Fractions, decimals and percentages

Fractions, decimals and 4 percentages

Master

Check P93

Strengthen P95

Extend P99

Test P103

4.1 Equivalent proportions

You will learn to: ? Convert between fractions, decimals and percentages ? Compare fractions, decimals and percentages ? Write a fraction as a decimal.

CONFIDENCE

Why learn this? Information about proportions can be given as fractions, decimals or percentages. You need to be able to convert between different types.

Fluency

What is the decimal

equivalent of ? _12 ? _34 ? _11_0 ? _17_0

Explore Can you write the number as a fraction?

Warm up

Exercise 4.1

1 Write as a percentage. a 72 out of 100 b 25 out of 50

c 0.73

d 1.29

2 Work out a 34 ? 10

b 286 ? 1000

c 9021 ? 100

3 Write each decimal as a fraction.

a 0.75

b 0.3

c 0.9

4 Write < or > between each pair of decimals.

a 0.15 u 0.172 b 0.62 u 0.603 c 4.049 u 4.5

5 Sort these into five sets of equivalent proportions.

17

7

3

100

10

4

4

1

5

20

0.75 0.05 0.17 0.7

0.8

5% 80% 17% 70% 75%

6 Write each fraction as a decimal and as a percentage.

a _21_0= 0.u = u% c _38= 0.u = u% e _78= 0.u = u%

b _230_= 0.u = u% d _270_= 0.u = u% f _1210_= 0.u = u%

79

Topic links: Measures

Key point

You can write a fraction as a decimal by dividing the numerator by the denominator. _42= 2 ? 4 = 0.5

Q5 hint Rewrite fractions so they have denominator 100 to work out the percentage.

Q6 hint

Use the S D key on your calculator

to change the fraction into a decimal.

7 Write each fraction as a decimal and as a percentage.

a _430_2= 0.u = u%

b _8701_= 0.u = u%

c _19_200_= 0.u = u%

d _11_0550_= 0.u = u%

e _11_16= 0.u = u%

f _11_285_= 0.u = u%

8 Problem-solving Which shape has the longer perimeter?

A 0.1 m

0.3 m 0.4 m

0.1 m

B

7 20

m

3 20

m

1 20

m

8 20

m

9 Use the decimal equivalent of each fraction to write the sets in order,

smallest to largest.

a _12_30 _35 _58

b _38 _27_0 _4106_

10 Put these values in order, smallest to largest. _12_30 0.62 64.5% 9 out of 16

Worked example

Write 0.245 as a fraction. 0.245 = _10_2_04_50__

There are 3 decimal places so 245 has been divided by 1000

= _2_409_0_

Divide numerator and denominator by 5 to simplify. 49 and 200 don't have any common factors so it

cannot be simplified further.

11 Write each decimal as a fraction. Simplify where possible.

a 0.85 = _108__50_=

b 0.375 = _10_u_0_0_ =

c 0.84

d 0.125

e 0.23

f 0.875

g 0.19

h 0.444

12 Write each percentage as a fraction. Simplify where possible.

a 35% = 0.35 = _10_u_0_= _2u_0_

b 6% = _10_u_0_= _uu_

c 88%

d 5%

e 12.5% = 0.125 = _10_12_05_0_ =

f 37.5%

g 45.8%

h 1.2%

13 Explore Can you write as a fraction? Choose some sensible numbers to help you explore this situation. Then use what you've learned in this lesson to help you answer the question.

14 Reflect This lesson uses a lot of mathematical terms, such as ? terminating ? equivalent ? percentage ? common factor. Write down what each of these terms means, in your own words. Which terms are new, and which ones have you met before?

Key point

A terminating decimal ends after a definite number of digits, for example 0.39 and 1.042. You can write any terminating decimal as a fraction.

Q11 hint Use the number of decimal places to decide whether the denominator should be 100 or 1000.

Explore

Reflect

Pi 3, Section 4.1

Unit 4 Fractions, decimals and percentages 80

Master

Check P93

Strengthen P95

Extend P99

Test P103

4.2 Recurring decimals

You will learn to: ? Write recurring decimals as fractions.

Why learn this? Lots of calculations in real life have an answer that is a recurring decimal.

Fluency Round each number to 2 decimal places. ? 1.454 ? 6.087 ? 3.3642

? 10.4985

Explore How can you tell if a fraction will be a terminating or a non-terminating decimal?

CONFIDENCE

Warm up

Exercise 4.2

1 Work out a 726 ? 6

b 7125 ? 5

c 345 ? 4

d 842 ? 8

2 a i Jarred thinks of a number. Half of it is 240. What is _14of his number?

ii What number did Jarred think of? b i Sophie thinks of a number. _16of it is 12. What is _13of her number?

ii What number did Sophie think of? c i Amee thinks of a number. _110_of it is 250. What is _15of her number?

ii What number did Amee think of?

3 Reasoning a Use your calculator to match each fraction to its equivalent decimal.

Q2b hint

12 11 66

1 3

3

5

7

5

7

2

5

8

12

9

11

3

0.625

0.5

0.6

0.6

0.63 0.583

b Use the decimals to order the fractions in part a, smallest to largest.

Worked example

Write _19as a decimal.

0__

9 doesn't go into 1 so write a 0 in the units column.

9)1.1000

0_ .1111..._

There are now 10 tenths.

9 )1.1010 1010

_91_= 0.1

9 goes into 10 once with remainder 1. There are now 10 hundredths. Continue like this and the decimal recurs.

Key point

A recurring decimal contains a digit, or sequence of digits, which repeats itself forever. A dot over the digit shows it recurs. For example, 0.11111.... = 0.1

4 Write each fraction as a decimal.

a _18 e _16

b _112_ f _56

c _152_ g _29

81

d _78 h _89

5 Problem-solving / Reasoning Nira works out 12 ? 18 and gets

an answer of 0.6666667 on her calculator.

What is the equivalent fraction? Discussion What has the calculator done?

6 Use a written method to work out each division. Write your answers

as recurring decimals using dot notation.

a 823 ? 3

b 375 ? 9

c 37564 ? 3

d 6385 ? 9

e 97 ? 12

f 1756 ? 12

Investigation

Reasoning / Problem-solving

Caroline says, `Some fractions can be written as terminating decimals but some are recurring decimals'.

1 Write each fraction as a decimal. _21_13_14_15_61_17_18_19_110__111_ _112_

2 Sort them into terminating decimals and recurring decimals.

3 Jack says, `If the denominator of a fraction is even, it will be a recurring decimal'.

Find an example to show that Jack is wrong.

4 Which denominators give terminating decimals?

Discussion Caroline says, `I think it's to do with the fact that 2 ? 5 = 10.' What do you think she means?

5 Investigate what happens if the numerator is not 1.

7 a Which bag has the greater proportion of red counters? b What is the proportion of blue counters in each bag?

A

B

8 The pie chart shows the first International School Languages

language of people working in a summer school. Write the proportion of each language as a fraction and as a percentage.

120 150

60 30

Chinese English French Russian

Q8 hint

How many degrees represent the whole pie chart?

9 Explore How can you tell if a fraction will be a terminating or a nonterminating decimal? Is it easier to explore this question now you have completed the lesson? What further information do you need to be able to answer this?

10 Reflect A shorthand way of writing a recurring decimal is to use a dot, or dots, over the numbers that repeat. For example, 0.2222... is written as 0.2 Write down five other short forms that you use in maths. Do you think these short forms are useful or not?

Reflect Explore

Pi 3, Section 4.2

Unit 4 Fractions, decimals and percentages 82

Master

Check P93

Strengthen P95

Extend P99

Test P103

4.3 Adding and subtracting fractions

You will learn to: ? Add and subtract fractions ? Add and subtract mixed numbers.

CONFIDENCE

Why learn this? Statisticians add and subtract fractions to work out the probably of different events happening (or not happening).

Fluency

What is ? _12+ _41 ? _12? _14 ? _35? _13_0 ? _38+ _12

Explore How many years ago did people start writing fractions?

Warm up

Exercise 4.3

1 Write each improper fraction as a mixed number.

a _15_4

b _28_0

c _172_

d _233_

2 Write each mixed number as an improper fraction.

a 2_14

b 1_23

c 5_38

c 4_130_

3 Work out a _152_+ _13

b _56? _13

c _34? _38

d _31+ _25

4 Work out each calculation. Give your answer as

a mixed number where necessary.

a _58+ _38+ _18

b _172_+ _11_2+ _1112_ c _59? _29+ _79

d _12? _13+ _14

e _45+ _130_? _34

f _23? _49+ _16

5 Work out a _12+ _13+ _14

b _13+ _14+ _15

c _14+ _15+ _16

6 Problem-solving Work out the missing number.

a _45+ _34+ u = 2 b _65+ _35? u = 1

7 Problem-solving In a clothes shop, _15of clothes are suits, _23are trousers and the rest are tops. What fraction of the clothes are tops?

Q4a hint

5

3

1

8

8

8

111111111 888888888

8 8

5

1

Worked example

a Work out 3_56+ 1_34

Add the whole number parts and

3_65+ 1_34= (3 + 1) + (_65+ _34)

add the fraction parts separately.

= 4 + (_1102_+ _192_)

Convert the fractions to equivalent

= 4 + _1192_

fractions with a common denominator.

= 4 + 1_172_

Change the improper fraction to a mixed number

= 5_172_

so you can add the whole number parts.

83

b Work out 2_12? 1_56 2_21? 1_65= (2 ? 1) + (_21? _65) = 1 + (_63? _65) = 1 + (?_62) = 1 ? _31 = _23

Subtract the whole number parts and the fraction parts separately.

8 Work out a 1_15+ 2_13_0 d 3_45+ 1_13 g 1_11_12+ 1_56

9 Work out a 2_17_0? 1_25 d 3_45? 1_13 g 4_11_12? 2_65

b 2_14+ 3_25 e 4_23+ 2_56 h 3_34+ 1_45

b 2_34? 1_38 e 4_23? 2_16 h 3_34? 1_35

c 2_13+ 2_25 f 2_170_+ 3_45 i 2_59+ 3_65

c 4_56? 3_23 f 3_170_? 3_35 i 3_56? 2_59

10 Yazdi uses 2_14litres of white paint and 2_35litres of blue paint. How many litres of paint did he use in total? Give your answer as a mixed number.

11 Peter walked 4_56km, Brenda walked 3_45km. How much further did Peter walk?

12 A farmer needs 3_25metres of netting for his chickens. He already has 1_12_10metres. How much more does he need?

13 A box of apples weighs 3_57kg and a box of pears weighs 2_35kg. How much do they weigh altogether?

InvestigationReasoning

1 Write down six fractions that are between

a 0 and 1

b 0 and _12

c _12and 1

0

1

1

2

2 Write down two fractions that are between

a _14and _34

b _28and _38

c _170_and _180_

d _35and _45

Discussion In how many different ways can you answer the questions in this investigation?

14 Explore How many years ago did people start writing fractions? What information do you need to start answering this question?

15 Reflect Choose one of the parts of Q9 that you felt confident in answering. How would you explain the method you used to a classmate who had missed this lesson?

Reflect Explore

Pi 3, Section 4.3

Unit 4 Fractions, decimals and percentages 84

Master

Check P93

Strengthen P95

Extend P99

Test P103

4.4 Multiplying fractions

You will learn to: ? Use strategies for multiplying fractions.

CONFIDENCE

Why learn this You multiply fractions when working out the distance you can travel with half a tank of fuel.

Fluency

What is ? _37of 21 ? _25of 15 ? _29of 36 ? _56of 72

Explore How many times can you cut a piece of paper in half?

Warm up

Exercise 4.4

1 Work out a 3 ? _14 d 4 ? _29

b 2 ? _37 e _56? 4

c _25? 2 f _13? 6

2 Write each improper fraction as a mixed number in its simplest form.

a _31_20

b _24_2

c _35_5

d _36_4

3 What is the inverse operation of a multiplying by 10 b dividing by 8?

4 Simone says, `I'm thinking of a number. I multiply it by 8 and then divide the answer by 8, and get 5.' What number was she thinking of?

Q3 Literacy hint

An inverse operation is the opposite operation.

5 Reasoning Work out

a 8?1?4

b 8 ? _14

c 8?4?1

d Explain why the answer is the same in parts a, b and c.

6 Work out a 6 ? _13 c _23? 3

b 5 ? _45 d _57? 7

7 Work out a _25? 250

b _23? 360

c _23of 360

8 Real / Problem-solving A car has 45 litres of fuel in the tank. The driver uses _35of the fuel. How many litres of fuel are left?

9 Reasoning Which of these products will have an answer less than 1?

a 5 ? _23

b _19? 7

c 2 ? _14_5

d _34? 6

Q6b hint Work out one-fifth of 5, then multiply it by 4.

Q7a hint Work out one-fifth of 250, then multiply by 2.

85

Worked example

Work out _14? _32

_41? _23= _41 _??_2_3 = _122_ = _61

2 3

1 4

of

2 3

_14of _23= _61

Key point

To multiply two fractions, multiply their numerators and multiply their denominators.

10 Work out each calculation. Simplify your answer where needed.

a _23? _34= _23_??__43= _uu_= _uu_

b _14? _45

c _23? _25

d _34? _34

e _34? _161_

f _49? _37

g _35? _172_

11 Work out

a _12? _14

b _12? _13

c _12? _15

d _12? _12

Discussion What happens to the denominator when you multiply a

fraction by _12?

12 Work out

a (_14)2 = _41? _41= d (_16) 2

b (_13)2 e (_23) 2

c (_15)2 f (_38)2

Worked example

Work out _38? _29

_83? _92= _83_??__92_ = _82_?_?_93_ = _28? _93 = _41? _31 = _112_

Rewrite the calculation with a fraction that can be simplified. 2 is a factor of 8 and 3 is a factor of 9.

Simplify the fractions before multiplying.

Discussion How could you work out the multiplication using fewer steps?

13 Work out a _58? _35 d _49? _38

b _34? _27 e _195_? _56

c _34? _185_ c _56? _230_

14 Holly drank _23of a _21litre bottle of juice. How much did she drink?

15 Explore How many times can you cut a piece of paper in half? Choose some sensible numbers to help you explore this situation. Then use what you've learned in this lesson to help you answer the question.

16 Reflect Look again at Q8. Write down the steps you took to work out the answer. Work out the answer again using a different method. Did you get the same answer? If not, check your working.

Key point

Sometimes you can rearrange fractions so they can be simplified before multiplying.

Reflect Explore

Pi 3, Section 4.4

Unit 4 Fractions, decimals and percentages 86

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download