22 calculations with ratiowestsidemathematics.wikispaces.com/file/view/ch22.pdf/...1 a in a class of...

$: 22 Calculations with ratiowestsidemathematics.wikispaces.com/file/view/ch22.pdf/...1 a In a class of 35 pupils 21 are boys. What fraction of the class are girls? b What fraction of$
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Using mathematics: real-life applications

Converting between different currencies, working out which packet of crisps is the best value for money, mixing large quantities of cement and scaling up a recipe to cater for more people all involve reasoning using ratios.

Before you start …

Ch 10 You need to be able to identify and simplify fractions.

1 a In a class of 35 pupils 21 are boys. What fraction of the class are girls?

b What fraction of this shape is shaded? Write your answer in its simplest form.

Ch 10 You need to be able to find a fraction of a quantity.

2 Find 2 __ 3

of 42.

Ch 10 You need to be able to find an original amount given a fraction.

3 There are 51 parents of students in the audience at a school play.

These parents make up 3 __ 4

of the audience. How many people are in the audience?

“Every day customers bring me paints to match. I have to understand how changing the ratio of base colours affects the colour of the paint and how to scale the quantities up and down for larger or smaller amounts of paint. If I get it wrong, customers will have patches of different colours and their walls would look quite strange.” (Paint technician)

For more resources relating to this chapter, visit GCSE Mathematics Online.

In this chapter you will learn how to …

• work with equivalent ratios.

• divide quantities in a given ratio.

• identify and work with fractions in ratio problems.

• apply ratio to real contexts and problems, such as those involving conversion, comparison, scaling, mixing and concentrations.

22 Calculations with ratio

402

GCSE Mathematics for OCR (Higher)

Assess your starting point using the Launchpad

Go toSection 1: Introducing ratios

Go toSection 2: Sharing in a given ratio

✓

✓

Go toSection 3: Comparing ratios

Go to

Chapter review

6 Order the following paints from lightest to darkest shade.

Nectarine night 3 : 2 red to yellow

Satsuma delight 8 :15 red to yellow

Amber 24 : 30 red to yellow

7 The cost of hiring a van in terms of hours to cost in pounds is in the ratio 1 : 11, draw a graph to show the relationship. What kind of relationship is it?

STEP 3

STEP 1

1 Write the ratio 12 : 21 in its simplest form.

2 In a class of 14 girls and 16 boys what is the ratio of boys to girls?

3 In every 80 minutes of television broadcast, a quarter of an hour of adverts is shown and the rest is programming. What is the ratio of adverts to programming?

STEP 2

4 Share 35 in the ratio 2 : 5.

5 The dry ingredients for chocolate brownies are dark chocolate, cocoa powder, plain flour, caster sugar and muscovado sugar in the ratio 17 : 5 :17 : 20 :10. I have 85 grams of dark chocolate. What weight of dried mixture can I make?

✓

© Cambridge University Press. This document is for personal use in accordance with our terms and conditions: gcsemaths.cambridge.org/terms



Section 1: Introducing ratios

Many colours of paint can be mixed from the four base colours: blue, yellow, red and white.

Think about mixing green paint. You would need to know which base colours to mix. You would also need to know how much of each colour to mix to get dark green or light green. The amount of each colour is important for getting the same shade of green each time you make it.

Paint technicians can mix the same shade of green over and over by mixing yellow and blue paints in a particular ratio.

Artists and designers use a special chart with thousands of numbered shades of colours to make sure they get the exact shade they want. The number allows the colour to be mixed using the correct ratio of base colours.

Ratio describes how parts of equal size relate to each other. A ratio of yellow to blue paint of 1 : 3 means one unit of yellow for every three units of blue. This gives a very dark green.The order in which a ratio is written is important. A ratio of 2 : 5 means 2 parts to 5 parts. Each part is equal in size.

A ratio of yellow to blue paint of 5 : 1 means five units of yellow for every one unit of blue. This would give a much lighter green.

The diagram shows a ratio of yellow to blue of 3 : 9.

Dividing by three simplifies the ratio of 3 : 9 to give 1 : 3.

Mixing paint in the ratio 3 : 9 would give the same colour as mixing it in the ratio 1 : 3 because the colours are mixed in the same ratio. The yellow paint makes up the same proportion of the mix in both cases.

The ratios 3 : 9 and 1 : 3 are equivalent ratios.

The difference between ratio and proportion

A ratio compares two or more quantities with each other. A proportion compares a quantity to the ‘whole’ of which it is a part.

For example, in the dark green paint mixture, the ratio of yellow paint to blue paint is 3 : 9 or 1 : 3. The proportion of yellow paint in the dark green paint

mixture is 3 ___ 12

, 1 __ 4

or 25%.

ratio: the comparison between two or more amounts in relation to each other.

Key vocabulary

With many ratio questions, drawing a picture of the situation can help you work it out.

Tip

3

1

:

:

9

34343

proportion: a comparison of a part, or amount, to the whole; often expressed as a fraction, percentage or ratio.

equivalent: having the same value; two ratios or fractions are equivalent if one is a multiple of the other because they will cancel to the same simplest term.

Key vocabulary

The ‘whole’ of a ratio 3 : 9 is 3 1 9 5 12; the ‘whole’ of the ratio 1 : 3 is 1 1 3 5 4.

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EXERCISE 22A

1 36 girls, 45 boys and 9 teachers went on a school trip.

a What is the ratio of boys to girls?

b What is the ratio of pupils to teachers?

c What is the ratio of pupils to people on the trip?

d The school policy is that each teacher can be responsible for no more than 10 pupils. Does this trip meet this requirement?

2 Look at each diagram. What is the ratio of shaded squares to unshaded squares in each? Write the answers in simplest form.

a b c

3 Look at each diagram. What is the ratio of shaded squares to total squares in each? Write the answers in simplest form.

a b c

4 The ratio of shaded to unshaded squares in this diagram is 1 : 3. How many more squares need to be shaded to make the ratio 2 : 3?

5 The distance between the post office and the bank on the local high street is represented as 5 cm on a map. In real life this distance is 20 m. What is the scale of the map (as a ratio)?

6 On a scale drawing of a cruise ship, a cabin is 8 cm from the restaurant. On the actual ship the distance is 76 m. Express the distances as a ratio.

7 A natural history programme lasts 90 minutes. The crew recorded 60 hours of footage. What is the ratio of footage used to footage recorded?

8 a Use the diagram to find the ratio of:

i side AB to side AC. ii side EB to side DC. iii side AE to side AD.

b What does this tell you about triangles ABE and ACD? c On this basis, what is the ratio of the angle AEB to angle EDC?

Think about what the ratio would have been before it was simplified to 1 : 3, and how many parts there are in the whole.

Tip

Ratios do not include units. To compare measured amounts you need to make sure they are written in the same units.

Tip

8 cm

4 cm 5 cm

18 cm

12 cm

10 cm

A

E

D C

B

You will learn more about similarity in Chapter 29, but for now think back to work you have done in earlier school years.

Tip




9 A jam recipe uses 55 g of fruit for every 100 g of jam. The rest is sugar. What is the ratio of fruit to sugar?

10 An adult ticket for the cinema is one and a half times the price for a child’s ticket. What is the ratio of the price of an adult ticket to the price of a child’s ticket? Write the ratio in its simplest form.

11 According to recent statistics 3 __ 5

of 16 year olds have a mobile phone.

What is the ratio of 16 year olds with mobiles to those without?

12 After an increase of 20% in the number of boys in a school, the ratio of boys to girls is 3 : 4. If there are now 630 pupils in the school, how many boys were there originally?

13 If 1 __ 5

of chocolates in a box are dark chocolate, 1 __ 2

are milk and the rest are

white, what is the ratio of dark : milk : white chocolate?

Section 2: Sharing in a given ratio

Often you will be given a ratio and asked to share an amount using that ratio.

For example, a group of three office workers form a lottery syndicate. Together they buy eight lottery tickets a week.

Simon pays £1 a week, Oliver £3 and Lucy £4. They win £32 000.

Should they each get an equal share of the winnings? If not, how should they share their winnings? What is the fairest way?

The fairest way would be for each member to receive winnings in the same ratio as they bought tickets.

The winnings should therefore be distributed between Simon, Oliver and Lucy in the ratio 1 : 3 : 4. This can be represented using a diagram, where each box represents the number of parts of the whole each individual should receive.Simon Oliver Lucy Every box has to have the same quantity in it. In

total we have 8 boxes (1 1 3 1 4), in which we have to share £32000.

Each box gets £32 000 4 8 5 £4000.

So: Simon receives £4000.Oliver receives 3 3 £4000 5 £12 000.Lucy receives 4 3 £4000 5 £16 000.

You can also think in terms of fractions and use what you know about finding fractions of a quantity.

1 : 3 : 4 gives 8 parts, so each person gets the following:

Simon receives 1 __ 8

3 32 000 5 £4000.

Oliver receives 3 __ 8

3 32 000 5 £12 000.

Lucy receives 4 __ 8

3 32 000 5 1 __ 2

3 32 000 5 £16 000.

The final step is to double check that the shared quantities sum to the original amount:

£4000 1 £12 000 1 £16 000 5 £32 000

Ratios can include decimal numbers, but not when they are written in their simplest form; the simplest form always uses integers.

Tip

The box method shown in the example is useful for working out shares in a given ratio problems.

Tip

You learnt how to find a fraction of a quantity in Chapter 10.

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EXERCISE 22B

1 Share 144 in each of the given ratios.

a 1 : 3 b 4 : 5 c 11 : 1

d 2 : 3 : 1 e 1 : 2 : 5 f 2 : 7 : 5 : 4

2 To make mortar you mix sand and cement in the ratio of 4 : 1.

a How much sand is needed to make 25 kilograms of mortar?

b What fraction of the mix is cement?

3 The first bi-colour £2 coin was issued in 1998. The inner circle is made of cupronickel. This is copper and nickel in the ratio 3 : 1. The inner circle weighs 6 grams. How much copper is used to make the centres of ten £2 coins?

4 Flaky pastry is made by mixing flour, margarine and lard in the ratio 8 : 3 : 3 and then adding a drizzle of cold water.

a How much of each ingredient is needed to make 350 g of pastry?

b What fraction of the pastry does the margarine and lard make together?

5 The sides of a rectangle are in the ratio of 2 : 5. Its perimeter is 112 cm.

a What are the dimensions of the rectangle?

b Use these dimensions to calculate its area.

6 Orange squash is made by mixing one part cordial to five parts of water. How much squash can you make with 750 ml of cordial?

7 Two-stroke fuel is used to power small engines. It is produced by mixing oil and petrol in the ratio of 1 : 20. How much oil needs to be mixed with 10 litres of petrol to make two-stroke fuel?

8 Tiffin is a sweet made by crushing biscuits and mixing them with dried fruit, butter and cocoa powder. The ratio of biscuit to dried fruit to butter to cocoa powder is 5 : 6 : 2 : 2. How much of each ingredient is needed to make 600 g of tiffin?

9 In a music college, the ratio of flute to oboe to string to percussion players is 7 : 2 : 15 : 1. If the college has 175 students. How many play an oboe?

10 The ratio of red to green to blue to black to white pairs of socks in a drawer is 2 : 3 : 7 : 1 : 4. If there are 8 pairs of white socks, how many pairs are there altogether?

11 Potting compost is made by mixing loam, peat and sand in the ratio of 7 : 3 : 2. If a gardener has 4.5 kg of peat and plenty of loam and sand, how much potting compost can she make?




Section 3: Comparing ratios

It is often useful to write ratios in the form 1 : n (where n represents a number) so that they are in the same form and you can compare them directly by size.

WORKED EXAMPLE 1

Red and white paints can be mixed to make pink paint.

Which of the mixes below will give the lightest shade of pink?

A B C

The ratios of red to white paint are: A 4 : 3 B 3 : 2 C 6 : 4

A 4 __ 4

: 3 __ 4

5 1 : 0.75 B 3 __ 3

: 2 __ 3

5 1 : 0.67 C 6 __ 6

: 4 __ 6

5 1 : 0.67.

Paint A has the greatest amount of white paint per unit of red paint, i.e. 0.75 tins of white for 1 tin of red, so this will be the lightest shade of pink.

First, work out the ratio of red to write paint in each diagram.

Change these to form 1 : n. For each diagram, divide both parts of the ratio by the first part. Give the answers as decimals to make the comparison simpler.

Ratios in the form of 1 : n are also useful for converting from one unit to another. For example, the ratio of inches to centimetres is 1 : 2.54.

This means that 1 inch is equivalent to 2.54 cm.

So, 2 inches 5 2 3 2.54 cm and 12 inches 5 12 3 2.54 cm.

This is a linear relationship and it can be shown as a straight-line graph. The equation of the line is y 5 2.54x. Notice that the ratio of inches to centimetres (the ratio of x : y) is 1: 2.54. The equation y 5 2.54x and the ratio 1: 2.54 both state that for every value of x, y is 2.54 times larger.

The scale of maps is given as a ratio in the form of 1 : n. For example, 1 : 25 000.

Did you know?

Ratios in the form of 1 : n can also be written as n : 1.

Tip

You learnt about linear graphs in Chapter 18.

Tip

20

2

0

4

6

8

10

12

cm

inches

14

16

18

20

22

24

26

28

30

4 6 8 10

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Golden ratio

The golden ratio, 1 : 1 1 √ __

5 ______ 2

5 1 : 1.618 has been studied and used for

centuries. Artists, including Leonardo Da Vinci and Salvador Dali often produced work using this ratio. The ratio can also be seen in buildings, such as the Acropolis in Athens. The golden ratio is said to be the most attractive way to space out facial features.

The diagram shows how the golden ratio can be worked out using the dimensions of a ‘golden’ rectangle. The large rectangle ACDF is similar to BCDE. Hence the ratio of a : a 1 b is equivalent to b : a.

A B C

F E D

ba

a

An approximate numerical value for this ratio can be found by measuring.

WORKED EXAMPLE 2

How golden are your hands?

A

B

C

Distance B 5 22 mmDistance C 5 40 mmDistance A 5 13 mmLength of hand 5 170 mmWrist to elbow 5 240 mmB : C 5 22 : 40 5 1 : 1.81A : B 5 13 : 22 5 1 : 1.69Hand : wrist-elbow 5 170 : 240 5 1 : 1.41All the ratios are close to 1 : 1.618 which is the golden ratio.

Measure the distances A, B and C on your hand.

Now calculate these ratios and write them in the form 1 : n.

Distance B : Distance C

Distance A : Distance B

Length of your hand : Distance from your wrist to your elbow

Can you see anything special about these ratios?

The closer your results are to 1.1618 the more golden your hand!




You worked with the sequence of Fibonacci numbers in Chapter 4. This sequence follows the golden ratio. If you calculate the ratio of consecutive numbers in the Fibonacci sequence you will find that the ratio gets closer

and closer to the actual golden ratio 1 : 1 1 √ __

5 ______ 2

as you get further along the sequence.

EXERCISE 22C

1 Different types of coffee are made by mixing espresso shots, hot water and milk in specified ratios.

Espresso 1 : 0 : 0

Double espresso 2 : 0 : 0

Flat white 1 : 2 : 1

Cappuccino 1 : 0 : 2

Latte 1 : 0 : 4

Put the drinks in order of strength, weakest first.

2 When Jon was going on holiday he used this graph to convert between pounds and euros.

50

5

0

10

15

20

25

30

euros

pounds10 15 20 25 30

a What is the ratio of pounds to euros? Express this in the form 1 : n.

b What is the ratio of euros to pounds? Express this in the form 1 : n.

3 This graph shows the relationship between ounces and grams.

0

20

0

40

60

80

100

120

140

160

180

200

220

240

grams

ounces2 4 6 8

a What is the ratio of ounces to grams? Express this in the form 1 : n.

b What is the ratio of grams to ounces? Express this in the form 1 : n.

The weakest coffee has the least espresso by volume.

Tip

Ounces is an imperial measurement of mass; you worked on metric units in Chapter 12.

Tip

410


4 The ratio of litres to gallons is 1 : 1.8.

a Draw a conversion graph to show this relationship.

b What is the ratio of gallons to litres in the form 1 : n?

5 Three siblings; Daisy, aged 5, Patrick, 8, and Imogen, 12, share sweets in the same ratio as their ages. Imogen gets 21 more sweets than Daisy.

a How many sweets were there to begin with?

b What fraction of the sweets did Patrick get?

6 The ratio of kilometres to miles is approximately 8 : 5. A car travels at 45 miles per hour for 20 minutes. Approximately how many kilometres does it travel?

7 This recipe for sausage casserole serves 6.

Find the quantities of ingredients needed to serve 4 people. Show all the steps in your answer.

8 Gill and her sister Bell share a box of chocolate. Bell gets 1 __ 3

of the box.

Gill shares her chocolates with her best friend Katy in the ratio 4 : 3. Katy gets 12 chocolates. How many chocolates were there in the box?

9 A quarter of a box of chocolates are white chocolate. The ratio of dark to milk chocolates is 2 : 5. There are 7 white chocolates. How many more milk chocolates than dark chocolates are there?

10 A, B and C are three pulley wheels. For every 3 turns A makes, B makes 4 turns. For every 2 turns B makes C makes 3 turns.

a What is the ratio of the turns A makes to the turns B makes?

b What is the ratio of the turns C makes to the turns B makes?

c What is the ratio of the turns A makes to the turns C makes?

d Pulley wheel A makes 24 turns. How many turns does C make?

e Pulley wheel C makes 36 turns. How many turns does A make?

11 A 210 cm ribbon is cut into two sections. The longer piece is 2 and a half times the length of the shorter piece.

a What is the ratio of the longer piece to the shorter piece of ribbon?

b How long is each piece?

12 Each month a sunflower’s height increases by 40%. What was the ratio of the height of the sunflower on 1st May to its height on 1st August?

13 Which is a better deal, 325 g jar of chocolate spread for 66p or a 1 kg tub for £1.99?

14 The ratio of Moisha’s height at age 3, to her height at age 4 is 15 : 16.

a What percentage increase is this?

b During this time Moisha grew 7 cm. If Moisha keeps growing at the same rate how tall will she be when she is 8?

Sausage casserole(serves 6)

12 sausages

3 tins of tomatoes

450 g potatoes

9 tsp mixed herbs

600 ml vegetable stock




Checklist of learning and understanding

Notation

A ratio compares two or more quantities with each other.

The order in which a ratio is written is important. A ratio of 2 : 5 means 2 parts to 5 parts. Each part is equal in size.

A proportion compares a quantity to the ‘whole’ of which it is a part; this can be written as a fraction, percentage or ratio.

Simplifying ratios

Ratios can be simplified by dividing both parts of the ratio by a common factor.

Two ratios are equivalent if one is a multiple of another, i.e. if they can be cancelled to the same simplest terms.

Expressing ratios in the form 1 : n makes it easy to compare ratios. If three different ratios of a : b were converted to 1 : n and compared, then the ratio with the highest value of n has the greatest amount of b per amount of a.

Ratios can be useful for converting from one unit to another where the relationship is linear. The relationship can be represented by a straight-line graph.

Sharing in a given ratio

The box method can be used to tackle problems that involve sharing a quantity in a given ratio. To share quantity Q in the ratio a : b : c, divide the quantity evenly into a 1 b 1 c boxes.

You can also work out the fraction each part is of the whole, and multiply the quantity by the fraction.

Chapter review

1 What is the ratio of the diameter of a circle to its circumference in the form 1 : n?

2 The three angles of a triangle are in the ratio 3 : 3 : 4. What information can you give about the triangle?

3 The ratio of the five angles in a pentagon are 1 : 1 : 1 : 1 : 1, what information does this tell you about the pentagon? How do you know this?

4 The ratio of an exterior angle to an interior angle of a regular polygon is 1 : 3. How many sides does the polygon have?

5 The ratio of the angles in a triangle is 1 : 2 : 1. What information can you give about the triangle? If the longest side is 10 cm, how long are its other two sides?

For additional questions on the topics in this chapter, visit GCSE Mathematics Online.

To answer the questions about circles, triangles, pentagons and rectangles you may need to look again at Chapter 5. For the questions on area and volume you may need to look again at Chapters 16 and 21.

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412


6 a Joe and Pam planted crocus bulbs in their gardens.

They shared a bag of 250 crocus bulbs.

The table shows the colour of the flower from each bulb.

Yellow Purple White totalsJoe 64 40 125Pam 56 32 125

totals 120 250

i Complete the table. (3 marks)

ii Write the ratio 64 : 56 as simply as possible. (1 mark)

b Sumita bought a pack of 60 crocus bulbs which produced Yellow, Purple or White flowers. The ratio Yellow : Purple : White was 7 : 5 : 3.

How many of the 60 bulbs produced White flowers? (3 marks)

© OCR 2012

7 The ratio of the sides of a rectangle is 3 : 4. After an enlargement by a scale factor of 5 what is the ratio of the same two sides?

8 Gareth and John share a box of chocolates. Gareth gets 3 __ 5

of the box. The

ratio of white to milk to dark chocolates in John’s share is 1 : 2 : 1. John gets 4 white and dark chocolates in total. Gareth gets twice as many white chocolates as John, and has an equal number of dark and milk. How many types of each type of chocolate were in the box?

9 The ratio of the sides of two squares is 3 : 4. What is the ratio of their areas?

10 The ratio of the edge of two cubes is 5 : 2.

a What is the ratio of their surface areas?

b What is the ratio of their volumes?

11 What is the ratio of vowels to consonants in the English alphabet?

12 What is the ratio of prime numbers to square numbers between (and including) 1 and 20 in its simplest form?

13 Share 360 in the ratio 3 : 5 : 1.

14 Using the graph to the left express the ratio of miles to kilometres in the form 1 : n.

15 In a car park, three-quarters of the cars are not silver, but are blue, red, black or yellow. The proportion of blue to red to black to yellow cars is 6 : 2 : 3 : 1. There are six more black cars than yellow cars. How many cars of each colour are in the car park?

Don’t forget about the silver cars!

Tip

00

2

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8

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km

miles2 4 6 8


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