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To divide a quantity in a given ratio, add the parts of the ratio together, divide the total by that sum to get the value of one part, then multiply each part of the ratio by it. Sharing 200 in the ratio 3:2 gives 3 + 2 = 5 parts, 200 ÷ 5 = 40 for a single part, so the two shares are 120 and 80.
The shares always add back to the amount you started with, so checking 120 + 80 = 200 catches almost every slip before you commit to an answer.
This page covers sharing an amount out in a given ratio, working backwards when you know one share rather than the total, and the proportion questions that ask you to scale a quantity up or down. If your ratio is simply untidy and you want it reduced to its lowest terms, that is a different job: see how to simplify a ratio to its simplest form.
Ratio division is one of the most common question types in numerical reasoning tests, usually dressed up as a budget, a staffing split, a recipe or a share of profit.
How to divide a ratio, step by step
Dividing an amount in a ratio means splitting it into equal parts, then handing out those parts in the proportions the ratio describes. The three steps below work whether the ratio has two parts, three parts or more.
Step 1: Add the parts of the ratio together
Start with the ratio, not the total. If a company employs men and women in the ratio 2:5 and has 49 staff altogether, add the parts together: 2 + 5 = 7.
That tells you the 49 staff are being split into 7 equal parts.
Step 2: Divide the total by the sum of the parts
Divide the amount you are sharing by the number of parts you found in step one.
49 ÷ 7 = 7
That result, sometimes called the quotient, is the value of one part. Every share in the answer is a multiple of it.
Step 3: Multiply each part of the ratio by the value of one part
Multiply each number in the ratio by the value of one part.
2 × 7 = 14 men
5 × 7 = 35 women
Then check the shares add back to the original total: 14 + 35 = 49.
Worked examples
Example 1: splitting a budget in the ratio 2:3
Mo has £1,500 to spend on clothes for a new job and an armchair for his flat. He splits the money in the ratio 2:3, armchair to clothes. What is his armchair budget?
Add the parts: 2 + 3 = 5.
Divide the total: £1,500 ÷ 5 = £300, the value of one part.
The armchair takes 2 parts, so 2 × £300 = £600. The clothes take 3 parts, so 3 × £300 = £900.
Check: £600 + £900 = £1,500.
Example 2: a three-part ratio
Harriet is splitting a £3,900 bonus pot between three staff in the ratio 1:2:3, with the newest member of staff receiving the smallest share. How much does each of them get?
Add the parts: 1 + 2 + 3 = 6.
Divide the total: £3,900 ÷ 6 = £650, the value of one part.
The newest member of staff gets 1 × £650 = £650.
The next member of staff gets 2 × £650 = £1,300.
The longest-serving member of staff gets 3 × £650 = £1,950.
Check: £650 + £1,300 + £1,950 = £3,900.
Example 3: working backwards from one known share
A recipe uses flour and butter in the ratio 5:2. You have measured out 350g of flour. How much butter do you need?
Here you are given one share rather than the total, so find the value of one part from the share you already know: 350g ÷ 5 = 70g.
Butter takes 2 parts, so 2 × 70g = 140g.
If the question also asks for the total mixture, that is all 7 parts: 7 × 70g = 490g, which matches 350g + 140g.
Example 4: dividing work between two contractors
A project needs 84 hours of work, shared between two contractors in the ratio 3:4. How many hours does each of them do?
Add the parts: 3 + 4 = 7. Divide the total: 84 ÷ 7 = 12 hours for one part.
The first contractor works 3 × 12 = 36 hours and the second works 4 × 12 = 48 hours.
Check: 36 + 48 = 84.
Dividing one part of a ratio by another
Some questions do not ask you to share anything out. They ask how many times bigger one quantity is than another, and for that you divide one part of the ratio by the other.
In the ratio 3:2, dividing the first part by the second gives 3 ÷ 2 = 1.5, so the first quantity is 1.5 times the second. The same ratio written as fractions of the whole gives 3/5 and 2/5, which is why sharing 200 in the ratio 3:2 produces 120 and 80: 3/5 of 200 is 120.
The “if more, less” rule in proportion questions
A ratio question is really a proportion question when one quantity moves in the opposite direction to the other. The rule of thumb is: if more of one thing means less of the other, multiply by the smaller number over the larger one, which has the same effect as dividing.
If 4 people take 12 days to finish a job, how long do 6 people take? More people means fewer days, so multiply by 4/6 rather than 6/4:
12 × 4 ÷ 6 = 8 days.
The check is the total amount of work involved: 4 × 12 = 48 person-days, and 48 ÷ 6 = 8 days.
When more of one thing means more of the other, such as more hours worked meaning more pay, scale the other way instead and multiply by the larger number over the smaller one.
Common mistakes to avoid
- Dividing by one part of the ratio rather than the sum of the parts. Sharing 200 in the ratio 3:2 means dividing by 5, never by 3.
- Matching the shares to the wrong side of the ratio. Read carefully which quantity the question names first.
- Skipping the check. Adding the shares back up should return the original total every time.
- Mixing units, such as splitting a total given in hours and answering in minutes.
Practice dividing ratios under test conditions
In a real assessment the method has to be automatic, because the pressure comes from the clock rather than the arithmetic. Work through full-length numerical reasoning tests to see ratios in the context of tables and charts, use basic numeracy tests if you want the arithmetic on its own first, and try a set of free aptitude tests to see where your timing stands.