To calculate a ratio, write the two quantities in the order the question asks for, separated by a colon, then divide both by the largest number that goes into each of them. For example, a team of 12 managers and 18 analysts has a ratio of managers to analysts of 12:18, and dividing both by 6 gives 2:3. That means there are 2 managers for every 3 analysts.
Ratio questions appear in almost every free numerical reasoning test, where you are asked to compare quantities, share a total, or scale an amount up or down. If you are new to the idea, start with our guide to what a ratio is.
How to work out the ratio of two numbers
A recruitment campaign receives 45 applications from graduates and 60 from experienced hires. What is the ratio of graduate to experienced applications?
- Write the numbers in the order asked. Graduates come first, so 45:60.
- Find the highest common factor. The largest number that divides both 45 and 60 is 15.
- Divide both sides by it. 45 ÷ 15 = 3 and 60 ÷ 15 = 4, so the ratio is 3:4.
If you cannot spot the highest common factor straight away, divide by any common factor and repeat. 45:60 divided by 5 is 9:12, and 9:12 divided by 3 is 3:4. You arrive at the same answer. Our guide on how to simplify ratios covers harder cases, including ratios with decimals and fractions.
Before you calculate, check the units match. A ratio of 30 minutes to 2 hours is not 30:2. Convert to the same unit first: 30 minutes to 120 minutes is 30:120, which simplifies to 1:4.
How to calculate a ratio of three or more numbers
The method is the same. A budget is split £24,000 on salaries, £36,000 on equipment and £60,000 on premises. The highest common factor of all three is 12,000, so the ratio 24,000:36,000:60,000 simplifies to 2:3:5.
Writing a ratio as 1 to n
Some questions ask for a ratio in the form 1:n, which makes comparisons easier. Divide both sides by the first number.
A warehouse has 84 operatives and 12 supervisors. The ratio of supervisors to operatives is 12:84. Divide both sides by 12 and you get 1:7, so there is one supervisor for every seven operatives.
The answer will not always be a whole number. A ratio of 3:4 becomes 1:1.33 when you divide both sides by 3.
Using a ratio to find a missing amount
If you know one side of a ratio and one real quantity, you can find the other quantity. Work out what one part is worth, then scale up.
A firm keeps trainees and permanent staff in the ratio 3:5. It has 27 trainees. How many permanent staff does it have?
- Match the known quantity to its parts. 27 trainees are 3 parts.
- Find one part. 27 ÷ 3 = 9.
- Scale to the other side. Permanent staff are 5 parts, so 5 × 9 = 45.
Sharing a total in a ratio
When a question gives you a total rather than one side, add the parts first.
A courier splits 4,900 parcels between three depots in the ratio 3:4:7. How many parcels go to the largest depot?
- Add the parts. 3 + 4 + 7 = 14.
- Find one part. 4,900 ÷ 14 = 350.
- Multiply by the share you need. The largest depot takes 7 parts, so 7 × 350 = 2,450.
Check by adding all three shares: 1,050 + 1,400 + 2,450 = 4,900. Our guide on how to share an amount in a ratio has more examples of this type.
Ratio questions in disguise
Many test questions are ratio questions without using the word. Look for “for every”, “per” and “out of”.
Three machines produce 7,200 units in four hours. How many units would five machines produce in six hours, working at the same rate?
- Reduce to one machine for one hour. 7,200 ÷ 3 ÷ 4 = 600 units.
- Scale back up. 600 × 5 × 6 = 18,000 units.
Reducing to a single unit first is slower to write down but much harder to get wrong.
Mistakes the answer options are built from
Multiple choice options in ratio questions usually come from the same few errors.
- Reversing the order. If the question asks for analysts to managers, the answer is 3:2, not 2:3. Read the order twice.
- Stopping at one part. In the parcel example, 350 is one part, not anyone’s share, but it will often be offered.
- Dividing by one side instead of the total. Sharing 4,900 in the ratio 3:4:7 means dividing by 14, not by 7.
- Answering for the wrong share. Check whether the question wants the largest share, the smallest, or the difference between two of them.
Ratios sit close to fractions and proportion, and tests move between them freely. To practise all of these under timed conditions, try a numerical reasoning test with a worked solution for every question.