Ratio: the method you forgot, worked step by step
To simplify a ratio, divide both sides by the same number until nothing more divides. To share a total in a ratio, find what one part is worth, then count the parts you need.
Ratios slipped out of your head years ago, and now they are back: in a pre-hire numerical test, or on your child’s homework sheet. Try one question now, or split a total in a ratio through ten drill questions.
Simplifying ratios: how do you do it without a calculator?
A ratio keeps its meaning when you divide both sides by the same number. Shrink it until no whole number divides both sides.
Formula
a : b = (a ÷ n) : (b ÷ n)
- n is any whole number that goes into both a and b.
Worked example
A team has 24 analysts and 36 managers. Write analysts to managers as a ratio in its simplest form.
Stopping at 12 : 18 is the usual slip. Both sides still divide by 6, so the ratio is not simple yet.
- Halve both sides: 24 ÷ 2 = 12 and 36 ÷ 2 = 18
- Divide both by 6: 12 ÷ 6 = 2 and 18 ÷ 6 = 3
Analysts to managers is 2 : 3.
How to calculate a ratio from two numbers, and which comes first?
A ratio follows the order of the words in the question. It compares one part with another part, or one part with the whole, so check which one is asked.
Formulas
Part to part
A : B
Part to whole
A : (A + B)
- A is the amount named first in the question.
Worked example
A team has 12 women and 8 men. What is the ratio of women to all staff, in its simplest form?
Writing 12 : 8 answers a different question, women to men. It simplifies to 3 : 2, and it earns nothing here.
- Find the whole: 12 + 8 = 20
- Write part to whole, 12 : 20, and divide both by 4: 12 ÷ 4 = 3
Women to all staff is 3 : 5.
What is the ratio formula, and when do you need it?
You need it when a total is split in a ratio. Count the parts first, find what one part is worth, then multiply for the share you want.
Formulas
One part = (Total) divided by (Parts)
Share = parts × one part
- Parts is the ratio's numbers added together.
Worked example
Two flatmates split a 540 euro bill in the ratio 5 : 4. How much does the larger share pay?
Dividing 540 by 5 gives 108. That treats the bill as five parts, and the other flatmate's four parts have gone missing.
- Add the parts: 5 + 4 = 9
- Find one part: 540 ÷ 9 = 60 euros
- Multiply by the larger share's parts: 60 × 5 = 300 euros
The larger share pays 300 euros.
Check it: the smaller share is 60 × 4 = 240 euros, and 300 + 240 = 540.
Ratio questions in a test: where do people lose the mark?
Most marks go in the same place: the question gives one share, not the total, and the sum is read from the wrong end of the ratio.
Formulas
One part = (Known share) divided by (Its parts)
Total = part × all parts
- Its parts is the ratio number that belongs to the share you were given.
Worked example
Two shifts split overtime pay in the ratio 3 : 2. The shorter shift gets 120 euros. How much was paid in total?
Dividing 120 by 3 gives 40 a part and a total of 200 euros. It is a tidy number, so nothing warns you, and the mark is gone.
- Find one part from the share you know. The shorter shift holds 2 parts: 120 ÷ 2 = 60 euros
- Add all the parts: 3 + 2 = 5
- Multiply: 60 × 5 = 300 euros
The total paid was 300 euros.
Want a mixed set instead of one skill? The button below opens the numerical practice page. Its 12-question diagnostic runs on a 12-minute clock, and the clock starts only when you press start on that page.
Ratio and proportion: when is it a ratio and when a rate?
A ratio compares amounts of the same kind. A rate compares different kinds, such as square metres covered per litre, and proportion keeps that rate fixed as the amount changes.
Formulas
Rate = (Amount) divided by (Units)
New = rate × new units
- Units is what the rate is per, such as litres.
Worked example
Eight litres of paint cover 56 square metres. How many square metres do 5 litres cover at the same rate?
Taking the missing 3 litres off the 56 gives 53. That treats each litre as one square metre, when each one covers more.
- Find the rate per litre: 56 ÷ 8 = 7 square metres
- Multiply by the new amount: 7 × 5 = 35 square metres
Five litres cover 35 square metres.
Want a mixed set instead of one skill? The button below opens the numerical practice page. Its 12-question diagnostic runs on a 12-minute clock, and the clock starts only when you press start on that page.
How we write, check and mark the practice questions is set out on the Methodology page.