Maths
Ratios
Ratios
A ratio is a way to compare amounts of something. Recipes, for example, are sometimes given as ratios. To make pastry you may need to mix 2 parts flour to 1 part fat. This means the ratio of flour to fat is 2 : 1.
If pastry is 2 parts flour to 1 part fat, then there are 3 parts (2 + 1) altogether. Two thirds of the pastry is flour; one third fat.
Ratios are similar to fractions; they can both be simplified by finding common factors. Always try to divide by the highest common factor. Have a look at this question.
- Question
There are 15 girls and 12 boys in a class. What is the ratio of girls to boys? Give your answer in its simplest form.
- Answer
The ratio of girls to boys is 15:12
However, both sides of this ratio are divisible by 3
Dividing by 3 gives 5:4
5 and 4 have no common factors (apart from 1).
So the simplest form of the ratio is 5:4
This means there are 5 girls in the class for every 4 boys.
Now practise ratios for yourself in this activity.
Now try some more difficult questions below.
- Question
A newspaper includes 12 pages of sport and 8 pages of TV. What is the ratio of sport to TV? Give your answer in its simplest form.
- Answer
The answer is 3:2
You can divide both sides of 12:8 by 4.
If your answer was incorrect, try to fill in the blanks here:
The ratio is 12:8.
Both of these numbers are divisible by 4.
Dividing by ? gives 3:2.
You have to be sure that the things you are comparing are measured in the same units. Look at the following problem:
- Question
Anna has 75p.
Fiona has £1.20.
What is the ratio of Anna's money to Fiona's money, in its simplest form?
- Answer
One amount is in pence, the other in pounds. We have to convert Fiona's amount into pence first:
£1.20 = 120p.
Now the ratio is 75:120. Both sides are divisible by 15.
Dividing both sides by 15 gives 5:8
So the ratio is 5:8.
Writing a ratio in the form 1:n or n:1
We have already covered that to write a ratio in its simplest form we divide both sides by their highest common factor (just as we divide the top and bottom of a fraction).
For example, 12:15 becomes 4:5, and 4:8 becomes 1:2.
When a ratio is in its simplest form, all the numbers are whole numbers.
However, it is sometimes useful to write a ratio in the form 1:n or n:1 (where n is any number, possibly a fraction or decimal). This means we will not necessarily be dealing with whole numbers.
For example, if we are asked to write the ratio 2:5 in the form 1:n, we need to make the left-hand side of the ratio equal to 1. We do this by dividing both sides of the ratio by 2.
2:5 = 2/2 : 5/2
= 1 : 2.5
If we were asked to write 2:5 in the form n:1, we would need to make the right-hand side equal to 1. So we would divide both sides by 5:
2:5 = 2/5 : 5/5= 0.4 : 1
- Question
Write the ratio 6:9 in:
- a) the form 1:n
- b) the form n:1
- Answer
a) Did you get 1 : 1.5? You divide both sides by 6.
b) You should have divided both sides by 9 to get 0.6667 : 1
or written as a fraction this would be 2/3 : 1
Using ratios
Ratios can be used to solve many different problems - for example, with recipes, scale drawings or map work. A typical question will expect you to change a ratio - the reverse of cancelling down.
- Question
Sam does a scale drawing of his kitchen. He uses a scale of 1:100. He measures the length of the kitchen as 5.9m.
How long is the kitchen on the scale drawing? Give your answer in mm.
- Answer
The answer is 59mm.
You need to convert 5.9m to mm, then divided by 100 to give the answer.
If you did not get the correct answer, remember that the scale of 1:100 means that the real kitchen is 100 times bigger than the scale drawing.
5.9m = 590cm (multiplied by 100) = 5900mm (multiplied by 10)
So the scale drawing would be 5900 ÷ 100 = 59mm.
Another typical question will give you a recipe to modify for a different number of people.
- Question
A recipe to make lasagne for 6 people uses 300 grams of minced beef. How much minced beef would be needed to serve 8 people?
- Answer
The best way to approach a problem like this is to find out how much 1 person would need first.
- Six people need 300g.
- So 1 person needs 50g (300 ÷ 6).
- So 8 people need 50g × 8 = 400g.
Try this one.
- Question
A recipe for flapjacks requires 240g of oats. This makes 18 flapjacks.
What quantity of oats is needed to make 24 flapjacks?
- Answer
The answer is 320g.
You divide 240 by 18, then multiply the answer by 24.
If you had problems working out the answer, try to fill in the blanks here:
- Eighteen flapjacks need 240g.
- So 1 flapjack needs ?g (240 divided by 18).
- So 24 flapjacks need 13.3333 × ? = 320g.
Dividing in a ratio
Ratios are also used when dividing up amounts. The basic method is:
- Simplify the ratio, if possible (not essential, but it makes life easier in the long run).
- Add the numbers in the ratio together (to get the total number of parts needed).
- Divide the amount by this total number of parts.
- Multiply the answer by each of the numbers in the ratio.
- Question
Amit is 12 years old. His brother, Arun, is 9.
Their grandfather gives them £140, which is to be divided between them in the ratio of their ages. How much does each of them get?
- Answer
The ratio of their ages is 12:9
We can simplify this. Dividing by 3 gives 4:3.
So Amit gets 4 parts, and Arun gets 3.
This means that the money has to be divided into 7 parts (4 + 3).
£140 ÷ 7 = £20, so 1 part is £20.
- Amit gets 4 parts: 4 × £20 = £80.
- Arun gets 3 parts: 3 × £20 = £60.
(Check that they add to make the total amount: £80 + £60 = £140.)
A ratio can have more than 2 numbers in it. Here is another typical question.
- Question
The angles in a triangle are in the ratio 1:2:9. Find the size of the largest angle.
- Answer
1 + 2 + 9 = 12 so in total, there are 12 parts.
The angles in a triangle add up to 180 degrees.
So 1 part is 180 ÷ 12 = 15
The largest angle in the ratio is the 9.
9 × 15 = 135 degrees.
Try this one.
- Question
In a certain town, the ratio of left-handed people to right-handed people is 2:9. How many right-handed people would you expect to find in a group of 132 people?
- Answer
Did you get the answer 108?
You divide 132 by 11, and multiply by 9.
If you did not get the correct answer, see if you can fill in the blanks here:
2 + 9 = 11
132 ÷ 11 = ?
9 × ? = 108 people.
Direct proportion
Two quantities are in direct proportion when they increase or decrease in the same ratio. For example, you could increase something by doubling it, or decrease it by halving.
Here is a typical problem:
- Question
Twelve pencils cost 72p. Find the cost of 30 pencils.
- Answer
To solve this problem, we need to know the cost of one pencil.
We know that 12 pencils cost 72p, so if we divide 72 by 12 to give us the cost of one pencil:
72 ÷ 12 = 6
So 1 pencil costs 6p. Now we need to know the cost of 30 pencils. We multiply 6p by 30.
6 × 30 = 180p.
So 30 pencils cost £1.80.
If you had a problem working out the answer, the basic method to remember is to divide by how many you know, then multiply by what you want to know.
Try this one:
- Question
Jenny buys 15 felt-tip pens. It costs her £2.85. How much would 20 pens have cost?
- Answer
The answer is £3.80.
You divide 2.85 by 15, then multiply the answer by 20.
If you had problems working out the answer, see if you can fill in the blanks here:
- 15 pens cost £2.85
- 1 pen costs £2.85 ÷ ? = £0.19
- So 20 pens would cost £0.19 × ? = £3.80.
Now try a Test Bite