ratio and proportion

Ratio and proportion

Learn 1 Finding and simplifying ratios

Learn 2 Using ratios to find quantities

Learn 3 Ratio and proportion

Learn 4 Calculating proportional changes

Learn 5 Direct and inverse proportion

57

Learn 1

Solve simple ratio and proportion problems,such as finding the ratio of teachers tostudents in a school

Learn 2

Solve more complex ratio and proportionproblems, such as sharing out moneybetween two groups in the ratio of theirnumbers

Learn 3

Solve ratio and proportion problems using theunitary method

Learn 4

Calculate proportional changes using amultiplier

Learn 5

Solve direct and inverse proportion problems

Interpret the graphs of direct and inverseproportion relationships

S P E C I F I C A T I O N R E F E R E N C E

Use ratio notation, including reduction to its simplest form and its various links to fraction notation H2.2f

Divide a quantity in a given ratio H2.3f

Solve word problems about ratio and proportion, including using informal strategies and the unitary method of solution (F2.3n)

Represent repeated proportional change using a multiplier raised to a power H2.3k

Calculate an unknown quantity from quantities that vary in direct or inverse proportion H2.3l

Set up and use equations to solve word and other problems involving direct and inverse proportion and relate algebraic solutions of problems involving direct and inverse proportion to graphicalrepresentations of the equations H2.5h

OB

JE

CT

IV

ES

D

C

C

B

A

8V

OC

AB

UL

AR

Y Definitions for these words can be found in the Students’ Book

Constant

Ratio

Unitary ratio

Proportion

Unitary method

Direct proportion

Inverse proportion

AQA Mathematics for GCSE

58

1 a On a flashcard or on a whiteboard, interactive board or OHP, show the Learn 1

students a fraction such as one from the list below. Ask the students to show you the simplest version of the fraction as quickly as they can. Check for omissions, errors, incomplete answers, etc. and deal with difficulties before giving the next fraction to simplify. Make the next fraction harder, easier or the same level of difficulty according to responses to the first one.

b On a flashcard or on a whiteboard, interactive board or OHP, show the students a proportion such as one from the list below. Ask the students to show you three fractions equivalent to the proportion given. Check for omissions, errors, incomplete answers, etc. and deal with difficulties before giving the next proportion. Choose the next proportion according to responses to the first.

2 On a flashcard or on a whiteboard, interactive board or OHP, show the Learn 1

students a number such as one from the list below. Ask the students to list all the factors of the number shown. Check for omissions, errors, incomplete answers, etc. and deal with difficulties before giving the next number. Choose the next number according to responses to the first. Discuss divisibility: how can you tell when a number is divisible by 2, 10, 5, 3, ...?Examples of possible numbers: 36, 70, 80, 108, 144, 1000

3 On a flashcard or on a whiteboard, interactive board or OHP, show the Learn 2

students a calculation such as one from the list below. Ask the students to work out the required amount. Check for problems, deal with difficulties and discuss different ways of working out the answers before giving the next calculation.Examples of possible calculations:

half of 150, of 150, of 24, of 120, of 4500, of 250

4 Give students an example of quantities in proportion, such as the number of Learn 3

hours worked and the amount earned, and ask them to suggest other examples. A good range of correct and incorrect examples should emerge and can be discussed to provide a rich background to the work of Learn 3.

5 Give each pair or small group of students in the class a word or phrase from Learn 3

the list below and ask them to develop an explanation of the word or phrase, using examples, pictures, etc. to illustrate their explanation. (This could be set as homework.) Ask a group to give their word and its explanation. Ask other students to comment. Discuss the issues raised. Continue for as many words as you have time for.Examples of possible words: fraction, ratio, unitary ratio, simplifying ratios, factor, multiple, unitary method, teacher : student ratio, map scales, male : female ratio, two pairs of numbers in the same ratio, simplifying fractions, profit : cost price ratio, dividing a number in given ratio, proportion, best buy.

17100

23

35

512

23

12, 23, 1

10, 310, 59, 50%, 30%

612, 50

100, 501000, 25

650, 108324, 25

35, 2.53.5

ST

AR

TE

R

Examples: a A school has 1400 students and 70 teachers. Write down the student to teacher ratio and express it in its simplest form.

The student to teacher ratio is 1400 to 70 or 1400 : 70.

Make sure students appreciate that the order is important. 1400 : 70 is not the same as 70 : 1400.

In its simplest form:1400 : 70

# 20 : 1

Students may be able to suggest ways in which this could be simplified in a similar way to simplifying fractions. They should be able to spot that 10 is a common factor and may be able to see that 70 is a factor. No matter how many steps are used to simplify the ratio, the final result, 1 : 20, is the same.

Emphasise that the numbers obtained at each stage are in the same proportion as they were at the beginning.

b A dealer in second-hand books buys a paperback for £1.20 and sells it for £1.50What is the ratio of profit : cost price in its simplest form?

The ratio of profit to cost price# 30p to £1.20# 30p to 120por 30 : 120

In its simplest form:30 : 120

# 1 : 4

Students need to understand that the two amounts of money must be expressed either both in pence or both in pounds (they should try them both and see that the result is the same).


59

6 Give students incomplete statements, such as those below, and ask them to Learn 4

write the missing percentages or multipliers. Discuss differences and errors before going on to the next example.

a Increasing by 10% is the same as finding ... %.

b Decreasing by 5% is the same as finding ... %.

c Increasing by 1% is the same as finding ... %.

d Decreasing by 20% is the same as multiplying by ...

e Increasing by 10% is the same as multiplying by ...

f Decreasing by 5% is the same as multiplying by ...

g Increasing by 1% is the same as multiplying by ...

h Decreasing by 20% is the same as multiplying by ...

7 Put numbers with indices, such as those below, on the board for students to Learn 4

gain some quick practice of using the index functions on a calculator.

3�5, 2�10, 4�5, 5�4, 3�0.5, 1.05�3, 1.5�6

8 Ask students to discuss and make a list of all the mathematical relationships Learn 5

and formulae they can think of. Choose some from the list and discuss them with the group, for example, for A # πr�2, A increases as r increases, but it is not a linear relationship; the graph is a parabola; A is directly proportional to r�2, etc.

Learn 1 Finding and simplifying ratios

The ratio of 30p to 120p is the same as the ratio 30 to 120, which simplifies to 1 : 4.

Discuss with the students the different ways in which this could be expressed, such as:�–� The profit is one quarter of the cost price.�–� The profit is one fifth of the selling price and the cost price is four fifths

of the selling price.�–� The profit is 0.25 of the cost price and 0.2 of the selling price.�–� The profit is 25% of the cost price and 20% of the selling price.

You may also wish to discuss the fact that some ratios can have units, such as 1 litre : 7.5 km to express a rate of fuel consumption.

It is important that students are helped to make sense of the ideas of ratio and how it is linked with fractions, decimals and percentages.

Most ratio questions can be solved in a number of ways; encourage students to explore these methods, explain them to others, and discuss their advantages and disadvantages.


60

Common errors:

● In simplifying ratios, students may make the same errors that they make withsimplifying fractions, such as ‘crossing out numbers that are the same’ andnot spotting common factors to use. Students could usefully be taught howto use the fraction button on the calculator to simplify fractions and ratios.

● Students may fail to deal appropriately with units and should be allowed toexplore the inconsistencies that this leads to.

● Students often do not understand that ratios represent multiplicativerelationships, tending to add the same amount to both parts of a ratioinstead of multiplying appropriately. Questions 2d and 8b in Apply 1 drawtheir attention to this potential problem.

Apply 1 answers

1 a 1 : 2 e 1 : 6 i 2 : 3 m 3 : 8

b 1 : 3 f 1 : 7 j 1 : 4 n 1 : 3

c 1 : 4 g 1 : 3 k 3 : 2 o 1 : 4

d 1 : 5 h 3 : 4 l 3 : 5 p 1 : 8

2 a For example, 2 : 4, 3 : 6, 4 : 8, 5 : 10 etc.

b For example, 2 : 8, 3 : 12, 4 : 16, 5 : 20 etc.

c The first number can be anything you like.The second number must be four times the firstnumber.

d Pippa has added 3 to each number in the ratioeach time instead of multiplying both numbers bythe same thing. The pairs should be 6 and 9, 9 and13.5, 12 and 18.

3 a 1 : 2; 2 : 1 b 100 g c 60 g d A half

4 1 : 10

5 a 300 m� milk, 50 g grated cheese, 20 g flour, 20 g butter, seasoning

b Multiply the amounts for 2 people by 5 ormultiply the amounts for 4 people by

6 a

b i 1000 ii 100

c School 4

7 a

b The profit is of the selling price.The cost price is of the selling price.

c You could change them to decimals orpercentages or express them as unitary ratios.

910

110

212

School 1School 2School 3School 4School 5

School

1 : 201 : 161 : 171 : 151 : 21

Ratio

Litre of petrolCarCalculatorBookMagazineSandwich

Item

1 : 171 : 92 : 31 : 4

3 : 192 : 3

Profit:cost price ratio


Homework 11 a 1 : 2 e 1 : 6 i 5 : 7 m 3 : 2

b 1 : 3 f 1 : 7 j 3 : 4 n 1: 3

c 1 : 4 g 1 : 2 k 9 : 10 o 1 : 3

d 1 : 5 h 1 : 4 l 5 : 7 p 3 : 8

2 a For example, 6 : 2, 9 : 3, 12 : 4

b For example, 10 : 2, 15 : 3, 20 : 4

c Any pair of numbers where the second is onefifth of the first.

d No, Harry is wrong. He has subtracted 2 fromeach of the numbers so they are no longer in thesame ratio. The number pairs should be 16 and20, 14 and 17.5, 12 and 15.

3 2 : 1

4 3 : 1

5 a Self-raising flour 480 gramsSalt 1 teaspoonSultanas 150 gramsButter 80 gramsCaster sugar 50 gramsEggs Two largeMilk 40 m�

b 600 grams

6 a 1 : 6

b 5 nurses

7 a

b Anything where the second amount is two and ahalf times the first

c No; the ratio is unchanged if all the quantitiesare measured in different units.

8 a 4.7 m b 1.70

9 a 27 b 8 c 3 d 10, 45

10 Students’ own questions and answers.

11 a

b i 600 ii 100

answers

5

6

4

2

10

15

2

2

3

3

5

4

6

1

Amount of red

paint (litres)

Amount of yellow

paint (litres)

Yes

No

No

No

Yes

Yes

Yes

Hot orange

12

School 1School 2School 3School 4School 5

School

1 : 152 : 351 : 215 : 911 : 16

Ratio

8 a i 8ii No; if both quantities are doubled, you can

just divide both numbers by two and you havethe original ratio.

b i 4 men and 6 womenii The ratio becomes 6 : 8 or 3 : 4. The number of

men is 0.75 of the number of women; beforethat, the number of men was 0.67 of thenumber of women, so the ratio has increased.

9 a 8 b 9 c 1.5 d 15 and 20

10 Students’ own questions and answers.

11 a

The ratio is clearly increasing, since the numberof green squares has overtaken the number ofyellow squares.

b Many possible examples.

1

2

3

4

1 92549

Number of green

8162432

Number of yellow

1: 8 9 : 1625 : 2449 : 32

Ratio

Explore

● 2 cm to 1 km is 1 : 50 000 in ratio form.● 3 cm represents 3 km.

61

Example: In a school, there are 150 students in Year 10, with boys and girls in the ratio 7 : 8.How many boys and how many girls are there in Year 10?

You need to divide 150 students in the ratio 7 : 8.The total number of ‘parts’ is 7 ! 8 # 15Each ‘part’ is worth So in 150 students there are 7 " 10 boys and 8 " 10 girls.

Students will have ideas about how this can be done – explore and build on these so that students can gain understanding and confidence.

The ratio 7 : 8 can be interpreted as ‘for every 15 students there are 7 boys and 8 girls’ and thus lead to ‘for 150 students there are 7 " 10 boys and 8 " 10 girls’.

The ‘standard’ method is to say that there are 15 parts altogether, you find one part by dividing 150 by 15, and then you multiply that by 7 and by 8 to find the numbers of boys and girls.

Another method is to interpret the ratio as fractions: the number of boys is of the total of 150 and the number of girls is of 150. Then all you have to do is work out the fractions.

Some students may find it useful to complete a table such as this, building up until the total reaches 150.

There are number patterns to observe and discuss in the table; doing so will support students in developing skills in proportional reasoning.

815

715

15015 = 10


Learn 2 Using ratios to find quantities

8162432

7142128

Number of

boys

Number of

girls

153045 60

Total number

of students

Common errors:

● If students are doing these questions byfractional methods, they need to beable to work out fractions of numbersand quantities, so common errors suchas not being sure what to multiply anddivide by may occur.

● More generally, students may not easilyunderstand what dividing quantitiesaccording to a given ratio is all about, sothey will need the opportunity toexplore this idea in familiar situations.

Apply 2 answers

1 a 50, 100 d 2 litres, 4 litres

b 100, 200 e 50p, £1

c £1.50, £3 f 0.5 litre, 1 litre

2 a 60, 90 d 2.4 litres, 3.6 litres

b 120, 180 e 60p, 90p

c £1.80, £2.70 f 0.6 litre, 0.9 litre

3 a 45, 105 d 1.8 litres, 4.2 litres

b 90, 210 e 45p, £1.05

c £1.35, £3.15 f 0.45 litre, 1.05 litres

4 a 15, 45, 90 d 0.6 litre, 1.8 litres, 3.6 litres

b 30, 90, 180 e 15p, 45p, 90p

c 45p, £1.35, £2.70 f 0.15 litre, 0.45 litre, 0.9 litre

5 a 100 g b 2 ounces c 180 g

6 a

b See last column of table.

c School E; it is the only school where there aremore boys than girls.

7 a

b Salmon, yoghurt, milk

c 400 g

d Taco chips, bread

375400800612602

School ASchool BSchool CSchool DSchool E

Number of

boys

375500

1000714582

Number of

girls

1 : 11 : 1.251 : 1.251 : 1.171 : 0.967

Ratio in form

1:nSchool

Chicken sandwichGrilled salmonYoghurt (whole milk)Taco chipsBreadMilk

Food

1 : 1 : 10 : 1 : 11 : 2 : 1

10 : 4 : 17 : 2 : 12 : 3 : 2

Carbohydrate : fat : protein

50 g75 g75 g40 g30 g65 g

Amount of

fat in 150 g

62

Example: Jean earns £52 for working 8 hours. How much does she earn for working 11 hours at the same rate of pay?

For working 8 hours Jean earns £52

So for working 1 hour she earns

For working 11 hours Jean earns

Students may suggest a variety of methods for solving this, including building up to 11 hours by adding up the pay for 8 hours and 2 hours (obtained by halving and halving again) and 1 hour (obtained by halving again). This build-up method is easily understood and quite versatile, but the majority of GCSE students should be able to understand the unitary method and should be encouraged to use it, as it can be applied to so many situations.

The calculations can be done in stages or at the end.

11 ×£528

= £71.50

£528


Homework 21 a 40, 160 d 1.6 litres, 6.4 litres

b 70, 280 e 50p, £2

c £1.30, £5.20 f 0.7 litre, 2.8 litres

2 a 80, 120 d 3.2 litres, 4.8 litres

b 140, 210 e £1, £1.50

c £2.60, £3.90 f 1.4 litres, 2.1 litres

3 a 140, 60 d 5.6 litres, 2.4 litres

b 245, 105 e £1.75, 75p

c £4.55, £1.95 f 2.45 litres, 1.05 litres

4 a 20, 80, 100

b 35, 140, 175

c 65p, £2.60, £3.25

d 0.8 litres, 3.2 litres, 4 litres

e 25p, £1, £1.25

f 0.35 litres, 1.4 litres, 1.75 litres

5 a 75 m� b 0.1 litre c 80 m�

6 a

b See last column of table.

c School B – the ratio in the form 1 : n makes thisclear.

7 a

b 286 g

c 1.200 kg

8 £5240, £4760 (nearest £)

9 a 500 g copper, 250 g gold, 250 g silver

b 580 g

c 0.5 g

answers

School ASchool BSchool CSchool DSchool E

1000400744280500

Number of

boys

100014001116245555

Number of

girls

1 : 11 : 3.51 : 1.51 : 0.8751 : 1.11

1 : nSchool

Fudge biscuitsStrawberriesScrambled eggsChilli con carneItalian sausage

Food

8.3g7.7g40g35g40g

Amount of protein in 100g

Learn 3 Ratio and proportion

8 Jamil gets £2833.24

Jane gets £4516.76

9 a 950 g copper, 40 g tin, 10 g zinc

b 9.5 kg copper, 400 g tin, 100 g zinc

c 475 g copper, 20 g tin, 5 g zinc

d 0.06 g

63


64

Common errors:

● Using adding�subtracting instead of multiplying�dividing (for example,thinking that 3 : 5 # 4 : 6).

● Keeping the total of the quantities constant, instead of keeping the ratioconstant.

● Not being sure what to multiply or divide by because of a lack ofunderstanding of how the quantities will vary – what will get bigger andwhat will get smaller?

Apply 3 answers

1 The ratio is £4.60 : £27.60 : £69 # 460 : 2760 : 6900

# 46 : 276 : 690 # 23 : 138 : 345 # 1 : 6 : 15.

2 a £37.50

b

c i Working for 0 hours earns £0 so the line goesthrough (0, 0); the amount earned goes up bythe same amount for each extra hour worked,so the line is straight.

ii The gradient represents the hourly rate of pay.

iii Vertical line drawn from 5 hours up to graph,then horizontally to y-axis to read off amountearned as £18.75

3 a 75g b 4 days c 437.5 g

4 Travel size: 0.533 g per penny; large size: 0.556 g perpenny. So large size is better value.(Or travel size costs 1.875p per gram; large size costs1.8p per gram. So large size is better value.)

5 55 g size: 0.917 g per penny; 100 g size: 0.952 g perpenny. So 100 g size is better value.(Or 55 g size costs 1.09p per gram; 100 g size costs1.05p per gram. So 100g size is better value.)

6 a 112 miles

b 21 miles

c 2.143 hours – that is, 2 hours 9 minutes to thenearest minute

7 a 20 b £20

8 a 9 b 16 c 44, 40

9 a 200 pounds

b 60 kg

c

d 6 : 1 : 15

Number of hours worked

Money earned (£)

27.50

00

415

622.50

830

1037.50


Mon

ey e

arne

d (£

)

Sajid’s pay

21 4 6 8753 9 10

10

20

30

5

15

25

3540

O

Explore

There are many possible explorations of Fibonaccisequences; this one focuses on the ratio of successiveterms, which tends to the golden ratio, approximately1 : 1.618Students should be able to get an informal sense ofwhat is happening whilst gaining some useful ratiocalculation practice. The first few ratios are 1 : 1, 1 : 2,1 : 1.5, 1 : 1.666 ..., 1 : 1.6, 1 : 1.625

Earth/Jupiter

Earth/Moon

Weight on Earth

Weight onother planet

Examples: The number of cells in a developing organism is increasing by 5% every hour. At the start, the number of cells is 500.

a What is the number of cells after 1 hour?

b What is the number of cells after 2 hours?

c What is the number of cells after 3 hours?

d What is the number of cells after 24 hours?

a Number of cells after 1 hour is 500 " 1.05 # 525

Students should be able to find the number of cells after 1 hour, at the lowest level by working out 5% and adding it on. Discussion should reveal the more sophisticated and useful one-step method, multiplying by 1.05, but students may need to get to this point by first working out 105% by finding 1% then finding 105% – that is, multiplying by , which is 1.05

b After 2 hours, the number of cells is 500 " 1.05 " 1.05 # 500 " 1.05�2 # 551.25

It is easy to find the number of cells after 2 hours and 3 hours by repeatedly calculating 5% and adding it on, but for calculating the number after 24 hours the more sophisticated method is needed, so students should be helped to see that the number after 2 hours is 500 " 1.05�2 and the number after 3 hours is 500 " 1.05�3.

105100


65

Homework 31 £6

2 a £140.80

b

c i Working no hours earns no money.

ii The hourly rate of pay.

iii Horizontal line from 100 on y-axis to graph,then down to x-axis at 5.7 hours.

3 a 400 g b 3 c 2 people

4 Small size: 1.91p per gram; large size: 1.93p pergram. So small size is better value.

5 a 192 miles

b 36 miles

c 8 hours 20 minutes

6 a Box of 22 chocolates: 81.6p per chocolateBox of 44 chocolates: 63.5p per chocolateBox of 66 chocolates: 54.5p per chocolate. Sothe box of 66 chocolates is the best value formoney.

b The delivery charge and the packaging areprobably relatively more expensive on thesmaller boxes.

7 a 8 b £8

8 a 9 b 20 c 54 and 45

9 a 40 pounds

b 18.75 kg

c

d 15 : 6 : 16

answers


Money earned (£)

235.20

00

470.40

6105.60

8140.80

10176

Learn 4 Calculating proportional changes

Weight on Earth

Weight on Mercurycompared with Earth

Wei

ght o

n M

ercu

ry

5010 20 30 40

10

20

30

O

c After 3 hours, the number of cells is 500 " 1.05 " 1.05 " 1.05 # 500 " 1.05�3 # 578.8125

d After 24 hours, the number of cells is 500 " 1.05�24 Q 1610

After working slowly through the early parts of this question and discussing different ways of finding the answers, students should then feel reasonably confident about finding the number of cells after 24 hours.

Calculating repeated proportional changes using a multiplier is a powerful method with many applications in real life. There are glimpses of advanced work for those students who may continue mathematics beyond GCSE.

Some students will need to gain familiarity with the ideas by working out the total after one time period, then using that to work out the next total and so on before being able to see that they are just multiplying by the same thing each time.

Working on this section should strengthen students’ understanding of percentages and help them when doing reverse percentage problems; it should also develop their understanding of, and familiarity with, indices.

The section requires competent use of various calculator functions.


Common errors:

● Not understanding the difference between increases�decreases that aredirectly proportional to the time and those that are a fixed proportionof the current amount.

● Not ‘seeing’ that multiplying by, for example, 1.1 increases somethingby one tenth (or 10%). Starter 6 should help to provide fluency withthese ideas.

● Lack of understanding of indices.

Apply 4 answers

1 a 1460 c 1070

b 1270 d 9850

2 a £520 c £608.33

b £540.80 d £108.33

3 a i 8 million

ii 32 million

iii 1.414 million

b Between 3 hours 19 minutes and 3 hours 20minutes

4 a £1537.50 b Yes, £20 365.35

5 a 3% c 4 years

b £530.45

6 a £80 525.50 c £x " 1.1�5

b £47 619.78

7 a £832.32

b 32p

c £65.95; £65.28 Difference is 67p

d With simple interest, the interest paid justdepends on the initial amount. So it makes nooverall difference how often it is paid.

8 £2296 or £2300

9 a 3722 b 367

c The number each year is 99% of the number inthe year before, not 99% of the initial number.

10 a 41 400 b 1.6 million square kilometres

11 a 21% approximately

b £3600

12 a 7.4% approximately

b The percentage rate obtained by that method isthe percentage of the original price not thepercentage of the price that year. The ratecalculated by that method is 8.6% which wouldgive a price of nearly £74 000.

Explore

This activity gives students an opportunity toinvestigate a situation being pushed to its limit.Students may wish to put this on a spreadsheet anddraw graphs to show what happens as the frequencyof adding interest increases. You may or may notwant to talk about the number e at this stage.

66

Examples: The area of a rectangle is constant. (This means that the length is inversely proportional to the width.) When the width is 9 cm, the length is 16 cm.

a Find an equation expressing the width of the rectangle in terms of its length.

b Use your equation to find the length of the rectangle when the width is 24 cm.

c Sketch a graph of length against width.

a The fact that the length of the rectangle is inversely proportional to its width can be expressed as:

where l cm is the length, w cm is the width and k is a constant.

Students should be helped to gain a clear understanding of the fact that, if two quantities are inversely proportional, their product is constant.

Not all students will be confident that is the same as and lw # k.

There are useful parallels in trigonometrical ratios, the distance�speed�time formula, etc.

Since l is 16 when h is 9, , so k # 16 " 9 # 144

So an equation connecting l and w is

The various other possible forms of this, and the links with the students’ common sense methods of solving this problem should be discussed.

It should now be a relatively simple task to find l when w is 24.

l = 144 ×1w

16 = k ×19

l =kw

l = k ×1w

l��∝��1w

��or��l = k ×1w


67

Homework 41 a £765.90 c £169.65

b £909.65

2 a £21 762.80b Her salary would be £3.46 per year more – it

makes very little difference as the annual rate ofincrease is so small.

3 a 3.4% c 12 years

b £4018.66

4 a 550 c 520

b 540 d 1610

5 a i 100 g ii 3200 g iii 35.4 g

b 1.41 ( )

c 12.5 g

d 25 " 2�t

6 a £85 995 b £95 238c Not very – there will be many variations over

time so it is unlikely to stay at the same rate forlong. Also, different houses will rise by differentpercentages.

7 a £105.06 c 21p to nearest penny

b £105

8 a £5200 b £3328

9 a 2.44 g b 2.28%

10

√2

answers

Time5700 years

Mass

Learn 5 Direct and inverse proportion

b When w # 16,

So, when the width is 24 cm, the length is 6 cm.

c The graph of length against width is of the form:

The sketching of this graph will need exploration; students may want to start by plotting some pairs of length and width values – and this should help them to realise the symmetry of the graph and its asymptotic properties.

This topic provides an opportunity for students to consider the mathematical relationships they have encountered in their studies so far and classify them into proportional and non-proportional relationships and, if proportional, into types of proportionality.

Real-life quantities are often related in various ways – for example, the perimeter of a square is always four times the length of one side.

Consideration of the graphs of the various relationships should deepen understanding.

l = 144 ×124

= 6


68

Width (cm)

Len

gth

(cm

)

Common errors:

● It is common for students to say that, if one quantity increases as anotherincreases, the quantities are proportional.

● Students may make algebraic errors when working with equations expressing

proportional relationships, for example, a frequent error in solving

would be to say that .k = 24

144 =k

6

Apply 5 answers

1 a P # 5.5h

i £44 ii 20 hours

b Straight line with positive gradient passingthrough the origin.

c The number of pounds Emma is paid per hour.

2

a 4.8 b 36 c

3 a C # 3.14d b 37.68 cm c 31.8 cm

4 175 seconds

5

6 62.5 g

7 a P # 3.75n

b £37.50

c Straight-line graph with positive gradient passingthrough the origin.

d It’s the same, but presented in a different way. It’s easier in Apply 3!

8 17.5 days

9 a 6 c 5 e 1.91 (to 2 d.p.)

b 6 d 5

10 It is halved. For example, in a rectangle with fixedarea, if the length is doubled, the width is halved.

P =E

1.62

Q =P 2

0.36

P = 0.6√Q


Homework 51 a 4 b 16

2 a y # kx�2 b k # 1

c i 8 ii 7.75 (to 3 s.f.)

3 a C # 0.16d

b i £16.80 ii 625 miles

4 a 3 hours 23 minutes

b 54 m.p.h.

5 a

b 6 days c 15

d

6 a False c False

b True d True

7 a

b 5.71 litres

c The number of litres is directly proportional tothe number of pints.

8 a Circumference of circle and diameter – directproportion

b Time taken to complete a task and number ofpeople available to do it – inverse proportion

c Area of square and square of its perimeter –direct proportion

d Fahrenheit temperatures and correspondingCelsius temperatures – neither (graph does notgo through origin)

e Number of weeks and the correspondingnumber of days – direct proportion

f Number of glasses of water that can be pouredfrom a bottle and the amount of water pouredinto each glass – inverse proportion

L =P

1.75

d =30n

answers

11 a Neither

b Inversely proportional

c Directly proportional

d Inversely proportional

e Directly proportional

f Neither (area is proportional to square of length)

12

13

14 i y is proportional to x: Graph D

ii y is proportional to x�2: Graph B

iii y is proportional to x�3: Graph E

iv y is inversely proportional to x: Graph A

v y is proportional to the square root of x: Graph C

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5n

d 10

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2

4

6

1

3

5

7

O P

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69

y is proportional to the cube root of x

y is inversely proportional to x cubed

√xk

y � kx�3

y � x13

y � kx3

y is inversely proportional to x

y is proportional to x3

y is inversely proportional to the square root of x

y � kx

1 Discuss the following question, exploring the many possible ways in which it Learn 1

could be solved:One school has 800 students and 47 teachers.Another school has 1600 students and 95 teachers.Which school has the higher ratio of students to teachers?

2 Take two pieces of A4 paper (or a larger A size if available). Two different Learn 2

colours provide opportunities for an attractive display. Cut one piece in half parallel to the shorter sides and turn one of the halves round to show that its sides are in the same ratio as the original piece. Check by measurement also. Repeat the halving, showing that the ratio is always the same. Encourage students to discuss what they see, explain the relationships between the lengths and the areas of the different sizes, etc. (There are possibilities for some more advanced work here if any students are working at a higher level – students can show algebraically that the ratio of the lengths of the sides of A sizes of paper is . The additional information that A�0 is one square metre enables the measurements of each size to be calculated.)

3 Ask the students to spend a couple of minutes thinking of some hard examples Learn 2

of splitting quantities according to a given ratio. Choose students to come out to work through their examples on the board and discuss the issues that arise.

4 Best buy questions such as those in Apply 3 can be done in many different ways. Learn 2

A useful plenary activity would be to discuss the various approaches and for students to identify which methods they like best, and whether some are better in certain circumstances.

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70

9 a y # kx

b xy # k

c y # kx�2

d

10 Yes, if they are experimental results – the numbersare approximately proportional (the ratiosbetween each pair of coordinates areapproximately the same) but not exactly.

11 a i 314.25 cm�2 ii 5028 cm�2

b

12 12.6 cm (to 3 s.f.)

13 a For example, time taken and distance travelled.

b

Graph does not go through origin.

y =kx

˚C

˚F

A

r

PL

EN

AR

Y


x

y

x

y

x

y

x

y

5 Put the word ‘ratio’ in the centre of the board and, with the class, create a Learn 3

‘mind map’ of everything they have learnt about ratio.

6 Give each pair or small group of students in the class a word or phrase from Learn 3

the list below and ask them to develop an explanation of the word or phrase, using examples, pictures, etc. to illustrate their explanation. (This could be set as homework.) Ask a group to give their word and its explanation. Ask other students to comment. Discuss the issues raised. Continue for as many words as you have time for.Examples of possible words: fraction, ratio, unitary ratio, simplifying ratios, factor, multiple, unitary method, teacher : student ratio, map scales, male : female ratio, two pairs of numbers in the same ratio, simplifying fractions, profit: cost price ratio, dividing a number in given ratio, proportion, best buy.

7 (This could also make a good starter discussion before doing proportional growth.) Learn 4

Discuss this simple mathematical model of population growth with the class:At the end of the eighteenth century, Thomas Malthus predicted that the population would double every 25 years. At that time, there were about 7 million people in the UK; now there are about 60 million. How accurate was Malthus’ prediction for the UK?

8 Return to the mathematical relationships list identified in the starter, discuss Learn 5

some of them in detail and add further examples to the list. Encourage the students to make links between the work they have done on ratio and proportion and other aspects of their maths (and maybe other subject) studies.


71


72

Ratio and proportionA

SS

ES

Sansw

ers 1 a 2 : 3 b 7 : 10

2 a 600, 720, 1080 b 625 m�, 125 m�, 250 m�

3 a 15 eggs

b ‘Lessprice’ ratio # 125 : 5 # 25 : 1; ‘Lowerpay’ ratio # 144 : 6 # 24 : 1.‘Lowerpay’ gives the greater value.

c i 5 : 6 ii 25 : 36

d i 7 seconds ii 16 seconds

4 a 13 " 1.1�3 # 17.3 stones (3 s.f.) b 50 " 1.2�3 # 86.4 cm�2 c 10 000 " 0.7�4 # £2401

5 a i A is proportional to r�2 iii y is inversely proportional to x�2

ii v is proportional to r�3 iv T is proportional to

b i 44.1 km (3 s.f.) ii 123 m (3 s.f.) iii 39.6 0 32.4 # 7.2 km

c 1.125 " 10�5 Nm�02

6 i C ii B iii D

√l

ratio and proportion

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