Hundreds of years ago, it was traditional to put dragons on maps in places that were unknown, dangerous or poorly mapped. Ratios are one of those places…
A ratio is just a model of the real world, shown in the lowest terms, but answering ratio questions can be just as scary as meeting dragons if you don’t know what you’re doing. The key to understanding ratios is to work out the scale factor. This is just like the scale on a map. If a map is drawn to a scale of 1:100,000, for instance, you know that 1cm on the map is the same as 100,000cm (or 1km) in the real world. To convert distances on the map into distances in the real world, you just need to multiply by the scale factor, which is 100,000 in this case. (You can also go the other way – from the real world to the map – by dividing by the scale factor instead.)
To work out the scale factor in a Maths question, you need to know the matching quantities in the real world and in the model (or ratio). Once you know those two numbers, you can simply divide the one in the real world by the one in the ratio to get the scale factor. For example:
If Tom and Katie have 32 marbles between them in the ratio 3:1, how many marbles does Tom have?
To answer this question, here are the steps to take:
- Work out the scale factor. The total number of marbles in the real world is 32, and the total in the ratio can be found by adding the amounts for both Tom and Katie, which means 3 + 1 = 4. Dividing the real world total by the ratio total gives 32 ÷ 4 = 8, so the scale factor is 8.
- Multiply the number you want in the ratio by the scale factor. If Tom’s share of the marbles in the ratio is 3, then he has 3 x 8 = 24 marbles.
The matching numbers in the real world and the ratio are sometimes the totals and sometimes the individual shares, but it doesn’t matter what they are. All you need to do is find the same quantity in both places and divide the real world version by the ratio version to get the scale factor. Once you’ve done that, you can multiply any of the ratio numbers to get to the real world number (or divide any real world number to get to the ratio number). Different questions might put the problem in different ways, but the principle is the same.
One complication might be having two ratios that overlap. In that case you need to turn them into just one ratio that includes all three quantities and answer the question as you normally would. For example:
If there are 30 black sheep, and the ratio of black to brown sheep is 3:2, and the ratio of brown to white sheep is 5:4, how many white sheep are there?
This is a bit more complicated, but the basic steps are the same once you’ve found out the ratio for all three kinds of sheep. To do this, we need to link the two ratios together somehow, but all the numbers are different, so how do we do it? The answer is the same as for adding fractions with different denominators (or for solving the harder types of simultaneous equations, for that matter): we just need to multiply them together. If we were adding fifths and halves, we would multiply the denominators together to convert them both into tenths. Here, the type of sheep that is in both ratios is the brown one, so we simply have to make sure the numbers of brown sheep in each ratio (2 and 5) are the same by multiplying them together (to give 10). Once we’ve done that, we can combine the two ratios into one and answer the question. Here goes:
- Ratio of black sheep to brown sheep = 3:2
Multiply by 5
- Ratio of black sheep to brown sheep = 15:10
- Ratio of brown to white sheep = 5:4
Multiply by 2
- Ratio of brown to white sheep = 10:8
- Therefore, ratio of black sheep to brown sheep to white sheep = 15:10:8
Now that we have just one ratio, we can answer the question by following exactly the same steps as before:
- Work out the scale factor. The total number of black sheep in the real world is 30, and the total in the ratio is 15. Dividing the real world total by the ratio total gives 30 ÷ 15 = 2, so the scale factor is 2.
- Multiply the number you want in the ratio by the scale factor. If the number of white sheep in the ratio is 8, then there are 8 x 2 = 16 white sheep.
Here are a few practice questions:
- One hundred paintings have to be selected for an art exhibition. If the ratio of amateur paintings to professional paintings has to be 2:3, how many amateur paintings and professional paintings have to be selected?
- The ratio of brown rats to black rats is 3:2. If there are 16 black rats, how many brown rats are there?
- Peter has 20 blue pens. How many red pens must he buy if the ratio of blue to red pens has to be 2:3?
- There are 35 children in a class and 15 are boys. What is the ratio of boys to girls?
- 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.
- 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.
- Anna has 75p, and Fiona has £1.20. What is the ratio of Anna’s money to Fiona’s money in its simplest form?
- 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.
- 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?
- A recipe for flapjacks requires 240g of oats. This makes 18 flapjacks. What quantity of oats is needed to make 24 flapjacks?
- 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?
- The angles in a triangle are in the ratio 1:2:9. Find the size of the largest angle.
- 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?
- Twelve pencils cost 72p. Find the cost of 30 pencils.
- Jenny buys 15 felt-tip pens. It costs her £2.85. How much would 20 pens have cost?
- If three apples cost 45p, how much would five apples cost?
- Sam is 16 years old. His sister is 24 years old. What’s the ratio of Sam’s age to his sister’s age? Give your answer in its simplest form.
- A map scale is 1:20000. A distance on the map is measured to be 5.6cm. What’s the actual distance in real life? Give your answer in metres.
- A recipe for vegetable curry needs 300 grams of rice, and it feeds 4 people. How much rice would be needed for 7 people?
- £60 is to be divided between Brian and Kate in the ratio 2:3. How much does Kate get?