. evise a plan
Bse addition.
3. Carr# out the plan
!r@un:s height
/ Salman:s height 0.12 m
1.(* m 0.12 m / 1.2- m
1 . ( *
0 . 1 2
1 . 2 -
Therefore, !r@un is 1.2- m tall.
-. Chec$
1 . 2 -
5 1 . ( *
0 . 1 2 ' The difference )
Wor$ed %&le
e"i collected +1.10 litres of late& on
onda#, + litres on Tuesda# and
+3.20 litres on Wednesda#. Find the
total amount of late& collected.
Solution
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The total amount of late& collected "as
2.20 litres.
C) Subtraction of ecimal
Wor$ed %&le
Calculate
'a) 0.* 5 0.3*
'b) 3 5 +.-+1
Solution
Wor$ed %&le
Calculate
'a) *(.3 5 3*.+ 5 +.3-
'b) 1 5 +-.20( 5 (.*
Solution
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) ?roblem Solving involving
Subtraction of ecimals
Wor$ed %&le
SuDiela "eighs *+.03 $g and !Dman "eighs 0.-
$g. Find the difference in their bod# mass.
Solution
1. Bnderstand the problem
iven information 4
SuDiela:s mass / *+.03 $g
!Dman:s mass / 0.- $g
Find 4 ifference in their bod# mass
+. evise a plan
Bse subtraction.
3. Carr# out the plan
0.- $g 5 *+.03 $g
Therefore, the difference in their bod#
mass is .3( $g.
-. Chec$
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* + . 0 3
. 3 (
0 . - 0
Wor$ed %&le
Eaiton bought .3 m of cloth. She used +.1* m
to ma$e a shirt and 3.+ m to ma$e a pair of
pants. Ao" much of the cloth remained
Solution
.3 m 5 +.1* m 5 3.+ m / 0.( m
Therefore, 0.( m of the cloth remained.
M5LTI3LICATION AND DI4ISION OF DECIMALS
A( M)lti#lication of Decimals
i - 6enerel m)lti#lication of decimals
1. To find t!e #rod)ct of decimals7 m)lti#l8 t!e
n)m*ers in t!e same a8 as for !ole n)m-
*ers first. T!en7 #)t in t!e decimal #oint.
&.T!e n)m*er of decimal #laces in t!e anser
m)st corres#ond to t!e total n)m*er of decimal
#laces in t!e decimals *ein m)lti#lied.
or+ed E$am#le
Calc)late
'a( ".9, $ ".9
Sol)tion
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ii( M)lti#l8in a decimal *8 a #oer of 1"
!en m)lti#l8in a decimal *8 1"7 1""7
1 """7 ect.7 e mo2e t!e decimal #oint 17
&7 ,7 ect. #laces res#ecti2el8 to t!e ri!t.
or+ed E$am#le
Calc)late
'a( 1.:& 'c( ".;
'*(
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".17 "."17 ".""17 ect.7 e mo2e t!e decimal
#oint 17 &7 ,7 ect. 3lace res#ecti2el8 to t!e
left.
or+ed E$am#le
Calc)late
'a( &:.; $ ".17
'*( =,.
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0( 3ro*lem Sol2in in2ol2in M)lti#lication
of Decimals
or+ed E$am#le
Dalina needs "./= + of flo)r to ma+e a ca+e.
Ho m)c! flo)r is needed to ma+e ; ca+es >
Sol)tion
1. 5nderstand t!e #ro*lem
6i2en information %
Amo)nt of flo)r to ma+e 1 ca+e ? "./=
Find % Amo)nt of flo)r to ma+e ; ca+es
&. De2ise a #lan
5se m)lti#lication.
,. Carr8 o)t t!e #lan
; $ "./= + ? 9.
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M)lti#l8 aain to 2erif8 t!e anser.
or+ed E$am#le
A a##le costs RM".9". T!e #rice of a @ac+fr)it
is :.; times t!e #rice of t!e a##le. !at is t!e
cost of = @ac+fr)its >
Sol)tion
Cost of a a##le ? RM".9"
Cost of a @ac+fr)it ? :.; $ RM".9"
Cost of < @ac+fr)its ? = $ :.; $ ".9"
? RM1
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or+ed E$am#le
Calc)late
'a( ".9 B 1"7 'c( ,;.:< B 1 """.
'*( &,1.: B 1""7
Sol)tion
ii - Di2idin a !ole n)m*er *8 a !ole n)m*er
or+ed E$am#le
Calc)late ,= B :
Sol)tion
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T!erefore7 ,= B : ? ;.9=
or+ed E$am#le
Calc)late ; B , and i2e 8o)r anser correct
to : decimal #laces.
Sol)tion
T!erefore7 ; B , ? &.
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'*( ".&& B &&
Sol)tion
&. !en di2idin a !ole n)m*er *8 a decimal7
con2ert t!e di2isor to a !ole n)m*er first *8
mo2in its decimal #oint to t!e ri!t.
or+ed E$am#le
Find t!e 2al)e of eac! of t!e folloin.
'a( < B ".&
'*( ,,".11
Sol)tion
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i2 - Di2idin a decimal *8 a decimal
!en di2idin a decimal *8 a decimal7 )se t!e
idea of e)i2alent fractions to con2ert t!e di2isor
to a !ole n)m*er. S!ift t!e decimal #oints t!e
same n)m*er of #laces in t!e di2idend and di2i-
sor to ma+e t!e di2isor a !ole n)m*er.
or+ed E$am#le
Find t!e 2al)e of eac! of t!e folloin.
'a( =.=& B ".<
'*( "."9& ".;
Sol)tion
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2 - Di2idin a decimal *8 a fraction and 2ice 2ersa
!en di2idin a dec$imal *8 a fraction7
c!ane t!e o#eration from di2ision to
m)lti#lication and in2ert t!e fraction at
t!e same time.
or+ed E$am#le
Find t!e 2al)e of eac! of t!e folloin.
'a( ,.9& B , :
Sol)tion
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D( 3ro*lem Sol2in in2ol2inDi2ision of Decimals
or+ed E$am#le
A+mal *)8s < tic+ets for a circ)s s!o
costin RM;=.;". Ho m)c! does eac!
tic+et cost >
Sol)tion
1. 5nderstand t!e #ro*lem
6i2en information %
N)m*er of tic+ets *o)!t ? <
Cost of < tic+ets ? RM ;=.;"
Find % Cost of 1 tic+ets
&. De2ise a #lan
5se di2ision
,. Carr8 o)t t!e #lan
RM;=.;" B < ? RM1:.,"
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T!erefore7 t!e cost of 1 tic+et is RM1:.,".
:. C!ec+
1 : . , " $ < ; = . ; "
COM0INED O3ERATIONS
OF 7 -7 $7 B OF DECIMALSA( Com*ined O#erations of Additionand S)*traction
or+ed E$am#le
Find t!e 2al)e of eac! of t!e folloin.
'a( 9.:1
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0( 3ro*lem Sol2in in2ol2in Addition and S)*traction
or+ed E$am#le
A 2essel ei!in "./; + contains &=.;
+ of rice. If 1".= + of t!e rice is )sed7
find t!e total mass of t!e 2essel it! t!e
remainin rice.
Sol)tion
1. 5nderstand t!e #ro*lem
6i2en information %
Mass of 2essel ? "./; +
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Mass of rice ? &=.; +
Mass of rice )sed ? 1".= +
Find % Mass of t!e 2essel and remainin rice
&. De2ise a #lan
3erform addition folloed *8 s)*traction.
,. Carr8 o)t t!e #lan
"./; &=.; - 1".=
? &
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Sol)tion
'a( 9 $ " . / B : =
?
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* & -. + G 11 +
/ * & -. + G 3
+
/ 12+. G 3
+
/ 12+. & +
3
/ 3*. + G 3
/ 1+. -
-. Chec$
3 & 1+. - / 12+.
+
12+. G 3
/ -. +
Wor$ed %&le
!m# bought +0 eggs costing 73. 0.
issa bought 3* of those eggs. Ao"
much did issa pa#
Solution
Cost of +0 eggs / 73.0
Cost of 3* eggs
/ 3.0 G +0 & 3*
/ 0.12 & 3*
/ .*
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Therefore, issa paid 7.*.
%) Combined 9perations of !ddition,
Subtraction, ultiplication and ivision
Wor$ed %&le
Calculate each of the follo"ing.
'a) 1+. * 5 0. + & +.
'b) 3. + & - 5 10. - G -
'c) *. 1+ 5 +. G 1
-
Solution