Self assessment ,f · Web viewComparing fractions and percentages 61 Fractions, Decimals and...

Self assessment ,f

Application of Number

Level 1

Contents

3Exercise 1

4Millions

5Exercise 2

8Addition

9Subtraction

10Multiplication

11Division

12Adding Decimals

15Subtracting Decimals

19Multiplying Decimals by Whole Numbers

21Dividing Decimals by Whole Numbers

24Multiplying, dividing by powers of 10

25Rounding Off and Approximation

26Rounding numbers to the nearest 10



29Rounding

30Rounding off amount of money

31Handling Data

31Charts

31Graphs

32Line graphs

32Pie charts

33Frequency tables

33Conversion graphs

34Mean and Range

37Range

40Averages (1)

41Working out equivalent ratios

42Writing ratios in common units

44Ratio

45Proportion

46Comparing Lengths

50Converting Between Units

51Conversions between Metric and Imperial Systems

52Conversions between Metric and Imperial Systems

53Distances

55Areas and Volumes

56Area of rectangles

58Area

60Volume

61Comparing fractions and percentages

62Fractions, Decimals and Percentages

64Finding fraction parts

68Percentages

Exercise 1

1) Write these numbers in words:

a) 6 320

b) 12 503

c) 163 900

d) 268 726

e) 402 150

2) Write these numbers in figures:

a) three thousand, four hundred and twenty

b) ninety six thousand, one hundred and thirty one

c) four hundred and twenty five thousand, seven hundred and two.

d) eight hundred and two thousand, and ninety three.

e) seventy thousand and sixteen.

f) two hundred and five thousand, and six.

For the following questions, select the appropriate answer A, B, C or D

3) The largest ever football crowd was 199 000 people. How is this number written in words ?

a) One hundred and ninety-nine million

b) One million nine hundred thousand

c) One hundred and ninety-nine thousand

d) Nineteen thousand nine hundred

4) Twenty thousand five hundred and six people went to a concert.

The number of people was

a) 2 056

b) 2 506

c) 20 506

d) 205 006

Millions

If you're talking millions, you need to have at least seven figures in there.

Examples

1) One million written in figures is

1 000 000

2) Two million, three hundred and forty two thousand

Write in the two, leave a space then write in three hundred and forty two as you normally would.

Leave another space then write in three zeros to show that there are no hundreds, tens or units.

2 342 000

3) Four million, two hundred and five thousand, one hundred and eighty

Write in four, leave a space then write in two hundred and five. Leave another space then write in one hundred and eighty.

4 205 180

2 500 000

Write in two then leave a space.

Whatever comes after the decimal point, in this case five, write it in and follow it with five zeros so that there are seven figures in total.

Exercise 2

This is really boring but excellent practice I


a) five million

b) nine million

c) two million, five hundred thousand

d) four million, two hundred thousand


a) 2 000 000

b) 4 000 000

c) 5 500 000


a) one million, two hundred and twenty two thousand

b) nine million, four hundred and four thousand

c) five million, forty two thousand


a) 5 757 000

b) 2 120 000

c) 1 304 000

d) 6 045 000

e) 7 002 000


a) six million, one hundred and seventy five thousand, two hundred and forty six

b) five million, three hundred and four thousand, two hundred and six

c) one million, eight thousand, one hundred and twenty


a) 1 234 567

b) 4 460 020

c) 2 009 009

d) 3 040 040

7) How much are the following amounts in figures ?

a) £8.5 million

b) £2.9 million

c) £0.5 million

For the following questions, circle either answer A, B, C or D

1) A total of four hundred and three thousand, five hundred and two people in Greater Manchester voted at the last general election. How many people is this in figures ?

a) 43 502

b) 430 502

c) 403 502

d) 4 003 502

2) A company made £308 010 profit in one month. How much is this amount in words ?

a) three million, eighty thousand and ten pounds

b) three hundred and eight thousand and ten pounds

c) thirty eight thousand and ten pounds

d) thirty thousand, eight hundred and ten pounds

3) The population of Cairo was nine million, five hundred thousand, four hundred and twenty when last counted. How do you write this number in figures ?

a) 9 500 000 420

b) 950 420

c) 95 420

d) 9 500 420

4) Jenny was lucky enough to win £6.2 million on the National Lottery. How much is this in figures ?

a) £620 000

b) £6 200 000

c) £62 000

d) £62 000 000

You can "use" numbers in four different ways..

WAYS

SIGNS AND WORDS USED

Addition

+ adding, sum of, total

Subtraction

- subtract, take-away, difference between, minus

Multiplication

x multiply, "times", product

Division

( divide, sharing

SEE IF YOU CAN DO THESE!

Addition

1) 76

+ 18

2) 89

+3

3) 625

+ 227

4) 206

498

+ 6

5) 625

+398

6) 462 + 87

7) Add these numbers; 274; 648

8) Find the total of 362 and 479

9) Find the sum of 462, 87 and 8

10) Add two hundred and seven, to six hundred and ninety four.

Subtraction

1) 462

- 120

2) 865

- 626

3) 300

- 212

4) Subtract 623 from 900

5) What is the difference between 174 and 310?

6) From 436 take 208

7) How many more than 297 is 642?

8) How many less than 700 is 285?

9) Seven hundred and four minus eighty nine.

Multiplication

1) 8 x 7

2) 24

x 2

3) 164

x 4

4) 208 x 8

5) 412

x 26

6) Multiply 245 by 4

7) Double four hundred and eighty six

8) Which number is 3 times greater than 297?

9) What is the product of 6 and 4?

10) 325 times 16

Division

1) 8 ( 4

2)

4

48

3)

5

365

4)

5

505

5)

26

1924

6) Divide 707 by 7

7) How many 6's are there in 912?

8) Divide nine hundred and forty-five by 9

9) Divide five thousand, five hundred and fifty-five by five.

10) 164

4

Adding Decimals

When you add money you are adding decimals

(

£1.75 +

£2.64

£4.39

1 (

You can see that the decimal point is separating the pounds from the pence.

All the decimal points are in a line.

This is the main thing to remember when adding decimals

Keep the decimal points in a column.

1) £2.35+

£1.40

2) £27.99+

3.67

3) £102.70+

23.56

4) £1.35 + £3.45

5) £28.53 + £3.55 + £7.00

6) Katie goes to Kwik Save and buys Sausages for £1.39, Cheese for £2.50 and Apples for £1.27. How much did she spend?

Have a look at this example.

(

3.76

7.51

11.27

1 (

You add and carry numbers in just the same way as when using whole numbers or money.

7) 3.24+

6.51

8) 5.25+

6.19

9) 11.35+

1.17

10) 5.75 + 16.89 + 4.81

11) Add up 14.67 metres and 3.09 metres

12) Karen's yard is 3.45 metres long and 2.57 metres wide. How far is it all around the edge (the perimeter)?

Sometimes it helps if you fill the spaces at the end of each number with zeros (0's)

(

3.76 +

3.760 +

7.5

7.500 (

0.375

0.375

11.635

11.635

It doesn't make any difference to the answer. It just makes it easier to add up without losing your place.

Only put zeros at the end of a number (not in the middle)

13) 3.25 +

1.5

14) 25.3 +

16.97

15) 10.5 + 0.375

16) 19.9 + 10.101 + 0.009

17) 0.425 + 7.6 + 3.48

18) Paula is making curtains. Her window is 1.7 metres high, and she wants the curtain 0.15 metres longer than the window. How long will the curtain be?

Sometimes you will have to add whole numbers to decimal numbers. To do this put a decimal point at the end of the whole number and then line it up with the rest of the sum.

For example: 23 + 2.35

23. +Put a decimal point at the end of the

2.3523 and line it up.

23 00 +You can Put zeros In to make it look more

2 35balanced. We see this in money ….

25 35£23 is the same as £23.00

19) 25 + 3.33

20) 102.5 + 25 + 3.4

21) 0.785 + 785

22) 2.92 + 0.292 + 292

23) There are 7 litres of water in the urn. Jacob adds 15.5 litres more. How much is in there altogether?

24) Yasmin is fitting a skirting board in her bedroom. The walls measure 3 metres, 1.6 metres, 1.75 metres, 3 metres, 0.95 metres and 0.85 metres. How much skirting board will she need?

Remember Line up your decimal points

Carry numbers as usual

You can use zeros at the end of each number if you want

Line up whole numbers with the point at the end

Subtracting Decimals

When you take away money you are subtracting decimals.

(

£3.75 +

£2.54

£1.21

(

You can see that the decimal point is separating the pounds from the pence.

All the decimal points are in a line. Always make sure the biggest number is on top

This is the main thing to remember when taking away decimals

Keep the decimal points in a column.

1) £2.75 -

£1.40

2) £27.99 -

£3.67

3) £162.78 -

£23.56

4) £16.35 - £3.45

5) £28.53 - £3.55

6) Arthur gets £67.56 a week. He spends £12.45 at the supermarket. What has he got left?

7) Bashir spends £3.35 at the newsagent. He pays with a £5.00 note. How much change will he get?

8) Jenny buys a bed for £199.99. She gets £19.98 discount. How much will she pay now?

Have a look at this example.

13.76 -

7.51

6.25

You subtract and borrow numbers in the same way as when using whole numbers or money.

9) 9.24 -

6.51

10) 17.25 -

6.19

11) 11.35 -

1.17

12) 16.89 - 4.91

13) 14.67 - 9.09

14) Yasmin has 3.7 metres of material. She needs 1.8 metres to make a dress. How much will she have left? Will she have enough material left to make another dress for her sister?

Sometimes it helps if you fill the spaces at the end of each number with zeros (O's)

9.76 -9.76 -

7.9 7.90(

1.861.86

It doesn't make any difference to the answer. It just makes it easier to take away without losing your place and helps when you are borrowing.

6.1 -6.10 -

3.453.45

2.652.65

Only put zeros at the end of a number (not in the middle)

15) 3.25 - 1.5

16) 25.3 - 16.97

17) 10.52 - 0.375

18) 19.92 - 10.101

19) 7.666 - 3.4

20) Janet is going to drive to Manchester. It is 34.9 miles. She stops after 13.0 miles to get some petrol. How much further does she have to go?

21) On Monday you can get 1.7 dollars for a pound. By Friday you get 1.61 dollars. What is the difference?

22) James and Edna share a bottle of wine (0.75 of a litre). Edna drinks 0.4 litres. How much did James drink?

Sometimes you will have to take away decimal numbers from whole numbers. To do this put a decimal point at the end of the whole number and then line it up with the rest of the sum. You can then add zeros (like you did above)

For example:23 - 2.35

23. -

2.35

Put a decimal point at the end of the 23 and line it up.

23.00

2.35

20.65

You can put zeros in to make it look more balanced. Take away as usual.

23) 102 - 25.4

24) 5.85 - 2

25) 2 - 0.292

26) 7.32 - 3

27) 27 - 3.3

28) Fiona buys a carpet which measures 7 metres. Her hall is 5.25 metres long. How much carpet has she got left? Will she have enough to carpet the landing which is 1.85 metres long?

29) Jamal can swim 2 lengths in 50.2 seconds. How much under a minute (60 seconds) is this?

30) A jug holds 2 litres. Lee pours a glass of juice which holds 0.125 litres.

How much is left in the jug?

How much is left if he pours himself another glass?

Remember

Line up your decimal points

Borrow numbers as usual

You can use zeros at the end of each number if you want.

Line up whole numbers with the point at the end.

Multiplying Decimals by Whole Numbers

When you have your answer check it by estimating and then with a calculator

1) £5.75 x 6

2) £9.40 x 7

3) £9.99 x 2

4) £107.44 x 5

5) Jan bought 6 towels. They were £4.99 each. How much did she spend altogether?

6) Dave went to buy Christmas cards. He got 7 packs at £3.15 each. How much did he spend?

7) 1.1 x

9

8) 5.4 x

2

9) 0.51 x

3

10) 107.42 x

5

11) 17.5 x

6

12) 0.999 x

2

13) Chris is building a fitted kitchen. He needs 5 units measuring 52.5cm and 3 units measuring 76.25 cm. How much does this come to altogether?

14) Cath is baking Christmas cakes for her family. She needs to make 4 cakes. For each one she needs 1.45 kg of dried fruit, 0.5 kg flour and 0.225 kg of almonds. How much of each ingredient will she need altogether?

Dividing Decimals by Whole Numbers

When we divide money we are dividing decimals.

If we share £5.16 by 3

1

3

£5.16

1

3

£5.216

You carry the remainder 2 onto the next number as usual.

1.7

3

£5.216

When you meet the decimal point in the sum, put it up in your answer. Now carry on dividing 3 into 21

1.72

3

£5.216

Finish off the sum and you have the answer. Does it look about right?

Have a go at dividing these sums of money.

6

£8.40

7

£2.24

2

£39.84

8

£16.56

1) Sharon buys 5 mugs for £9.95. How much is each one worth?

2) Eddie's electricity bill comes to £142.40 a year. How much is this each quarter (every three months)?

3) It costs £62.50 for 5 driving lessons. How much does 1 cost?

Now try these yourself.

5

9.175

57

2.8

2

7.156

3

10.2

6

6.390

9

70.38

Yasmin has a piece of wood measuring 1.72 metres. She wants to put 2 shelves in her hall. How long will each shelf be?

Evelyn has 7.36 metres of material. She wants to make 4 curtains out of it. Will she have enough material to make them if each one needs to be 1.8 metres long?

Sometimes you might get to the end of the sum and still have a number to carry (a remainder).

For example:

1

4

6.22

(

Start as normal

1.5

4

6.22

(

We divide 22 by 4, but still have a remainder of 2

1.5 5

4

6.2220

(

Add a 0 on the end and carry on as usual, until finished

You can carry on adding 0s for as long as you want, but usually it makes sense to ROUND OFF after 2 or 3 decimal places.

Multiplying, dividing by powers of 10

0.25 x 10 = 2.5, 3.197 x 100 = 319.7. In order to multiply by a power of ten, the digits must

be moved one place to the right (making the number bigger) for every zero in the divisor

(that is one place to multiply by 10, two places for 100 etc.)

6 ( 10 = 0.6, 8 ( 100 = 0.08. In order to divide by a power of ten, the digits must be moved

one place to the left for every zero in the divisor (that is, two places to divide by 100, three places for 1000 etc.)

1) 8.25 x 100 =

2) 4.927 x 1000 =

3) 0.05 x 10 =

4) 18.398 x 100 =

5) 6.2 x 1000 =

6) 9.41 x 100 =

7) 84 (10 =

8) 60 ( 100 =

9) 8.4 ( 1000 =

10) 27 ( 100 =

11) 1.06 (10 =

12) 2.41 ( 100 =

Rounding Off and Approximation

Example 1

The number 23.62 rounded to the nearest whole number

Temperature Readings on 14 November

0

2

4

6

8

10

1.002.003.004.005.006.00

Time (PM)

Temperature (oC)

23.62 is post half way so it is closest to 24.

Example 2

The number 23.62 rounded to one decimal place

Holidays

USA

Other

Europe

UK

23.62 is before half way so it is closest to 23.6.

Draw pictures to help you round the following numbers to the accuracy

given in each question.

1) 43.6 to the nearest whole number

2) 3.57 to one decimal place


4) 132 to the nearest 100

5) 58.99 to the nearest 10


7) 63.49 to the nearest whole number


Rounding numbers to the nearest 10

Round the following to the nearest ten:

Example: 46 is between 40 and 50 and would be rounded to 50.

1) 43 is between

and

and would be rounded to

2) 57 is between

and


3) 72 is between

and


4) 85 is between

and


5) 90 is between

and


6) 103 is between

and


7) A company made a profit of £7 821 in one year. What is this figure to the nearest £10?

8) A theme park had 9,462 visitors over a summer period. What is this to the nearest ten?


Round these numbers to the nearest hundred:

Example: 231 is between 200 and 300 and would be rounded to 200,

1) 384 is between

and


2) 946 is between

and


3) 162 is between

and


4) 853 is between

and


5) 211 is between

and


6) 7,423 is between

and


7) The cost of decorating a school was estimated at £25 670. What is this to the nearest £100?

8) Temi won £12 478 on the National Lottery. What is this figure to the nearest hundred pounds?


Round these numbers to the nearest thousand:

Example: 7 852 is between 7 000 and 8 000 and would be rounded to 8 000.

1) 1,963 is between

and


2) 2,243 is between

and


3) 4,625 is between

and


4) 8,463 is between

and


5) 9,271 is between

and


6) 19,253 is between

and


7) A distance between two countries is 12 870 km. What is this distance to the nearest 1000 km?

8) A newspaper sold 56 792 copies. What is this to the nearest thousand copies?

Rounding

1) Round the following football attendances to the nearest thousand:

17, 394

19, 801

35, 009

2) Round the following mileages to the nearest 10 miles:

369 miles

805 miles

466 miles

3) Round the following sums of money to the nearest £100:

£650

£8720

£299

4) Round the following weights to the nearest 10kg:-

637kg

3059kg

309kg

5) Round the following to the nearest £:-

£36.75

£6.99

£55.55

Rounding off amount of money

To the nearest pound

Remember that if the number after the decimal point is five or more, round up to the next pound.

Examples

£2.56 rounds up to £3.00

£8.49 rounds down to £8.00

To the nearest ten pence

Examples

£2.56 rounds up to £2.60 as 56p is closer to 60p.

£8.44 rounds down to £8.40 as 44p is closer to 40p.

Self assessment

Round these amounts of money off to the nearest pound.

£1.35

£2.89

£19.99

£111.49

£205.51

£3.50

Round the following amounts to the nearest £0.10 or 10p

£ 6.52

£ 13.18

£271.65

£215.55

£205.51

£989.89

Handling Data

· Data is information. Interpreting data means working out what information is telling you.

· Information in newspapers, on television, in books and on the internet is sometimes shown in charts, tables and graphs.

· It is often easier to understand the information like this rather than in writing, but it is important to read all the different parts of the graph or chart.

Charts

The title tells us what the chart is about

The column headings tell us what data is in each column:

· the name of the bike

· what colour it is

· its price

· and how many gears it has

You can use the chart to find out information about each bike by looking at each row in turn

· The Ranger is silver and has 5 gears

Bikes sold this week

Name

Colour

Price

Gears

Ranger

Silver

£140

5

Outdoor

Red

£195

10

Tourer

Blue

£189

15

Starburst

Silver

£215

15

Mountain

White

£129

5

Graphs

Graphs come in many different styles

Bar graph

Tip: With any graph, always look carefully at

· the title

· the scale

· the axis headings

Line graphs

Are made by joining the tops of bar-line graphs. This can make it easier to look at the shape of the graph.

This line graph shows that the temperature is falling each hour.

Pie charts

Pie charts are circular, like a pie! Each section of the pie shows a fraction of a total amount. This pie chart shows where 40 went on their last holiday.

One quarter of the people went to Europe. That means 10 people (40 ( 4) people went to Europe.

The UK was the most popular holiday destination.

Can you work out the second most popular?

Frequency tables

A frequency table shows information about a set of data. Sometimes there is so much data that the only way to show it all is to put it into groups called intervals.

This graph shows the heights of a class of children. The heights are grouped in equal intervals of 5cm. This means that 1.30 -1.34 includes children with heights of 1.30m, 1.31m, 1.32m, 1.33m

Childrens Heights

2

4

12

13

2

1

0

2

4

6

8

10

12

14

1.30 - 1.341.35 - 1.391.40 - 1.441.45 - 1.491.50 - 1.541.55 - 1.59

Height (Metres)

Number of Children

How many children are between the heights of 1.45m and 1.49m?

Conversion graphs

Conversion graphs are used to change one set of values to another.

This graph converts centimetres to inches. 5cm is approximately 2 inches.

Approximately how many centimetres are equal to 5 inches?

Mean and Range

After completing this unit, you will be able to:

· calculate the mean average for up to ten items of data

· calculate the range of up to ten items of data

Mean Average

The average value of a group of numbers is a number that can be used to represent the

whole group.

The MEAN AVERAGE is calculated as follows:

· Add up all the numbers.

· Divide your answer by the

· Number of numbers there are

Examples

1) Here are five numbers.

211863

Find the mean number.

Answer

2

11

8

6

3 +

30

there are five numbers, so divide 30 by 5.

30 ( 5 = 6 so the mean is 6.

2) The temperature of a science laboratory was recorded daily at noon over six days. The results are given in the table below.

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

9°C

11°C

14°C

12°C

8°C

0°C

What is the mean temperature at noon ?

Answer

9

11

14

12

8

0 +

54

2

there are six days so divide 54 by 6

54 ( 6 = 9 so the mean temperature is 9°C.

3) Find the mean of the following groups of numbers.

a) 1241437

b) 4cm3cm2cm2cm8cm4cm4cm3cm

c) £2.50£1.50£1.00£1.60

4) Sharon made a record of how much she spent each week on entertaining her daughter. What is the mean amount she spends per week ?

Week 1

Week 2

Week 3

Week 4

Week 5

Week 6

£18

£10

£15

£9

£12

£8

a) £10

b) £12

c) £15

d) £13

5) The table below shows the daily hours of sunshine in the Costa del Sol for the months of January to September.

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

5

7

8

9

10

11

12

10

9

What is the mean of the daily hours of sunshine for the months shown?

a) 5

b) 9

c) 12

d) 81

6) As part of a Health and Social Care assignment, a student researched the price of ten different drinks for children, five fizzy drinks and five fruit juices. Her task was to find out if, on average, it is cheaper to buy fizzy drinks or fruit juices.

Fruit juice

Price per 300 ml

Fizzy drink

Price per 300 ml

A

50p

A

55p

B

65p

B

85p

C

80p

C

55p

D

60p

&

65p

a) What's the mean price of fruit juice ?

b) What's the mean price of fizzy drink ?

c) What should the student's conclusion be ?

Range

The range of a group of numbers tells you how wide the numbers

are spread.

The RANGE is calculated as follows:

THE BIGGEST VALUE MINUS THE SMALLEST VALUE.

Examples

Find the range of the following numbers.

572986124

Answer

The biggest number is 12.

The smallest number is 2.

Range = 12 - 2 = 10.

The temperature of a science laboratory was recorded daily at noon over six days. The results are given in the table below.

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

9°C

11°C

14°C

12°C

8°C

0°C

What is the range of noon temperatures for the six days ?

Answer

The smallest number is now 0°C.

14 - 0 = 14, so the range is 14°C.

1) Find the range of the following groups of numbers.

a) 9151823

b) 10oc2oc22oc15oc7oc2oc

c) £3.00£1.20£4.50£6.30£2.00 £9.10

2) Mike is a car salesman. The table shows how many cars he has sold each month for a six month period

April

May

June

July

August

September

10

5

12

7

8

What is the range of number of vehicles he has sold from April to September

a) 5

b) 7

c) 9

d) 12

For the following questions, circle either answer A, B, C or D.

Five friends take part in a sponsored run and record the amounts they collected. The amounts are recorded in the table

Name

Ada

Bal

Cath

Dan

Ernest

Sponsorship (£)

10.00

24.00

23.50

42.50

60.00

Use the information above to answer questions 1 and 2.

1) What was the mean amount of sponsorship money collected per person ?

A £160 B £35 C £30 D £32

2) What was the range of the sponsorship money to be collected?

A £60 B £50 C £10 D £35

Willy is worried about his spending and decided to make a note of how much he withdraws from the cash machine on a daily basis. Over the past ten days he has made the following withdrawals:

£50£40£40£140£70

£0£50£90£130£50

Use this information to answer questions 3 and 4.

What is the mean amount he has withdrawn ?

A £66B £70C £50D £140

What is the range of Willy's withdrawals ?

A £100 B £66 C £140

Averages (1)

1) Find the average (mean) of 21, 17, 8, 3, 22

2) What is the total of the following mechanics' monthly wages:- £585, £836, £624, £453?

3) What is the mechanics' average wage?

4) What is the range of their wages?

5) The following distances are travelled on 40 litres of fuel by a fleet of taxis:-

56km, 87km, 98km, 72km, 86km, 76km.

What is the average length of journey?

6) What was the average fuel consumption in question 5?

7) The time taken by 6 mechanics to do a particular job was:-

10min, 13min, 15min, 17min, 10min, 9min.

What was the average time taken?

8) If I travel 300km in 3 hours, what is my average speed?

9) During a period of 9 months a car travels 7938km. What is the average distance travelled per month?

10) Five mechanics earn the following amounts as a weekly wage:- £98, £104, £106, £101, £96.

What is their average weekly wage?

What is the range of their wages?

Working out equivalent ratios

To make double this amount of pink paint you would need 8 white tins to 2 red tins.

This would give a ratio of 8:2

These two ratios are called equivalent ratios.

You can multiply or divide both parts of a ratio by any number to give an equivalent ratio. Both 8 and 2 can be divided by 2, so 8:2 is the same as 4:1

Simplifying ratios

e.g. Natasha and Rachel share a flat but because Natasha has a larger room they agree

to share the rent in the ratio 60 : 40. This can be simplified as follows:-

Divide both sides by 10 to get 6 : 4

Divide both sides by 2 to get 3:2

There are no further common factors of the two sides so this is the simplest form of the

ratio.

Self Assessment One - Writing ratios in their simplest form

Write each ratio in its simplest form

1) 6:2

2) 18 : 24

3) 80 : 20

4) 12 : 48

5) 12 : 25

6) 14 : 63

7) 75 : 25

8) 42 : 63

9) 4 : 8 : 6

10) 30 : 90 : 120

11) 14 : 28 : 35

12) 21 : 63 : 105

13) The label for a bottle of floor cleaning liquid says that one unit of fluid needs to be mixed with 20 units of water.

a) How much water is mixed with 5 units of cleaning fluid?

b) How much cleaning liquid is mixed with 180 units of water?

Converting to similar units

It is important to remember that when amounts are given in different units they must first

be converted into a common unit before they can be expressed as a ratio.

They can then be put into the simplest form.

E.G.Two TV programmes last 45 minutes and 2 hours.

Express the times of the two programmes as a ratio.

The times are45 minutes : 2 hours

Convert both times into minutes45 minutes : 120 minutes

This is now a ratio of45 : 120

Divide both by 153 : 8

This is the ratio in its simplest form.

Writing ratios in common units

Write each ratio in a common unit and then put it into its simplest form.

1) 20 minutes : 3 hours

2) 20mm : 3cm

3) 3kg : 850g

4) £3 : 60p

5) 250 ml : 1.75 litres

6) 2500m : 5km

7) An Orange drink is mixed in the ratio of 25 ml of orange cordial to 1 litre of water.

a) How much of this cordial would you mix with 4 litres of water?

b) What is the correct mixing ratio of cordial to water?

Write this ratio in its simplest form.

Ratio

Simplify the following:-

1) 3:6

2) 42 : 49

3) 40:50

4) 175 : 200

5) 50:60:70

6) 5000 : 7500

Simplify the following:

7) 500cm : 1m

8) £1.50 : 50p

9) 30p : 90p

10) 5 .5cm : 20mm

11) 2 kg : 500g

12) £12.50 : £2.50

13) If 2kg potatoes cost 32p, what would 7 kg cost?

14) 6 plates cost £7.20. What do 11 cost?

15) A machine makes 16 articles in half an hour. How many will it make in

a) 1 hour

b) 4 hours

c) 6 hours?

Proportion

1) 3kg potatoes cost 45p. What do 10kg cost?

2) 8 bolts cost 40p. What do 24 cost?

3) A model car is 50 times smaller than a real car.

a) What length on the real car is represented by 4 cm on the model?

b) 250cm on the real car is represented by how much on the model?

4) A mechanic earns £15 in 3 hours. Find how much she earns in

a) 1/2 hour

b) 4 hours

c) 7 hours

5) A mixture of paint used for spraying a car was made up of 2 litres blue with 6 litres grey. What is this, as a ratio, in the lowest terms?

6) Divide £50 in the ratio 1: 4

7) 4 lightbulbs cost £3.00. What would 5 cost?

8) 3 cans of cola cost £1.20. What would 5 cost?

Comparing Lengths

Sometimes you will have to compare different lengths and different units.

They become easier to compare once you have converted them to the same units.

You can convert any metric measurements like this:

Kilometres (km)

Millimetres (mm)

Using this chart change:

1) 9cm to mm

2) 21.5cm to mm

3) 30mm to cm

4) 95mm to cm

5) 4m to cm

6) 2.5m to cm

7) 400cm to m

8) 725cm to m

9) 7km to m

10) 10.675km to m

11) 5000m to km

12) 13cm to mm

13) 10.9cm to mm

14) 151mm to cm

15) 9mm to cm

16) 17m to cm

17) 19.3m to cm

18) 950cm to m

19) 87cm to m

20) 7.5km to m

21) 0.725km to m

22) 9990m to km

Which is bigger - 205cm or 2.5m?

We know 2.5m is the same as 250cm, so 2.5m is the longer length

Which is bigger?

1) 105cm or 1m

2) 10mm or 1.1cm

3) 700cm or 7m

4) 105mm or 1m

5) 100mm or 15cm

6) 3555m or 3.9km

7) 1.275m or 0.3km

8) Match up the measurements in column A with the one that is the same length in column B

A

B

30cm

7.6cm

9.5cm

300mm

107cm

950cm

3mm

0.237km

9.5m

3000mm

237m

760cm

76mm

0.3cm

7.6m

1.07m

3m

95mm

237cm

2.37m

Now try and use these skills to work out these perimeters.

1) Leanne measures her front room.

It is 6 metres long and 350 centimetres wide.

What is its perimeter?

2) She decides to extend the room.

When it is finished it is still 6m long but it is 150cm wider.

What is the new width and the new perimeter?

3) Leanne also has a bedroom with these measurements:

a) What are the missing measurements?

b) What is the perimeter of the room?

c) Will she be able to fit her furniture along the wall measuring 2.6m?:

bed (width 1.5m)

bedside table (45cm)

chest of drawers (70cm)

Converting Between Units

Length

1cm = 10mm

1m = 10cm

1km = 1000m

Weight

1kg = 1000g

Capacity

1litre=1000ml

How many millimetres are there in:

8cm

22cm

36cm

85cm

50m

How many centimetres are there in:

5m

28m

36m

90m

150m

How many metres are there in:

2km

12km

25km

65km

100km

How many grams are there in:

4kg

10kg

40kg

60kg

100kg

How many kilograms are there in:

6000g

15000g

25000g

82000g

10000g

Conversions between Metric and Imperial Systems

1) Convert the following to centimetres.-

a) 3 metres

b) 3.6m

c) 36mm

d) 0.065km

e) 5inches

f) 3.4ft

2) Convert the following to metres:-

a) 40cm

b) 213mm

c) 1.39km

d) 27inches

e) 5yards

3) Convert the following to grams:-

a) 300mg

b) 15.34kg

c) 12.31b

d) 0.231b

e) 17oz

4) Convert the following to litres. -

a) 3 gallons

b) 5 pints

5) (1 mile - 1.6km, 1km - 0.6miles)

Convert the following:-

a) 14 miles to kilometres

b) 26km to miles

6) 1 lb = 16oz

Convert the following to ounces:-

a) 2 1/2 lb

b) 1/2 lb

c) 2 lb 3oz

Conversions between Metric and Imperial Systems

1 inch - 25mm

1 ounce - 28g

1 foot - 30cm

1 pound (lb) - 450g

1 yard - 90cm

1 cwt (hundredweight) - 50kg

1m - 39 inches

(3ft 3in) or 3.25ft

100g - 3.5oz

250g - 8.8 oz

25cm - 10 inches

1 mile - 1.6 km

1kg - 2.2 lb

1 tonne - 1 ton

Complete the following:-

1) 600g =oz

2) 75cm =inches

3) 3 lb =g

4) 5 miles =km

5) A vehicle is travelling at 70 miles per hour. How many km/hr is this?

6) A mechanic weighs 80kg. How many pounds is this?

7) Write as a fraction of a pint in lowest terms :-

a) 6fl.oz

b) 15fl.oz

c) 8 fl.oz

8) Convert the following, if 1 fluid ounce = 30mh-

a) 2fl.oz= ml

b) 5 floz. = ml

c) 1/3 floz. =ml

Distances

Try these questions and see how you do

Check each of these answers and tick one of the boxes.

True

False

1) A map has a scale of 2 cm = 1km, so 4cm = 1/2 km

2) On a map with a scale of 3cm = 1 mile, 12 cm represents 4 miles

3) The distance between two towns is measured as 12cm. the map uses a scale of 4cm to 1km so the towns are 3km apart

4) A map uses a scale of 5cm to 1mile and the distance between two towns on the map is 20cm. this represents 100miles

Use the map below to answer the questions. (Scale 3 cm = 1 km)

5) The distance between D and E is km

6) The journey G to E to D iskm

7) Travelling from E to G, then to D and on to F is a total of km

8) The trip from F to D and on to E is a total of km

Distances


Check each of these answers and tick one of the boxes

True

False

1) A map has a scale of 1 cm = 5km, so 4cm = 20 km

2) On a map with a scale of 1cm = 8 mile, 4 cm represents 2 miles

3) The distance between two towns is measured as 12cm. the map uses a scale of 1cm to 4km so the towns are 3km apart

4) A map uses a scale of 1cm to 8km and the distance between two towns on the map is 5cm. this represents 40km

Use the map below to answer the questions, (Scale 1 cm = 8km)

5) The distance between A and C iskm

6) The journey A to B to C iskm

7) Travelling from A to C, then to B and on to D is a total ofkm

Now check yours answers with those on the answer sheet. How did you do

Areas and Volumes

Rules

Area of a rectangle = Length * Width and gives results in e.g. square metres or square cm or square feet etc.

Tip; get used to drawing a diagram and the dimensions

Area = L*W = 3cm * 5cm = 15cm squared, (change to same units where necessary)

E.g.1m= 100cm

1m= 1000mm

1cm = 10mm

10cm=100mm etc

Class questions

1) A rectangle has sides 15cm and 12cm. Find the area, what would be its perimeter distance?

2) A square has a side of 8cm 5mm, What is its area?

(Note before multiplying, the measurements must be in the same units).

Thus 8cm 5mm= 8.5cm or 85mm

A = L * W = 8.5cm*8.5cm - 72.25cm2

or

85mm *85mm = 7225mm2

Perimeter = 85 + 85 + 85 + 85

Area of rectangles

The area of a rectangle or square = length x width

Find the answers to the following problems:

1) area = 25 m2 length = 5 m width = ?

2) area = 36 m2 length = ? width = 3 m

3) If the area is 153 m2 and the width 9 m calculate the length.

4) 72 sq metres is available for a playground area in the local park. The length must be 12 metres because of the nearby houses. How wide must it be?

5) Jim bought a can of paint which will cover 50 sq metres of fencing. The height of the fence is 2 metres. How many metres length of fence will he be able to paint before running out of paint?

Jim has 70 metres of fencing to paint. How many tins of paint will he need to

buy?

6) 6. A lawn is being laid with new turf. Each roll of turf has an area of 1 sq metre and is 0.5 m wide.

How long is each roll?

The area of a rectangle = length x width

Jeff sees a carpet in the soles that he would like to buy.

area = 24 sq metres width = 4 metres

1) Calculate the length of the carpet.

2) Jeff's room measures 3 metres by 7 metres. Should he buy the carpet? Give reasons for your answer.

3) Extension activity:

If the original cost of the carpet is £95, how much is it in the sale?

Area

(Make sure you use the correct units of measurement for each answer.)

Find the area of the following rectangles.

1

2

3

4

5) A corridor has a rectangular floor of length 10 metres and width 1.5 metres. Find its area.

6) What is the area of a window 0.4 metres wide and 3 metres long?

(Make sure you use the correct units of measurement for each answer.)

Find the length of the following rectangles.

1) Area = 20 cm2, width = 4 cm

2) Area = 16 cm2, width = 4 cm

3) Area = 22m2, width = 10 m

Find the width of the following rectangles.

4) Area = 50 m2, length = 100 m

5) Area = 2.5 cm2, length = 10 cm

Volume


Work out the volume of each shape below, being careful to use the correct units.

1)

Volume

2)

Volume

Volume

3)

Volume

4) Work out the volume of a cuboid 16 cm long, 4 cm wide and 10 cm high.

Volume

5) What if the volume of a cube of side length 5cm?

Volume

Comparing fractions and percentages

Fractions to percentages

Write these fractions as percentages.

1) 1/2 =

2) 1/4 =

3) 1/5 =

4) 1/10 =

5) 3/4 =

6) 2/5 =

Percentages to fractions

Write these percentages as fractions, simplifying your answer.

1) 25% =

2) 50% =

3) 75% =

4) 10% =

5) 20% =

6) 30% =

Fractions, Decimals and Percentages

Complete the following table, converting fractions (in lowest terms), decimals and percentages

Percentage

Fraction (lowest terms)

Decimal

1/5

0.1

60%

0.8

3/4

70%

100%

1/2

0.12

17%

0.35

Simplify the following fractions

16/50

55/110

15/20

800/900

25/100

48/72

20/80

8/18

Work out the following:-

1) 1/2 of 7 minutes 16 seconds

2) 1/3 of 10 years 3 months

3) 2/7 of 364 days

4) 2/9 of £7.56

5) 3/10 of £81

6) 3/4 of £1000

7) 7/8 of 1000m

8) 3/4 of 84 litres

9) 7/8 of 104 gallons

10) 2/5 of 85km

Finding fraction parts

1) Find:

a) 2/3 of £15

b) 3/4 of £12

c) 4/5 of £10

d) 6/7 of £21

2) When a box of eggs is dropped, 2/3 of the eggs are broken. If the box holds 18 eggs altogether, how many are damaged?

3) Two hundred people take part in a survey. Three-quarters say that they like to go to the cinema. How many people is this?

4) Complete the table below:

Fraction wanted

Divide by

Multiply by

2/3

5/7

8

3

12

7

11/

15

Now try these

Exercise 1

Change these fractions to percentages.

1) 4/5

2) 1/3

3) 3/4

4) 1/5

Now try these

Exercise 2

Change these decimals to percentages.

1) 0.65

2) 0.375

3) 0.89

4) 0.6

5) 0.350

Now try these

Exercise 3

Change these to fractions

1) 25%

2) 30%

3) 75%

4) 4.33 1/3 %

5) 37 1/2 %

Exercise 4

Now convert these to decimals.

1) 63%

2) 4.7%

3) 51.65%

4) 3.1%

5) 7%

Copy out this table, then fill in the gaps

Fraction

Decimal

Percentage

1

1.0

100%

1/2

0.5

3/4

75%

1/4

0.25

1/8

1/3

2/3

Exercise 5

1) An apprentice earns £8600 p.a. He is allowed £4000 tax free and pays 20% income tax on the rest. How much income tax does he pay?

2) A company awards a 5% pay rise to all employees. Complete the table below:-

Name

Current wage

Increase

New wage

Paul

£9000

Ali

£9600

Sue

£9900

Pat

£12200

Find

3) 10% of £3.50

4) 10% of 60cm

5) 10% of 126m

6) 20% of £6.50

7) 6% of 1km

8) 40% of 35m

9) 33 1/3% of 9cm

10) 25% of 52 litres

Percentages

Change these percentages to fractions, in lowest terms:

a) 40%

b) 35%

c) 28%

d) 75%

e) 90%

1) 10% of 45kg

2) 30% of 70

3) 5% of 300

4) 12% of 30

5) 12.5% of 240

Now try these

Exercise 6

1) Increase £200 by 4%

2) Decrease £200 by 4%

3) Increase £420 by 10%

4) Decrease £420 by 10%

Now try these

Exercise 7

What percentage is:

1) 12 out of 48

2) £30 out of £150

3) £200 out of £700

4) 0.5 out of 2.5

5) 1000 out of 8000

23.5

23

24

23.65

23.7

23.6

350cm

6m

1.7 m

375cm

260cm

2.6m

?

?

D

F

8 cm

9 cm

E

6 cm

G

12 cm

B

C

A

D

5 cm

6 cm

2 cm

9 cm

L=5cm

W=3cm

8.5 cm

85mm

6cm

2cm

3cm

3cm

6mm

13mm

8.6m

9m

4 m

8 m

3 cm

3 cm

cube

8 m

15 m

4 cm

6 cm

cube

Self assessment ,f · Web viewComparing fractions and percentages 61 Fractions, Decimals and...

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