# Ratio Problems

## Topic Overview

The mathematical term '**ratio**' defines the relationship between two numbers of the same kind. The relationship between these numbers is expressed in the form "**a to b**" or more commonly in the form:

**a : b**

A ratio is used to represent how much of one object or value there is in relation to another object or value. For example:

If there are 10 apples and 5 oranges in a bowl, then the ratio of apples to oranges would be 10 to 5 or 10:5. This is equivalent to 2:1.

In contrast, the ratio of oranges to apples would be 1:2.

## Key Concepts

In the new linear GCSE Maths paper, you will be required to solve various mathematical problems involving ratios. The specific questions you will be expected to answer will vary depending upon which examination board with which you are registered, but as a rule you will be required to:

- Use ratio notation; including reduction to its simplest form and its various links to fraction notation
- Divide a quantity in a given ratio

Listed below are a series of summaries and worked examples to help you solidify your knowledge about ratios.

## Worked Examples

**1 - Dividing in a ratio**

Without realizing, you use ratios every day in order to divide and share out amounts fairly. As a result, there will be questions within your GCSE maths exam where you will be required to use ratios in order to share out amounts of money or other items:

*Example*

(a) - Tom and Stacy received a £200 cash prize in the ratio 2:3. Calculate the amount of the cash prize which Tom receives, and the amount which Stacy receives.

*Solution*

(a) - Firstly, you need to find the total number of parts in the ratio. You can do this by adding up the number values in the ratio to get a total.

For the ratio 2:3;

2 + 3 = 5 .

This means that you need to share the money into 5 equal parts.

Now you need to calculate the amount which one part will receive. In order to do this you need to divide the total amount of money being shared by the total number of parts in the ratio:

£200/ 5 = £40

Using this value, you can now calculate the share which each individual receives. To do this, you individually multiply each number in the ratio by the amount you have calculated for one part:

2 x 40 = £80

3 x 40 = £120

Therefore, out of the total cash prize, Tom receives £80 and Stacy receives £120

(Note: You can check your answers are correct by adding up the individual shares. If they add up to your original total, you know they are correct. For instance, Tom's share of £80 and Stacy's share of £120 both add up to make the original cash prize total of £200).

**2 - Equivalent ratios**

Equivalent ratios are ratios which all have the same meaning. For example :

1:4 , 2:8 , 10:40 , 2000:8000

All of these ratios have the same meaning: that the amount of variable 'b' is 4 times the amount of variable 'a'.

You can calculate equivalent ratios by multiplying or dividing both sides by the same number. In this way, ratios are very similar to fractions:

- Both ratios and fractions can be simplified by finding common factors. As with fractions, you should aim to divide by the highest common factor when simplifying ratios.
- Like a fraction, a ratio is in its simplest form when both sides are whole numbers and there is no whole number by which both sides can be divided.

*Example*

(a) - There are 21 boys and 18 girls in a classroom. Calculate the ratio of boys to girls and present your answer in its simplest form.

*Solution*

(a) - The ratio of boys to girls is 21:18

The highest common factor of 21 and 18 is 3

If you divide both sides by 3, the equivalent ratio is 7: 6

Therefore, the simplest form of this ratio is 7:6, meaning that there are 7 boys in the classroom for every 6 girls.

**3 - Scaling ratios**

By multiplying and dividing, you can use ratios to scale various objects. For example:

The height to width ratio of a piece of fabric is 2:3.

This means that, for every 2 units of height, there must be 3 units of width. Consequently, if the piece of fabric was extended to be 20m high, it must be 30m wide.

When scaling ratios up or down, always remember that the same unit of measurement must be applied to both sides; i.e. millimetres, centimetres or metres .

Moreover, the original ratio rules must be upheld. For example, if the piece fabric was made 80mm high, its width must be of the same unit of measurement and retain the rules of the ratio 2:3. As a result, the piece of fabric must be 120mm wide.

**4 - Writing a ratio in the form 1:n or n:1**

As well as being able to write a ratio in its simplest form, you must also be able to write a ratio in the form:

1:n

or

n:1

where 'n' can be any whole number, fraction or decimal.

In order to write a ratio in the form '1:n', you must make the left-hand side of the ratio equal to 1.

Alternatively, in order to write a ratio in the form 'n:1', you must make the right-hand side of the ratio equal to 1.

*Example*

(a) - Write the ratio 4:20 in the form 1:n

*Solution*

(a) - For the ratio 4:20 , you can make the left-hand side of the ratio equal to 1 by dividing both sides of the ratio by 4:

4:20 = 4/4 : 20/4 = 1:5

Therefore, the ratio 4:20 can be written in the form 1:5

*Example*

(b) - Write 4:20 in the form n:1

*Solution*

(b) - For the ratio 4:20 , you can make the right-hand side of the ratio equal to 1 by dividing both sides of the ratio by 20:

4: 20 = 4/20 : 20/20 = 0.2:1

Therefore, the ratio 4:20 can be written in the form 0.2:1

## Exam Tips

- When dividing in a ratio, remember to check your answers are correct by adding up the individual parts of the whole. If they add up to your original total, you know they are correct.
- Remember that equivalent ratios are ratios which all have the same meaning.
- Bear in mind that ratios and fractions can both be simplified by finding common factors. As with fractions, you should aim to divide by the highest common factor when simplifying ratios.
- A ratio is in its simplest form when both sides are whole numbers and there is no whole number by which both sides can be divided.
- When scaling ratios up or down, always remember that the same unit of measurement must be applied to both sides. Also bear in mind that the rules of the original ratio must be upheld.
- In order to write a ratio in the form '1:n', you must make the left-hand side of the ratio equal to 1. Alternatively, in order to write a ratio in the form 'n:1', you must make the right-hand side of the ratio equal to 1.
- Write down ALL of your working out in order to earn maximum method marks.

## Topic Summary

Although ratio problems may appear complex at first, with practice you will find that they are relatively simple to solve. When tackling ratio problems, it is advisable that you revise the main principles of 'Arithmetic with Fractions'. This is because fractions and ratios share many fundamental properties. Once you understand these properties, you can use ratios to solve various real-world problems.

Furthermore, when tackling ratio problems, it is always useful to write down all of your working out and double check your answers. Add up the values you have calculated for the ratio parts and if they make the original total value outlined in the question, then you will know you have answered the question correctly. This process will enable you to ensure you have earned the maximum amount of marks possible for the examination section on ratios, and will also help you develop confidence in your own ability to solve both simple and complex ratio problems.

## Related Topics

- Repeated Proportional Change
- Proportion: Numerical Problems
- Proportion: Algebraic Problems
- Problem Solving with Decimals and Percentages
- Compound Measure Problems
- Pre-calculus Skills