## Ratios and Proportion

### Introduction

Ratios occur in mixing things - such as concrete which is made of cement, sand and gravel in a definite ratio. For example, a ratio of 1:3:4 would mean that no matter what volume of concrete you have, 1 part is cement, 3 parts is sand and 4 parts is gravel. An alternative way of stating this is to say that $\frac{1}{8}$ is cement, $\frac{3}{8}$ is sand and $\frac{4}{8}$ $\left(\mbox{i.e. }\frac{1}{2}\right)$ is gravel.

The aspect ratio which is commonly used in describing the width to length ratios of aircraft wings, is also commonly seen nowadays to describe the ratio of width to height of a TV screen. "Old-style" TV screens feature a 4:3 (1.33:1) aspect ratio, but newer widescreen TVs have a 16:9 (1.78:1) ratio; and most feature films are shot in at least a 1.85:1 ratio.

### Simplification of Ratios

Conventionally ratios are stated as whole numbers, or at least decimal numbers (as used for some examples in the Introduction).

Whole number ratios would normally be stated in their lowest terms. A ratio of the form

3 : 6 : 15

can be reduced to

1 : 2 : 5

by dividing every number by 3 (note the similarity with reducing fractions to their lowest terms).

On the other hand, if a ratio was stated as

$\frac{1}{3},\ \ \frac{1}{6},\ \ \frac{1}{2}$

(which is much less common) then it would be usual to multiply every term by an appropriate number to achieve a ratio stated in whole numers only. Here we could multiply every term by 6 to get

2 : 1 : 3

or

1 : 2 : 3

To go back to the decimal representation mentioned in the first paragraph, you can see from the examples in the Introduction that when this is used then at least of one of the ratios is 1 .

This is the crux of the representation, the ratios have been divided through so that the lowest ratio is 1, and the others are either 1 or more than 1.

For example, the ratio

2 : 3 : 15
could be represented as

1 : 1.5 : 7.5

by dividing the original ratios through by 2.

You do come across ratios of this type where the numbers are quoted as decimals less than 1, contravening the rule I have just given. Mathematically, there is nothing wrong with this, but hopefully you can see that this type of represenation would not be as easy to understand.

Example

Of 200 customers of a cafe, 80 ordered and 120 ordered coffee. What is the ratio of tea drinkers to coffee drinkers ?

State the given figures as a ratio (in the right order)

80 : 120

and then reduce this to its lowest terms if applicable. Here we can see that both sides can be divided by 40, to give

2 : 3

Quick Quiz

 Simplify these ratios, keeping them in integer form $1) \ \ 3 : 6 : 9$ $2) \ \ 24 : 48 : 60$ $3) \ \ 3 : 27 : 36$ $4) \ \ 4 : 8 : 16$ Represent these ratios in decimal form $5) \ \ 4 : 7 : 12$ $6) \ \ 5 : 7 : 24$

### Calculation

Ratios are closely related to fractions, for example if two items ( A and B ) are connected in the ratio

3 : 4

then

• A will constitute

$\frac{3}{7} \ \mbox{of the whole}$

• B will constitute

$\frac{4}{7} \ \mbox{of the whole}$

Likewise, if three quantities are related in the ratio

1 : 9 : 15

then the quantities will constitute

$\frac{1}{25}, \ \ \frac{9}{25},\ \ \mbox{and } \frac{15}{25} \left(= \frac{3}{5}\right)\ \mbox{of the whole, respectively}$

Example

 If a line 10 cms long is to be divided in the ratio 3 : 5 then stated as fractions of the whole, the two lengths will be $\frac{3}{8} \ : \ \frac{5}{8}$ The two required lengths are then $\frac{3}{8} \times 10 = \frac{30}{8} = 3.75\ \mbox{cm}$ $\frac{5}{8} \times 10 = \frac{50}{8} = 6.25\ \mbox{cm}$

Example

 If £ 4.50 is to be divided between three people in the ratio 4 : 5 : 6 then stated as fractions of the whole, the three amounts will be $\frac{4}{15} : \frac{5}{15} \left(=\frac{1}{3}\right) : \frac{6}{15}$ So the required monetary amounts are then $\frac{4}{15} \times 4.50 = \frac{18}{15} = £ 1.20$ $\frac{1}{3} \times 4.50 = \frac{4.5}{3} = £ 1.50$ $\frac{6}{15} \times 4.50 = \frac{27}{15} = £ 1.80$

### Proportions

Consider a question like

A car travels for 300 kilometres on 35 litres of petrol. How far will it travel on 54 litres ?

 To explain how to do this in words - divide 300 by 35 to find how far it will travel on 1 litre multiply this figure by 54 to find how far it will travel on 54 litres

If needs be, you can start off doing calculations of this type in these two stages, but once you get more practised, you can start to do it in one step. For example, the calculations for the above question would be

$300 \times \frac{54}{35}$

= 462.9 (to 1 d.p.)

A car travels for 300 kilometres on 35 litres of petrol. How much petrol would be needed for a journey of 369 kilometres ?

Similar logic would be needed, but applied differently.

 To explain how to do this in words - divide 35 by 300 to find many litres would be required to travel for 1 kilometre (obviously this would be quite a bit less than 1 litre) multiply this figure by 369 to find how much is needed to travel 369 kilometres

As before, you can start off doing calculations of this type in these two stages, but once you get more practised, you can start to do it in one step. For example, the calculations for this question would be

$35 \times \frac{369}{300}$

= 43.05 litres

Quick Quiz
 1) 5 kgs of potatoes cost £ 2.20 - how much will 28 kgs cost? 2) If 34 items cost £ 45.67, how many will 83 items cost? 3) If a train takes 3.4 hours to travel 980 kilometres (obviously not a British train), how long will it need to travel 1 200 kilometres? 4) If a machine produces 34 items in 5 minutes, how many will it produce in 34 minutes? 5) If 56 items cost £ 120 pounds, how many can I get for £ 84?

### Inverse Proportion

Some problems have an inverse proportion.

Very common ones involve workers doing a particular job - the more workers you have, the less time the job will take (assuming an ideal situation where all workers produce exactly the same, at the same rate).

For problems like these, the technique would be the opposite to that used for 'proportions' previously. There you carried out a two-stage operation, first dividing and then multiplying. For inverse proportion, a two-stage operation is involved, first multiplying and then dividing.

Example

 4 workers build a wall in 12 days. How long would it take 7 workers ?

 To explain specifically how to do this in words - •  multiply 12 by 4 to find how long it would take 1 worker •  divide this figure by 7 to find how long it would take 7 workers

As explained in other sections, with practise you can conflate these two steps into one line of calculation

$4 \times \frac{12}{7} = 6.9\ \mbox{(to 1 d.p.)}$

Example

20 workers produce 3 000 articles in 15 days. How long would it take 13 workers ?

 Note that the figure of 3 000 articles does not enter into the calculation (previously the details about the wall that the workers were building did not enter into the calculation, apart from knowing that it had been finished). To explain specifically how to do this in words - •  multiply 15 by 20 to find how long it would take 1 worker •  divide this figure by 13 to find how long it would take 13 workers

Conflating these two steps into one line of calculation

$15 \times \frac{20}{13} = 23\ \mbox{(to nearest no. of whole days)}$

Quick Quiz
 1) If 12 workers take 11 days to harvest a crop, how long will 15 workers take? 2) When a set amount of fruit is distributed equally to 7 people, each receive 2.3 kgs. How much would each receive if the same amount had been divided among 12 people? 3) If it takes 12 workers 2 hours to dig a hole, how long would 7 workers take?

## Past Exam Questions

#### Jake is making 55 biscuits for the playgroup Christmas party. He has a recipe for 20 biscuits, which requires: 150g margarine 150g sugar 1 egg 300g self-raising flour 50g ground almonds

1. How much flour will he need to make exactly 55 biscuits?

• A    413g
• B    825g
• C    900g
• D    1 650g

2. What is the ratio of ground almonds to sugar to self-raising flour in the recipe?

• A    3 : 6 : 1
• B    6 : 1 : 3
• C    1 : 3 : 6
• D    1 : 6 : 3

#### 1. The border for the top of the walls costs £ 3.97 per metre. Which estimate is most accurate for the total cost of the border?

• A     (8 + 3) x 4 = £ 44
• B     (8 + 3 + 8 + 3) = £ 22
• C     (8 + 3) x 2 x 4 = £ 88
• D     (8 x 3) x 4 = £ 96

#### 2.    3.2 litres of paint are needed to decorate the bedroom. (1 litre = 1 000cm3). The decorator mixes 3 paint colours together. The amounts of paint are in the ratio: 15 parts of Honey Yellow to 12 parts of Aztec Orange to 5 parts of Ravishing Red. How much Ravishing Red will she need?

• A     500cm3
• B     720cm3
• C     1 067cm3
• D     1 200cm3

• A     3:8
• B     3:5
• C     5:8
• D     5:3

• A     6%
• B     12%
• C     25%
• D     125%

• A     70 000
• B     80 000
• C     90 000
• D     10 000

#### Here are the scores that 20 people get for a test

 8, 7, 5, 6, 9 4, 5, 7, 2, 1 6, 1, 9, 2, 1 5, 9, 7, 3, 8

• A     3%
• B     15%
• C     25%
• D     75%

• A     1:4
• B     2:5
• C     3:2
• D     1:3