Edexcel GCSE (9-1)

Edexcel GCSE (9-1)

MathematicsFoundationStudent Book

11 - 19 PROGRESSION

Confidence • Fluency • Problem-solving • Reasoning

A LWAY S L E A R N I NG

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Sample unit

Draft, subject to

endorsement

We are seeking endorsement for use with the Edexcel GCSE (9-1) Mathematics specification.

317

Unit 11 Ratio and proportion

11 RATIO AND PROPORTION

Masterp318

Checkp337

Problem solvep336

Strengthenp339

Extendp342

Testp345

The engine performance of cars can be compared by looking at the power-to-weight ratio.

Cars with a high power-to-weight ratio accelerate well.

The times taken for cars to accelerate from 0 to 100 km/h are given in the table.

Car make and model Time (0–100 km/h)

Ferrari 458 Italia 3.3 s

Lamborghini Aventador 2.7 s

Porsche 918 Spyder 2.4 s

Which car accelerated the fastest?

Numerical fl uency

1 Copy and complete

a 10 × = 60 b × 9 = 900

2 Find the highest common factor (HCF) of each pair of numbers.

a 25 and 35 b 24 and 40

3 Work out

a 24 kg ÷ 3 b 72 mm ÷ 8

c £2.80 ÷ 4 d 300 g ÷ 5

4 Work out

a 17.5 ÷ 5 b 5.4 × 6

5 Work out

a 40 ÷ 5 × 3 b 36 ÷ 4 × 8

6 What fraction of each diagram is shaded?

a b

7 Copy and complete

a 3 __ 5

= ___ 20

b 24 ___ 32

= 3 ___

Fluency with measures

8 How many

a g in 1 kg

b ml in 1 litre?

Geometrical fl uency

9 What is the scale factor of this enlargement?

B

10 cm

6 cmA

5 cm

3 cm

11 Prior knowledge check

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Draft, subject to endorsement

Contents

1 Number

2 Algebra

3 Graphs, tables and charts

4 Fractions and percentages

5 Equations, inequalities and sequences

6 Angles and polygons

7 Averages and range

8 Perimeter, area and volume 1

9 Graphs

10 Transformations

11 Ratio and proportion

12 Right-angled triangles

13 Probability

14 Multiplicative reasoning

15 Constructions, loci and bearings

16 Quadratic equations and graphs

17 Perimeter, area and volume 2

18 More number

19 Congruence, similarity and vectors

20 More algebra

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318


Graphical fl uency

10 Plot a line graph for the values in the table.

x 1 2 3 4 5

y 3 6 9 12 15

a What is the gradient of the line?

b What is the equation of the line?

Challenge11 Design a set of 20 loop cards. The fi rst two

have been done for you.

The last card must end with 3 __ 4

or 0.75. Explain why.

34 0.5 1

2 0.8

1 The bar chart shows the number of teenagers doing each activity at an outdoor pursuits centre on one day.

a How many teenagers are taking part in activities at the centre altogether?

b Which activity is the most popular?

War

m u

p

canoeing0

5

10

15

20

25

Nu

mb

er o

f te

enag

ers

Activity

Outdoor pursuits centre data

windsurfing jet skiing

Key point 1

A ratio is a way to compare two or more quantities.

11.1 Writing ratios

Fluency

• Work out 28 ÷ 4, 45 ÷ 5

• Chloe has 3 red beads and twice as many blue beads. How many blue beads does she have?

Objectives

• Use ratio notation.

• Write a ratio in its simplest form.

• Solve simple problems using ratios.

Why learn this?

Outdoor pursuits need to have the correct supervision ratios.

Questions in this unit are targeted at the steps indicated.

2 Write down each ratio of red tins to yellow tins.

a

b

3 Draw tins to show these ratios of red to yellow.

a 4 : 3 b 3 : 4Discussion Is the ratio 4 : 3 the same as the ratio 3 : 4?

Q2a hint There is 1 red tin and 2 yellow tins. Write the ratio ‘red : yellow’ using the numbers.

Homework, practice and support: Foundation 11.1

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4 A necklace has 30 beads. There is 1 purple bead for every 4 blue beads.

How many beads are

a purple b blue?

5 Write each ratio in its simplest form.

a 4 : 12 b 16 : 8

c 27 : 9 d 7 : 42

e 15 : 20 f 63 : 28

g 18 : 48 h 36 : 45

6 Problem-solving / Reasoning The bar charts show the numbers of gold medals and other medals won at a competition by each group from a gym club.

group A0

1

2

3

4

Nu

mb

er o

f g

old

med

als

Club group

Gold medals won

group B group C group D group A0

2

4

6

8

1

3

5

7

Nu

mb

er o

f o

ther

med

als

Club group

Other medals won

group B group C group D

a What is the ratio of gold medals won to other medals won? Write your answer in its simplest form.

b The club’s coach says that they won twice as many other medals as gold medals.

Is the coach correct? Explain your answer.

7 Real An a� er-school club at a primary school is attended by 32 children. It is run by 4 adults. The guidelines say that the adult-to-child ratio should be 1 : 8.

Does the club have enough adults?

8 Problem-solving Which of these ratios are equivalent?

A 36 : 16 B 135 : 60

C 28 : 16 D 126 : 56

E 49 : 28


a 20 : 25 : 15 b 36 : 24 : 30

c 56 : 42 : 35 d 16 : 40 : 56

Q4 hint How many sets of ‘1 purple, 4 blue’ are these?

Key point 2 You simplify a ratio by making the numbers as small as possible.Divide the numbers in the ratio by their highest common factor (HCF).

Q5a hint 4

1�4 �4

:

:

12Q5e hint

15�5 �5

:

:

20

Q6a hint What do you need to work out before you can fi nd the ratio?

Q7 hint Show your working.

Q8 communication hint Ratios are equivalent if they have the same simplest form.

Q9a hint The highest common factor of 20, 25 and 15 is 5, so divide all the parts by 5.

20�5 �5

:

:

:

:

25�5

15

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10 Communication Show that these ratios are equivalent.

10 : 15 : 25 12 : 18 : 30 18 : 27 : 45

11 STEM A recipe for shortbread uses 125 g of butter, 55 g of sugar and 180 g of fl our.

Write the ratio of butter : sugar : fl our in its simplest form.

12 Exam-style question

There are 80 marbles in a bag. Of these, 25 are china and the rest are glass. Write the ratio of china marbles to glass marbles in its simplest form. (2 marks)

Q12 strategy hint Start by writing C : G and the information you know.

11.2 Using ratios 1

Fluency

• 1 m = cm • 1 cm = mm

• 1 m = mm • 2500 mm = m

Objective

• Solve simple problems using ratios.

Why learn this?

You need to mix paints in the same ratio to get the same colour each time.

1 Copy and complete

a 2 × = 100 b 3 × = 75 c 2 × = 150 d 6 × = 300

2 Work out

a 1.5 × 10 b 2.45 × 100 c 3.71 × 10 d 9.37 × 100

3 Find the HCF of each pair of numbers.

a 24 and 30 b 35 and 49 c 18 and 45 d 64 and 80

4 STEM The ratio of oyster sauce : fi sh sauce in a stir fry recipe is 2 : 1.

Clare uses 5 tablespoons of fi sh sauce.

How many tablespoons of oyster sauce does she use?

War

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Example 1

To make orange paint Maria mixes yellow paint with red paint in the ratio 3 : 1.

She uses 4 tins of red paint. How many tins of yellow paint does she use?

Write down the ratio. Use Y for yellow and R for red.

3

12�4 �4

:

:

1

4

Y : R

Maria uses 12 tins of yellow paint.

Multiply each part by the same number to get an equivalent ratio.


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11.2 Using ratios 1

5 Real Thomas makes a model of the Eiff el Tower using a ratio of 1 : 1280.

The height of his model is 250 mm.

What is the height of the Eiff el Tower in metres?

Q5 hint 1 : 1280 means that 1 mm in the model represents 1280 mm in real life.

6 Write each ratio as a whole number ratio in its simplest form.

a 0.4 : 6 b 3.5 : 4.2 c 45 : 13.5 d 25.6 : 46.4

Discussion What should you multiply by if a number in a ratio has 2 decimal places?

7 Write each ratio as a whole number ratio in its simplest form.

a 0.25 : 3.1 b 1.4 : 0.28 c 1.62 : 1.8 d 4.8 : 11.2

8 STEM / Problem-solving Old computer monitors had a width : height ratio of 4 : 3. New computer monitors have a width : height ratio of 16 : 9.

Is each of these monitors old or new?

a 16.8 inches

12.6 inches

b 18.4 inches

10.35 inches

9 STEM Sterling silver is made from silver and copper in the ratio 92.5 : 7.5.

Write this ratio in its simplest form.

10 STEM The ratio of chick peas : broad beans in a hummus recipe is 2 : 9.

Jack uses 150 g of chick peas. How many grams of broad beans should he use?

11 STEM / Reasoning The ratio of okra : sweet potato in a korma recipe is 15 : 8.

Raj uses 160 g of sweet potato. He has 350 g of okra.

Will he use all of the okra?

Q5 hint

1 :

:

:

1280

Model Real

Key point 3

Ratios in their simplest form only have whole numbers.

Example 2

Write 1.5 : 8 as a whole number ratio in its simplest form.

1.5

15

3

�10 �10:

:

:

8

80

16�5 �5

1.5 has 1 decimal place so multiply both sides of the ratio by 10 to get a whole number.

The HCF is 5 so divide both sides by 5.

Q10 hint

2

150�75 �75

:

:

9

C : B

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12 Finance Anna splits her monthly net pay into rent money and money le� in the ratio 7 : 13.

Her rent is £420. How much money does she have le� each month?

13

Exam hintDeal with the information one paragraph at a time.

Exam-style question

5 schools sent some students to a conference.

One of the schools sent both boys and girls. This school sent 16 boys. The ratio of the number of boys it sent to the number of girls it sent was 1 : 2

The other 4 schools sent only girls.Each of the 5 schools sent the same number of students.

Work out the total number of students sent to the conference by these 5 schools. (4 marks)

March 2013, Q12, 1MA0/1H

Q13 hint Start by working out how many girls the school sent.

11.3 Ratios and measures

Fluency

• 1 kg = g, 1 m = cm, 1 litre = ml, 1 cm = mm, 1 day = hours

• 9 = 2, 25 = 2, 8 = 3

Objectives

• Use ratios to convert between units.

• Write and use ratios for shapes and their enlargements.

Why learn this?

You can use ratios to work out your money for holidays abroad.

1 Faaiz has 7 red marbles and 2 blue marbles.

a How many marbles does he have altogether?

b What fraction are i red ii blue?

2 Work out the area of this shape. 3 Work out the volume of this cube.

5 cm

7 cm

2 cm

4 Write these ratios in their simplest form.

a 15 minutes : 1 hour

b 400 g : 1 kg

c 2 hours : 30 seconds

5 A mug contains 250 ml of coff ee and a jug contains 1 litre of coff ee.

Write down the ratio of the amount of coff ee in the mug to the amount of coff ee in the jug.

Give your answer in its simplest form.

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Q4 hint Both parts of the ratio need to be in the same units before you simplify.

Q4a and c hint There are 60 seconds in 1 minute and 60 minutes in 1 hour.


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11.3 Ratios and measures

6 Complete these conversions.a 2.5 km = m b 34 000 g = kg

c 3.8 litres = ml d 95 mm = cm

7 1 mile ≈ 1.6 km. Converta 5 miles to km b 16 km to miles.c Real The length of the London Marathon is 26.219 miles. How far is this in kilometres?

8 A recipe needs 2 lb of fl our.a 1 lb = 16 oz (ounces). Convert 2 lb to ounces.b 1 oz ≈ 25 g. Convert your answer to part a into grams.

9

10 Real / Finance On the day Lucy buys some US dollars for her holiday, £1 buys US $1.68.How many dollars does Lucy get for £500?

11 Real / Finance Choose an amount in £ from the cloud. Choose an exchange rate from the table.

£200£350

£475 £690

Work out how much money you will get. Repeat for two more amounts and currencies.

Key point 4

You can use ratios to convert between units.

Example 3

Convert 8 m to cm.

1 m is 100 cm.

1

8�8 �8

:

:

100

800

m : cm

So 8 m is 800 cm.

The ratio of m : cm is 1 : 100.

Q6a hint km : m = :

Exam-style question

Linda is buying wool to knit a baby’s blanket.She needs 1800 yards of wool.

Linda chooses some balls of wool.There are 245 metres of wool in each ball of wool.

She knows that

• 1 yard is 36 inches

• 1 inch is 2.54 centimetres

How many balls of wool does Linda need to buy?You must show all your working. (4 marks)

Nov 2013, Q21, 1MA0/2F

Exam hintPresent your working clearly and remember to show the units next to each number. Explain what you are doing at each step so it is easy for an examiner to follow.

Q10 hint

1

500

:

:

:1.68

£ $

�500 �500

Currency £1Dollars 1.68

Euros 1.23

Rupees 99.12

Yen 172.2

Swiss francs 1.50

Q11 communication hint The exchange rate is the amount of money in another currency that your currency will buy or sell for.

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12 Real Use the table in Q11.a How many £ does Sue get for 300 Swiss francs?b Which is worth more, 49 560 rupees or 85 700 yen?

13 The diagram shows three rectangles.

2 cm

5 cm

6 cm

15 cm

8 cm

20 cm

A BC

a Write these ratios in their simplest form.i The width of rectangle A to rectangle B.

ii The length of rectangle A to rectangle B.iii The area of rectangle A to rectangle B.

Discussion What do you notice about the results?

Rectangle C is an enlargement of rectangle A.

The heights of rectangle C and rectangle A are in the ratio 4 : 1.

b Reasoning What do you expect the ratio of the area of rectangle C to rectangle A to be?Work it out. Were you correct?

c Write the ratio of the length of rectangle C to rectangle B in its simplest form.

d Reasoning Predict the ratio of the area of rectangle C to rectangle B.Work it out. Were you correct?

14 The diagram shows two cubes.

a Write these ratios in their simplest form.i Height of A to height of B.

ii Area of a face of A to area of a face of B.iii Volume of A to volume of B.

Discussion What do you notice?

b Reasoning Another cube C has a side length of 6 cm. What do you expect the ratio of the volume of cube A to cube C to be?

Check to see if you were correct.

15 A bag contains 11 red counters and 3 white counters.

a What is the ratio of red counters : white counters?

b What fraction of the counters are red?

c What fraction of the counters are white?

Discussion What do you notice about your answers?

16 In a box of mints and toff ees, 3 _ 4 of the sweets are mints and 1 _ 4 are toff ees.

a What is the ratio of mints : toff ees?

b There are 8 toff ees in the box. How many mints are there?

c How many sweets are in the box altogether?

Refl ect Is there more than one way to work out the answer to part c?

17 In a band, 3 _ 5 of the members are male and the rest are female.

a What is the ratio of males : females in the band?

b Maya thinks there are 18 people in the band. Explain why she must be wrong.

Q13a iii hint 32 = ?

3 cm A 5 cm B

Q16 hint

M M M T

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11.4 Using ratios 2

Fluency

Write down the ratio of green : blue : yellow counters on the dish.

B

B

B

B

GG

G Y

Y

Objectives

• Divide a quantity into 2 parts in a given ratio.

• Divide a quantity into 3 parts in a given ratio.

• Solve word problems using ratios.

Why learn this?

The ratio of gold to other metals in a piece of jewellery is used to work out the value of the item.

1 Work out

a 35 ÷ 7 b 10 ÷ 4 c 4.5 ÷ 3 d 2.8 ÷ 4

2 Round these fi gures.

a 12.468 to 2 decimal places b 4.973 to 2 decimal places

c 3.1584 to 3 decimal places d 0.012 439 to 3 decimal places

3 Write these ratios in their simplest form.

a 25 : 10 b 100 : 350 c 6000 : 2000 d 4.5 : 6

4 Share these amounts in the ratios given.

a £18 in the ratio 2 : 1 b £42 in the ratio 1 : 6

c £27 in the ratio 4 : 5 d 35 kg in the ratio 2 : 3

e 60 m in the ratio 5 : 7 f 7.5 litres in the ratio 2 : 3

5 STEM / Real Before 2012, 10p coins were made from copper and nickel in the ratio 3 : 1.

The coins had a mass of 6.5 g. What was the mass of

a copper b nickel?

Warm

up

Example 4

Share £25 in the ratio 3 : 2.3 + 2 = 5 parts

£25

£25 ÷ 5 = £5£5 × 3 = £15£5 × 2 = £10Answer: £15 : £10Check: £15 + £10 = £25

Work out how many parts there are in total.

Work out 1 part.

Work out 3 parts and 2 parts.

Check they add up to the correct total.

Q4 hint How many parts are there?


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6 Purple paint is mixed from red paint and blue paint in the ratio 5 : 3.

A painter needs 20 litres of purple paint.

a How many litres of red paint should he use?

b How many litres of blue paint should he use?

Discussion How did you check your answers?

7 A fruit drink is made from orange, pineapple and apple juice in the ratio 1 : 2 : 4.

Rita wants to make 35 litres of fruit drink. How much of each type of juice does she need?

8

9 Share these amounts in the ratios given.

a £72 in the ratio 2 : 3 : 4 b 100 g in the ratio 2 : 3 : 5

c 360 ml in the ratio 3 : 4 : 5 d £25 in the ratio 1 : 3 : 4

Discussion How should you round if the ratio is money? What about litres? Why?

10 Share these amounts in the ratios given. Round your answers sensibly.

a £80 in the ratio 2 : 5 b 70 litres in the ratio 2 : 7

11 Bob and Phil buy a greyhound for £4500.

Bob pays £3000 and Phil pays £1500.

The greyhound wins an £18 000 prize.

a Write the amounts they each pay as a ratio.

b Write the ratio in its simplest form.

c Divide the prize money in this ratio. How much does each of them get?

Discussion Is this a fair way to share the prize money?

12 Andrea and Penny buy a statue for £350.

Andrea pays £140 and Penny pays £210.

They sell the statue 3 years later for £475.

How should they share the money fairly?

13 Problem-solving / Reasoning Two numbers are in the ratio 1 : 3 and their diff erence is 12.

What are the numbers?

Q8 strategy hint You could work out how many kg of each he needs to make 180 kg.

Exam-style question

Talil is going to make some concrete mix.He needs to mix cement, sand and gravel in the ratio 1 : 3 : 5 by weight.

Talil wants to make 180 kg of concrete mix. Talil has

• 15 kg of cement

• 85 kg of sand

• 100 kg of gravel

Does Talil have enough cement, sand and gravel to make the concrete mix? (4 marks)

Nov 2012, Q29, 1MAO/1F

Q9 hint How many parts are there?

Q11 hint

:

:

:Bob Phil

��

Q13 strategy hint 1 : 3

and 2 5 5 12

:

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14 Rob and Simon share £50 in the ratio 7 : 3.

a How much do they each receive? b How much more does Rob get than Simon?

Discussion How could you check your answer?

15

Refl ect How can drawing a diagram help you to answer ratio questions?

Exam-style question

Pat and Julie share some money in the ratio 2 : 5Julie gets £45 more than Pat.

How much money did Pat get? (3 marks)June 2012, Q22, 1MA0/2F

Q15 strategy hint Draw a diagram.

1 What are the missing numbers?

a 3 __ 5

= ___ 10

b 1 whole = ___ 5

c 1 __ 6

+ ___ 6

= 6 __ 6

d 2 __ 7

+ 1 __ 7

+ ___ 7

= 7 __ 7

2 Work out

a 2.5 + 3.5 b 1 − 2 __ 9

c 3 __ 4

× 100 d 2 __ 5

× 20

3 Simplify these ratios.

a 2 : 2.5 : 1.5 b 5.5 : 4.5 : 3 c 3 litres : 500 ml d 70 cm : 1 m

4 In a canoeing lesson 2 _ 7 of the group are girls and the rest are boys.

What is the ratio of girls to boys in the group?

5 Copy and complete the table for diff erent groups of teenagers having canoeing lessons.

Fraction of group that are girls Ratio of girls : boys

5 _ 9

3 : 2

7 __ 10

4 : 7

Warm

up

Key point 5

A proportion compares a part with the whole.

Q4 hint

G G B B B B B


11.5 Comparing using ratios

Fluency

A bracelet has gold and silver links in the ratio 2 : 7. Which type of link is used most?

Objectives

• Use ratios involving decimals.

• Compare ratios.

• Solve ratio and proportion problems.

Why learn this?

You can use this to calculate and compare the performance of diff erent cars.

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6 STEM In a gluten-free pizza base, 7 __ 12 of the flour is rice flour,

3 __ 12 is potato flour and the rest is tapioca flour.

Work out the ratio of rice flour : potato flour : tapioca flour.

7 STEM / Real A 5p coin used to be made of copper and nickel in the ratio 3 : 1.

What fraction of the coin was

a copper b nickel?

8 Clare and Fiona share a cash prize in the ratio 4 : 3.

What fraction of the prize should a Clare get b Fiona get?

Discussion How can you check your answer?

9 STEM / Real A 20p coin is made of copper and nickel in the ratio 21 : 4.

What fraction of the coin is a copper b nickel?

10 The ratio of office staff to manufacturing staff at a company is 2 : 5.

Jo says that 2 _ 5 of the people are office staff.

Is she correct? Explain your answer.

11 STEM / Real Red brass is made from a mix of copper, zinc, lead and tin in the ratio 17 : 1 : 1 : 1.

a What fraction of red brass is i copper ii zinc?

b In 100 g of red brass, what are the masses of copper, zinc, lead and tin?

c Lily has plenty of copper and zinc, but only 80 g of lead and 60 g of tin. What is the maximum amount of red brass she can make?

d Reflect Look back at your working for part c. Have you clearly shown how you got your answer?

12 STEM A topping for shortbread uses milk, chocolate, butter and syrup in the ratio 4 : 3.5 : 1.5 : 1.

a What fraction of the topping is chocolate?

b In 500 g of the topping what are the masses of milk, chocolate, butter and syrup?

c You have plenty of milk and chocolate, but only 200 g of butter and 100 g of syrup. What is the maximum amount of topping you can make?

13 Copy and complete to write these as unit ratios.

a 5�3 �3

:

:

3

1

b 2�2 �2

:

:

7

Q7 hint

copper copper copper nickel

copper = ___ 4

Q10 strategy hint Draw a diagram.

Q12 hint Write the ratio with whole numbers first.

Key point 6

You can compare ratios by writing them as unit ratios. In a unit ratio, one of the numbers is 1.

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14 Write each of these in the form m : 1.

Give each answer to a maximum of 2 decimal places.

a 2 : 5 b 7 : 4 c 16 : 9

d 5 : 36 e 9 : 42 f 11 : 56

15 Anna makes orange squash by mixing 50 ml of squash with 850 ml of water.

Jeevan makes orange squash by mixing 60 ml of squash with 1110 ml of water.

Whose squash is the stronger? Explain your answer.

16 Josh makes purple paint by mixing 2 litres of red paint and 500 ml of blue paint.

Dexter makes purple paint by mixing 1.5 litres of red paint and 400 ml of blue paint.

Whose paint is the darker purple? Explain your answer.

17 Raj makes concrete using aggregate to cement in the ratio 1930 : 265.

Sunil makes concrete using aggregate to cement in the ratio 935 : 175.

Whose concrete has the higher proportion of cement?

18 Billy makes mortar by mixing 360 kg of cement with 90 kg of sand.Sam makes mortar by mixing 340 kg of cement with 85 kg of sand.Whose mortar has the higher proportion of cement?

19

Q14 hint Write each of these as a unit ratio, with the second value 1.

Example 5

Molly makes a blackcurrant drink by mixing 30 ml of blackcurrant with 450 ml of water.

Hope makes a blackcurrant drink by mixing 40 ml of blackcurrant with 540 ml of water.

Whose drink is the stronger? Explain your answer.

Molly Hope

30

1�30 �30

:

:

450

15

blackcurrant : water

40

1�40 �40

:

:

540

13.5

blackcurrant : water

Hope’s drink is the stronger because it uses less water for every millilitre of blackcurrant.

Simplify to a unit ratio.

Compare the quantity of water per ml of blackcurrant.

Q16 hint Darker purple paint has a higher concentration of blue paint.

Exam-style question

STEM Electrum is 2 _ 5 gold and 3 _ 5 silver.

Jessica says, ‘The amount of silver is one and a half times the amount of gold.’

Is she correct? Explain your answer. (3 marks)

Exam hintExplain your workings and answer by writing your reasons alongside your calculations rather than listing them at the end.

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11.6 Using proportion

Fluency

In each pair, which number is bigger? 3.28 or 3.7? 2.9 or 2.45? 1.06 or 1.009?

Objectives

• Use the unitary method to solve proportion problems.

• Solve proportion problems in words.

• Work out which product is better value for money.

Why learn this?

Working out which product is the better buy helps you get the best value for money.

1 Work out

a 840 ÷ 7 b 450 × 9 c 2.7 × 5 d 4.8 ÷ 3

e 10 × = 20 f 10 ÷ = 5 g 5 ÷ = 1 h 7.2 ÷ = 1

2 Write each of these as a unit ratio.

a 4 : 18 b 11 : 5 c 14 : 35 d 18 : 8

3 What is the HCF of each pair of numbers?

a 500 and 350 b 600 and 800

4 Which is better value if you are shopping?

a Paying 1.8p per gram or 2.1p per gram? b Getting 3.25 ml for 1p or 2.85 ml for 1p?

5 A recipe for 4 people uses 6 eggs. How many eggs are needed for

a 8 people b 2 people c 6 people d 10 people?

6 5 tickets to a theme park cost £125.

How much will 18 tickets cost?Discussion Is there more than one way to work out the answer? Which is the best method? Why?

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Example 6

A recipe for 6 people uses 900 g of mince. How much mince is needed for

a 12 people b 3 people c 9 people?

6

12�2 �2

:

:

900 g

1800 g

P : M

6

3�2 �2

:

:

900 g

450 g

P : M

6 people + 3 people = 9 people 900 + 450 = 1350 g

Discussion How would you work out the amounts for 18 people and 15 people?

Key point 7

In the unitary method you fi nd the value of one item before fi nding the value of more.

Q6 hint Find the cost of 1 ticket fi rst.

5

1� �

:

:

:

£125

Tickets Cost


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7 20 litres of fuel cost £25.38.

What do 35 litres cost?

8 Maria gets £31.55 for working 5 hours.

How much will she get for working for

a 9 hours b 26 hours c 30 hours?

Refl ect Did you do part c the same way as parts a and b?

Can you show a quicker way?

9 A gym coach orders a tracksuit for each of the 25 gymnasts in the club.

The total cost is £1247.50.

2 more gymnasts join the club so he orders an extra 2 tracksuits.

What is the total value of the order now?

10 Washing powder comes in two sizes.

a Copy and complete to fi nd the unit ratios.i 1.3 kg

kg�

:

:

£4.40

£1�

ii 2.6 kg

kg�

:

:

£6

£1�

b Which size is the better buy?

11 Jenna can buy her favourite blend of coff ee in two sizes: 500 g for £14.90 or 300 g for £8.85.

Which is better value for money?

Discussion How did you work it out?

12 David can buy blackcurrant squash in two sizes: 2 litres for £4.99 or 600 ml for £1.98.


13 Mary gets paid £31.50 for 5 hours of work. Alice gets paid £50 for 8 hours of work.

They both work 36 hours a week. Who earns more money?

14

Q7 hint Find the cost of 1 litre fi rst.

20

1� �

:

:

:

25.38

L £

Key point 8

You can use the unitary method to work out which product gives better value for money.

1.3 kg for £4.40 2.6 kg for £6Q10b hint Which gives more grams for £1?

Q12 hint Convert both to the same units.

Exam hintHighlight or underline key information in a question. What information not given in the boxes should you highlight here?

Exam-style question

Brenda works in an offi ce.She fi nds out the prices of folders from two companies, Offi ce Deals and Paper World.

Brenda needs to buy exactly 60 folders. She wants to buy the folders as cheaply as possible.

Which company should Brenda buy the folders from?You must explain your answer. (4 marks)

Nov 2013, Q16, 5MB3F/01

Packs of 20 folders£10.80

OFFICEDEALS

PAPERWORLD

Pack of 15 folders£8.40

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1 The graph changes between gallons and pints.

Use the graph to work out these conversions.

a 5 gallons = pints

b 9 gallons = pints

c 120 pints = gallons

2 Plot a line graph for the values in the table.

a What is the gradient of the line?

b What is the equation of the line?

3 Look at the graph in Q1.

Are gallons and pints in direct proportion? Explain your answer.

4 Modelling The table shows some temperatures in both Celsius and Fahrenheit.

Celsius 5° 10° 20° 25°

Fahrenheit 41° 50° 68° 77°

a Plot a line graph for these values.

b Are Celsius and Fahrenheit in direct proportion? Explain your answer.

c Ice melts at 0 °C. What is this temperature in Fahrenheit?

War

m u

p

200

406080

100120

50 10 15 20

Conversion graph

Gallons

Pin

ts

x 1 2 3 4 5

y 4 8 12 16 20

Key point 9

When two values are in direct proportion, if one value is zero so is the other. When one value doubles, so does the other.

Q3 hint Does doubling one value double the other? When one value is zero, is the other?

Key point 10

When two quantities are in direct proportion, plotting them as a graph gives a straight line through the origin. The origin is the point (0, 0) on a graph.

Q4a hint Plot Celsius on the horizontal axis and Fahrenheit on the vertical axis. Use sensible scales.


11.7 Proportion and graphs

Fluency

• What is the equation of a straight line through the origin (0, 0) with gradient 5?

• What is the gradient of the line y = 7x?

Objectives

• Recognise and use direct proportion on a graph.

• Understand the link between the unit ratio and the gradient.

Why learn this?

Knowing simple relationships allows us to make predictions.

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5 Modelling The table shows the price of pears at a market.

Mass of pears (kg) 0.5 1 5 10

Price of pears (£) 1 2 10 20

a Plot a line graph for these values.

b Is the price of the pears in direct proportion to the mass? Explain your answer.

6 Modelling This line graph shows the price of grapes by mass.

a Are price and mass in direct proportion?

b Work out the gradient of the line.

c How much does 1 kg of grapes cost?

Discussion How does your answer to b help you answer c?

7 Reasoning Which of these are in direct proportion? Give reasons for your answers.

a Inches and centimetres

b Pounds (lb) and kilograms

c Age and favourite music type

d Number of teens in swimming pool and time of day

e Euros and pounds

8 A plumber charges a callout fee of £50 plus £35 per hour he works.

Is his total charge (C ) in direct proportion to the number of hours (h) he works?

9 Which of these sketch graphs show one variable in direct proportion to another? Explain your answer.

O O O O O

A B C D E

10 Modelling The table shows the amount paid for diff erent numbers of litres of fuel on one day.

Number of litres (n) 5 10 20 30 50

Cost (C ) £6.60 £13.20 £26.40 £39.60 £66

a Draw the graph.

b Work out the equation of the line.

c Are C and n in direct proportion?

d Write a formula linking the number of litres (n) and the cost (C ).

e Use the formula to work out the cost of 82 litres of fuel on that day.

Refl ect Look at the equation of the line and the formula for C and n. What is the same? What is diff erent?

1

0

2

3

4

5

2000 400 600 800 1000 1200

Price of grapes

Mass (g)

Pri

ce (

£)

Q8 hint You might wish to draw a graph.

Q10d hint

C = × n

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11 Modelling Look back at the graph in Q1.

a Write a formula that links gallons (G) and pints (P).

b Use your formula to change 16.5 gallons to pints.

12 Modelling The ratio of miles to km is 5 : 8.

a Copy and complete this table of values for miles and kilometres.

Miles 0 5 10

Kilometres

b Draw a conversion graph for miles to kilometres.

c Write a formula linking kilometres ( y) and miles (x).

d Write 5 : 8 as a unit ratio.

e What do you notice about the unit ratio and the gradient of the line?

13

Q12a hint Are miles and km in direct proportion?

Q12b hint Put miles on the x-axis and km on the y-axis.

Exam-style question

STEM / Problem-solving A formula to change between temperature in kelvins (K ) and in degrees Celsius (C ) is

K = C + 273.15.

Are kelvins and degrees Celsius in direct proportion? Explain your answer. (2 marks)

Exam hintYou won’t get any marks for just ‘Yes’ or ‘No’. You must write ‘Yes’ or ‘No’ because ………….

1 Work outa £20 ÷ 8 b £36 ÷ 5 c £2.45 × 3 d £5.90 × 3

2 How many hours and minutes ina 315 minutes b 200 minutes c 2 3 _ 4 hours d 4 1 _ 3 hours?

War

m u

p

Key point 11

When two values are in inverse proportion, one increases at the same rate as the other decreases. For example, as one doubles (×2) the other halves (÷2).

Example 7

It takes 2 painters 7 days to paint a house. How many days does it take 1 painter to paint an identical house?

2 people take 7 days, so 1 person takes 2 × 7 = 14 days.

It takes 1 person twice (×2) as long as 2 people.


11.8 Proportion problems

Fluency

Will 5 men take more or less time to build a wall than 3 men?

Objectives

• Recognise diff erent types of proportion.

• Solve word problems involving direct and inverse proportion.

Why learn this?

A manager needs to understand proportion to allocate the right number of people to a job.

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11.8 Proportion problems

3 3 people dig a ditch in 7 hours. How long will it take

a 1 person b 5 people?

4 2 workers take 5 hours to tile a room. How long will it take

a 4 workers b 3 workers?

Give your answers in hours and minutes.

5 4 concert tickets cost £50.

How much will 9 concert tickets cost?

6 Real A forestry supervisor needs to transplant 420 tree seedlings.

He wants all the work done in 7 hours or less.

He knows that 1 person can transplant 8 seedlings in an hour.

How many people does he need for this job?

7 A band with 4 people can play a song in 4 minutes 30 seconds.

How long will it take a band with 8 people to play the same song?

8 It takes 5 hill walkers 1 hour 20 minutes to walk up a hill.

How long will it take 8 hill walkers?

Discussion Compare your answer with other people. Is this a proportion problem?

9 Real A farmer has enough food for 250 chickens for 20 days.

He buys 50 more chickens.

How many days will the food last for?

10 The weight of 30 identical coins is 150 g.

What is the weight of 800 of these coins? Give your answer in kg.

11 Real A contractor needs to decorate 20 rooms.

He knows that 2 people can decorate a room in 3 hours.

He needs all the work to be done in 8 hours.

How many people does he need for the job?

12 Frank is paid £43.61 for 7 hours of work.

How many hours does he work to earn £112.14?

13 It takes 5 shredders 4 hours to shred 625 sheets of paper.

Rick gets 3 more shredders.

How long will it take to shred 625 sheets of paper now?

14 Refl ect a Write and solve your own proportion problem.

b Give your question to a classmate. Mark their answer. Did they show their working clearly?

15

Q3a hint Will it take 1 man more or less time than 3 men?

Q4 hint 60 minutes = 1 hour

Q5 hint Do you expect them to cost more or less?

Q6 strategy hint How many could 1 person transplant in 7 hours?

Q9 strategy hint How many chickens does he have to feed?

Exam-style question

It takes 5 machines 2 days to complete a harvest.

How long would it take 4 machines? (2 marks)

Q15 strategy hint Start by deciding if you expect it to take more or less time. This will tell you whether it is direct or inverse proportion.

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• Use bar models to help you solve problems. Objective

Example 8

Miguel and Jean share a tin of biscuits in the ratio 3 : 5. Jean has 6 more biscuits than Miguel.

How many biscuits were in the tin?

all the biscuits

Miguel Jean

all the biscuits

Miguel

Jean

6 biscuits

1 section = 6 ÷ 2 = 3 biscuits

All the biscuits = 8 × 3 = 24 biscuits

Check: Miguel has: 3 × 3 = 9 biscuits

Jean has: 3 × 5 = 15 biscuits

Jean has 15 − 9 = 6 more biscuits

Total number of biscuits = 9 + 15 = 24 biscuits ✓

1 In an orchestra, 10% of the musicians play brass instruments, 15% play woodwind and 5% play percussion. 28 musicians play stringed instruments.

How many musicians are in the orchestra altogether?

2 The sizes of the angles in a triangle are in the ratio 1 : 3 : 5.

Sketch the triangle and mark the angles in degrees.

3 Hilary and Ruth share a fl at with a monthly rent of £885. Hilary ’s bedroom is twice the size of Ruth’s, so she pays twice as much rent as Ruth.

How much do they each pay?

Draw a rectangular bar to represent all the biscuits in the tin.

Split the bar into the ratio 3 : 5 for Miguel and Jean.

Compare the bars for Miguel and Jean.

Label diff erence between the bars as 6 biscuits.

2 sections represent 6 biscuits. Work out 1 section.

8 sections represent all the biscuits.

Check your answer works.

Q1 hint Draw a bar to represent all the musicians in the orchestra. Split the bar into 10% sections.One section = musicians.Total number of musicians =

Q2 hint Draw a bar to represent the sum of all the angles in a triangle. Split the bar into sections in the ratio 1 : 3 : 5. One section = °

Q3 hint£____

Hilary Ruth

11 Problem-solving

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4 Amjit buys 3 bunches of tulips and 1 bunch of roses. Bunches of roses cost twice as much as bunches of tulips.

Amjit hands over a £20 note and receives £5 change.

How much is a bunch of roses?

5 Toby and Isy share some money in the ratio 5 : 7. Isy gets £60 more than Toby.

How much money does Toby get?

6 A farmer has sheep, cows and pigs in the ratio 12 : 3 : 2. The farmer has 150 more sheep than pigs.

How many cows has she got?

7 8 boys and 12 girls go to swimming lessons. In one lesson, the mean number of lengths swum by the boys is 4.5, and the mean number of lengths swum by the girls is 2.

Work out the mean number of lengths swum by all the children in that lesson.

8 Refl ect How did drawing bar models help you?

Is this a strategy you would use again to solve problems?

Q4 hint

tulips tulips tulips

Q5 hint Draw a bar to show the ratio 5 : 7. Label the sections.How many sections represent £60?One section = £

Q7 hint Draw a bar to represent all the children.Split the bar into sections that show the numbers of boys and girls.Work out how many lengths the boys swam in total and how many lengths the girls swam in total.Work out how many lengths the children swam altogether.

11 Check up

Simple proportion and best buys

1 A recipe for 4 people uses 150 g of sugar. How much sugar is needed for

a 8 people b 2 people c 6 people?

2 Helen can buy 300 ml of shampoo for £3.50 or 75 ml of shampoo for £1.


Ratio and proportion

3 A necklace has 24 beads. There is 1 red bead for every 3 blue beads.

How many beads are

a red b blue?


a 24 : 32 b 27 : 18 : 54 c 2.5 : 3.5

Homework, practice and support: Foundation 11 Check up

Log how you did on your Student Progression Chart.

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5 Copy and complete these conversions.

a 4.2 km = m b 0.05 litres = ml

c £200 = euros (£1 = 1.23 euros) d 25 km ≈ miles (8 km ≈ 5 miles)

6 Iona makes a scale model of a sailing boat using the ratio 1 : 25.

Her model is 60 cm long.

How long is the real sailing boat? Give your answer in metres.

7 Hazel mixes pink paint using red paint and white paint in the ratio 1 : 4.

How much white paint should she use with 5 tins of red paint?

8 Naadim and Bal share £60 in the ratio 2 : 3.

a How much do they each receive?

b What fraction of the total amount does Naadim receive?

c Show how you checked your answer is correct.

9 Write each of these as a unit ratio. Give your answers to a maximum of 2 decimal places.

a 8 : 5 b 7 : 19

10 The ratio of primary school children to secondary school children in a club is 2 : 3. There are 12 primary school children in the club.

a How many secondary school children are there?

b What is the total number of children in the club?

11 Luke makes lemon squash using 40 ml of squash and 380 ml of water.

Joseph makes lemon squash using 50 ml of squash and 475 ml of water.

Who has made the stronger squash? Explain your answer.

Proportion, graphs and inverse proportion

12 The table shows the number of Barbadian dollars you get for diff erent numbers of US dollars.

a Draw a conversion graph with US dollars on the x-axis and Barbadian dollars on the y-axis.

b Are US dollars and Barbadian dollars in direct proportion?

c Write a formula linking US dollars and Barbadian dollars.

13 It takes 4 people 3 days to lay the cables for high-speed fi bre broadband.

How long would it take

a 1 person b 6 people?

14 It takes 3 machines 4 minutes to print 600 pages.

How long will it take 8 machines to print 600 pages?

15 How sure are you of your answers? Were you mostly

Just guessing Feeling doubtful Confi dent

What next? Use your results to decide whether to strengthen or extend your learning.

US dollars 5 10 50

Barbadian dollars 10 20 100

Refl

ect

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Challenge16 a How many diff erent ways can you divide this rectangle

into two parts in the ratio 1 : 3?

b How many of your answers show a symmetrical pattern?

11 Strengthen

Simple proportion and best buys

1 A bus ticket costs £3. Work out the cost of

a 2 tickets b 5 tickets.

2 It costs £32 for 4 teenagers to go to the cinema.

How much does it cost for

a 8 teenagers

b 2 teenagers

c 10 teenagers?

3 Harry can wash 6 cars in 1 hour. How many cars can he wash in 3 1 _ 2 hours?

4 Which is better value for money: 200 g of lotion for £2.50 or 300 g of lotion for £3.60?

Ratio and proportion

1 Andie makes a bracelet with green beads and yellow beads.

She uses 1 green bead for every 2 yellow beads.

The bracelet has 12 beads altogether. How many beads are

a green b yellow?

Q2a hint £32 £32

?

Q2b hint £32

?

Q3 hint

6�3

:

:

:

1

3

Cars Hours

12

12�31

2

1 hour

Q4 strategy hint Work out how much you pay for 100 g of each.

£2.50

100 g 100 g

£3.60

100 g 100 g 100 g

Q1 hint

G Y Y

G Y Y G Y Y G Y Y G Y Y

Homework, practice and support: Foundation 11 Strengthen

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a 2 : 6 b 6 : 9 c 10 : 15

Q2b hint You cannot divide by 2. Try dividing by 3.

d 24 : 36 : 60 e 2.5 : 6 f 2.4 : 1.6

Q2e hint Choose a number to multiply by to give whole numbers first.

3 Complete these conversions.

a 3.8 m = cm b 2.5 litres = ml

c £600 = euros (£1 = 1.23 euros) d 24 pints = gallons (1 gallon = 8 pints)

4 Jim mixes red paint and blue paint in the ratio 1 : 5.

He uses 3 tins of red paint.

How much blue paint should he use?

5 Jenny makes a scale model of a train carriage using the ratio 1 : 20.

Her model is 24 cm long.

a How long is the real train carriage in cm?

b Write the real length in metres.

6 The ratio of butter to sugar in a recipe is 2 : 1.

Ian uses 6 ounces of butter.

How many ounces of sugar should he use?

7 Isabel and Freya share £20 in the ratio 2 : 3.

a How much do they each receive?

b Show how you have checked your answer.

Q7b hint Add your answers for Isabel and Freya’s shares together.

8 A piece of cloth is cut into three pieces in the ratio 1 : 2 : 5.

The piece of cloth is 240 cm long.

a How long is each piece?

b Show how you have checked your answer.

Q2a hint

2

1�2 �2

:

:

6

Q3a hint

0

m

cm

0

3

1 2 3 4

100

m

cm4

Q3c hint

0

£

euros

0

3

1

1.23

4

Q4 hint 1 red for every 5 blue.

R B B B B B

Q5a hint

20 cm

model

real

1 cm

3

24 cm

Q6 hint

6 ounces

butter butter sugar

Q7a hint£20

Isabel Isabel Freya Freya Freya

Q8 hint240 cm

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9 A shop sells jam doughnuts and toffee doughnuts.

The ratio of jam doughnuts to toffee doughnuts sold is 2 : 3.

The shop sold 18 jam doughnuts.

a How many toffee doughnuts were sold?

b How many doughnuts were sold altogether?

10 The ratio of quad bikes to go-karts at an activity centre is 3 : 7.

a What fraction of the vehicles are go-karts?

b What fraction of the vehicles are quad bikes?

11 3 _ 4 of the bikes at a cycle shop are road bikes. The rest are mountain bikes.

a What is the ratio of road bikes : mountain bikes?

b There are 27 road bikes. How many mountain bikes are there?

c How many bikes are there in the cycle shop altogether?

12 Write each of these as a unit ratio. Give your answers to a maximum of 2 decimal places.

a 4

1�4 �

:

:

26 b 25� �

:

:

2

1

c 13 : 5 d 7 : 22

13 Luke makes lemon squash using 20 ml of squash and 180 ml of water.

Ben makes lemon squash using 50 ml of squash and 500 ml of water.

Who has made the stronger squash? Explain your answer.

Proportion, graphs and inverse proportion

1 The sketch graph shows an object moving at a constant speed.

Time (minutes)

Dis

tan

ce(m

iles)

O

a Is this graph a straight line?

b Does the graph go though (0, 0)?

c Is the distance travelled in direct proportion to the time taken?

2 An electrician charges a callout fee of £60 and £30 per hour she works.

Hours worked 0 1 2 3

Total charge (£) 60 90 120 150

a Draw a graph for the table of her charges.

b Is her total charge in direct proportion to the hours she works?

Q9 hint 18

J J T T T

Q10a hintquad bikes go-karts

Q11a strategy hint Draw a picture to show the fractions.

Q12 strategy hint Divide both numbers in the ratio by the smaller number.

Q13 strategy hint Write them both as unit ratios.

Q1c hint Did you answer ‘Yes’ to parts a and b?

Q2b hint Answer Q1a and Q1b for the graph you have drawn.

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3 A shop sells material for bridesmaid dresses at £4 per metre.

Is the price of material proportional to the length bought?

Answer Q1a and Q1b for the graph you have drawn.

4 The ratio of gallons to pints is 1 : 8.

Write a formula that shows the relationship between gallons ( g) and pints ( p).

5 It takes 3 people 4 hours to lay some pipes. Copy and complete the table.

Number of people Longer or shorter? Hours

3 — 4

1

2

6

6 It takes 4 washing machines 12 hours to wash the hotel laundry.

How long will it take 6 machines?

Q3 strategy hint Make a table of values and draw a graph.

Length (m)

Co

st (

£)

O

Q4 strategy hint Make a table of values.Gallons

Pints

0

0

1

8

2 3�

g = × p

Q5 strategy hint If longer, then multiply. If shorter, then divide.

Q6 hint Draw a table like the one in Q5.

1 Real Barry uses 50 g of lawn seed per square metre.

He needs to seed a lawn that is 8 m by 14 m. He has a 5 kg bag of lawn seed.

Does he have enough seed for the lawn?

2 Real / Problem-solving Freda is planning a trip for her gymnasts.

The recommended adult-to-child ratios are

• Age 4 to 8 – a minimum of 1 adult for every 6 children



Here are the ages of the gymnasts.

Age Number of gymnasts5 or 6 18

7 or 8 12

10 or 11 13

12 3

14 and over 23

Work out the minimum number of adults that need to go on the trip.

11 Extend

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3 Exam-style question

Ketchup is sold in three different sizes of bottle.

Small bottle Medium bottle Large bottle

£3.991500 g

£1.95570 g

88p342 g

A small bottle contains 342 g of ketchup and costs 88pA medium bottle contains 570 g of ketchup and costs £1.95A large bottle contains 1500 g of ketchup and costs £3.99

Which bottle is the best value for money?You must show your working. (4 marks)

June 2013, Q9, 5MB3H/01

4 The lengths of the sides of a triangle are in the ratio 3 : 4 : 5.

The perimeter of the triangle is 18 cm.

What is the length of the longest side?

5

6 3 painters take 12 hours to paint a house.

a How long would it take 4 painters to paint the house?

b The charge for the work is £18 per painter per hour. What is the total cost?

7 Problem-solving The sizes of the angles in a triangle are in the ratio 2 : 3 : 4. What size is each angle?

8 Real Potting compost is made using loam, peat and sand in the ratio 7 : 3 : 2.

Molly uses 2 1 _ 2 litres of sand.

a How much loam and peat should she use?

b What is the total volume of the compost?

9 Sunil and Karl shared some money in the ratio 3 : 5.

Sunil gave 1 _ 6 of his money to Lucas.

Lucas got £12.50.

How much money was shared originally by Sunil and Karl?

Q3 strategy hint Write your fi nal answer clearly ‘__ bottle is the best value for money.

Exam-style question

STEM Here is a list of ingredients for making cherry scones.Chen wants to make 20 cherry scones.

a Work out how much milk he will need.

Sophie has 80 grams of sugar and 300 grams of fl our.She has plenty of the other ingredients.

b What is the greatest number of cherry scones she can make?You must show all your working.

(5 marks)Nov 2013, Q5, 5MB2H/01

Makes 8 cherry scones

200 grams fl our

60 grams margarine

40 grams sugar

60 grams cherries

160 ml milk

Exam hintMost of the marks are for your method so write down all the steps of the working.

Q9 strategy hint Start with £12.50. How much did Sunil have?

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10 Modelling 3 boys did some gardening. They earned £129.60.They shared the money in the ratio of the number of hours they each worked.Ollie worked for 9 1 _ 2 hours, Sam worked for 8 1 _ 4 hours and Peter worked for 6 1 _ 4 hours.How much money did each boy receive?

11 Modelling Adam is tiling a room. He can fi t 80 tiles in 1 hour.He takes a 15-minute morning break and 3 _ 4 hour for lunch.He has to fi t 540 tiles. He starts work at 8:30 am. At what time will he fi nish work?

12 A box contains 480 counters.There are twice as many red counters as yellow counters. There are three times as many blue counters as red counters.The rest of the counters are green.The box contains 50 yellow counters.How many green counters are in the box?

13 The sizes of the angles in a quadrilateral are in the ratio 2 : 4 : 5 : 7.

What size is the largest angle?

14 Reasoning / Problem-solving Sasha wants to buy a new camera.

She can either buy it from a local dealer or order it from the USA online.

Local dealer £399

Imported from USA $529 plus taxes and duties of 20%

Both prices include postage

The exchange rate is £1 = $1.68. She wants to pay the cheaper total cost.

Which option should she choose?

15 The chart shows the proportions of sand and cement needed to mix mortar for brickwork.A builder uses 78 kg of cement to make his mortar.How much sand should he use?

Q15 hint Write down the ratio of sand : cement.

16 50 people go on a trip to the cinema. The ratio of adults to teenagers on the trip is 2 : 3.30 per cent of the adults are senior citizens. The table shows the ticket prices.What is the total cost for all 50 people to visit the cinema?

Q12 strategy hint Start with the 50 yellow counters. What can you work out next?

80°

280°

sand

Key:

cement

Adult £8.45

Teenager £6.75

Senior or child £6.35

11 Knowledge check

◉ A ratio is a way to compare two or more quantities. Mastery lesson 11.1

◉ You simplify a ratio by making the numbers as small as possible. Divide the numbers in the ratio by their highest common factor (HCF). Mastery lesson 11.1

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◉ Ratios in their simplest form only have whole numbers. Mastery lesson 11.2

◉ You can use ratios to convert between units. Mastery lesson 11.3

◉ A proportion compares a part with the whole. Mastery lesson 11.5

◉ You can compare ratios by writing them as unit ratios. In a unit ratio, one of the numbers is 1. Mastery lesson 11.5

◉ In the unitary method you fi nd the value of one item before fi nding the value of more. Mastery lesson 11.6

◉ You can use the unitary method to work out which product gives better value for money. Mastery lesson 11.6

◉ When two values are in direct proportion, if one value is zero so is the other. When one value doubles, so does the other. Mastery lesson 11.7

◉ When two quantities are in direct proportion, plotting them as a graph gives a straight line through the origin. Mastery lesson 11.7

◉ When two values are in inverse proportion, one increases at the same rate as the other decreases. For example, as one doubles (×2) the other halves (÷2). Mastery lesson 11.8

Refl ect For each statement, A, B and C, choose a score:

1 – strongly disagree; 2 – disagree; 3 – agree; 4 – strongly agree

A I always try hard in mathematics

B Doing mathematics never makes me worried

C I am good at mathematics

For any statement you scored less than 3, write down two things you could do so that you agree more strongly in the future

Refl ect

11 Unit test Log how you did on your Student Progression Chart.

1 At the 2014 Commonwealth Games, Australia won 42 silver medals and India won 30 silver metals.

Write the ratio of the number of silver medals won by Australia to the number of silver medals won by India. Give your ratio in its simplest form. (2 marks)

2 1 kg of cheddar cheese costs £6.40. How much does 100 g cost? (2 marks)

3 There are 45 lemons and limes in a box.

The ratio of the number of lemons to the number of limes is 1 : 4.

Work out the number of limes in the box. (3 marks)

4 Greg changed £475 into euros. The exchange rate was £1 = 1.2 euros.

How many euros did he get? (2 marks)

Homework, practice and support: Foundation 11 Unit test

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5 At Mini Mart, 4 tubs of ice cream cost £9.40.

At Dave’s Deli, 3 tubs of the same ice cream cost £7.50.

At which shop is ice cream better value for money? (3 marks)

6 A school with 900 students has 150 computers. Write the ratio of the number of students to the number of computers in the form m : 1. (1 mark)

7 There are 250 marbles in a tin. 40 marbles are metal, 60 are stoneware and the rest are glass or china.There are twice as many glass marbles as china marbles.

How many are a china b glass? (4 marks)

8 Sam, Jack and Ali share £45 in the ratio 2 : 3 : 4.

a What fraction does Sam get?

b How much does Ali get? (4 marks)

9 James, Isaac and Lucas share £30 in the ratio of their ages.

James is 10 years old, Isaac is 8 years old and Lucas is 7 years old.

Isaac gives a third of his share to his Dad.

How much money does Isaac have now? (4 marks)

10 Here are the ingredients needed to make a beef pie for 6 people.

a How much beef is needed to make a beef pie for 15 people? (3 marks)

b If you have 5 eggs and plenty of the other ingredients, can you make a beef pie for 8 people? (3 marks)

11 A company makes fruit drinks.

A machine fi lls crates of 12 bottles with drink. In 1 hour the machine can fi ll 18 000 bottles.

How many seconds does the machine take to fi ll a crate of 12 bottles? (4 marks)

12 The graph shows a car moving at a constant speed.

50

0

100

150

200

10 2 3 4

Distance–time graph

Time (T hours)

Dis

tan

ce (D

mile

s)

a Does the graph show distance and time in direct proportion? Explain your answer. (1 mark)

b Write down the equation of the line. (1 mark)

Beef pie (serves 6 people)180 g fl our

320 g beef

80 g butter

4 eggs

160 ml milk

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13 3 machines can print a batch of leafl ets in 2 hours.

How long would it take 4 machines to print the same batch of leafl ets? (3 marks)

14

Sample student answers

Which student gives the best answer and why?

Student A Farm shop: £9 for 12.5 kg

Supermarket: £1.83 for 2.5 kg

£9.15 for 12.5 kg�5 �5

The potatoes are be� er value at the farm shop.

Student B 12.5 ÷ 2.5 = 5 £9 ÷ 5 = £1.80

Student C Farm shop: 12.5 ÷ 9 = 1.388… kg for £1Supermarket: 2.5 ÷ 1.83 = 1.366 kg… for £1

The potatoes are be� er value at the supermarket.

Exam-style question

Each day a company posts some small letters and some large letters.

The company posts all the letters by fi rst class post.

The tables show information about the cost of sending a small letter by fi rst class post and the cost of sending a large letter by fi rst class post.

Small letter Large letter

Weight First class post Weight First class post0–100 g 60p 0–100 g £1.00

101–250 g £1.50

251–500 g £1.70

501–750 g £2.50

One day the company wants to post 200 letters.

The ratio of the number of small letters to the number of large letters is 3 : 2

70% of the large letters weigh 0–100 g.

The rest of the large letters weigh 101–250 g.

Work out the total cost of posting the 200 letters by fi rst class post. (5 marks)

Nov 2013, Q11, 1MA0/1H

Exam-style question

Potatoes cost £9 for a 12.5 kg bag at a farm shop.The same type of potatoes cost £1.83 for a 2.5 kg bag at a supermarket.

Where are the potatoes the better value, at the farm shop or at the supermarket? You must show your working. (4 marks)

June 2012, Q23, 1MA0/2F

Exam hintRemember that there are marks for showing your method and for communicating your answer clearly, as well as for getting the answer right.

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Documents

Edexcel GCSE (9-1)