How to Simplify Ratios

To simplify a ratio, divide every part of it by the highest common factor, the largest whole number that goes into all of the parts exactly. For 12:15 the highest common factor is 3, so 12 ÷ 3 = 4 and 15 ÷ 3 = 5, and the simplest form is 4:5. A ratio is fully simplified when 1 is the only whole number that divides every part. If the ratio contains decimals or fractions, turn it into whole numbers first, then simplify in the same way.

Simplifying never changes what a ratio means, only how it looks: 12:15 and 4:5 describe the same comparison. Ratio questions are a regular part of a free numerical reasoning test, usually built on figures from a table or chart.

How to simplify a ratio, step by step

1. Find the highest common factor (HCF). A factor is a whole number that divides into another exactly. The factors of 12 are 1, 2, 3, 4, 6 and 12. The factors of 15 are 1, 3, 5 and 15. The largest one they share is 3.
2. Divide every part of the ratio by the HCF. 12 ÷ 3 = 4 and 15 ÷ 3 = 5, so 12:15 becomes 4:5. Divide every part, never just one, or you change the comparison.
3. Check nothing else divides every part. For 4:5 nothing larger than 1 does, so 4:5 is the simplest form. If you do find another common factor, you did not start with the highest one. Divide again.

If ratios are new to you, our guide to what a ratio is explains the basics first.

Worked examples with whole numbers

Simplify 700:500. Both numbers end in two zeros, so they both divide by 100. 700 ÷ 100 = 7 and 500 ÷ 100 = 5. Nothing else divides both 7 and 5, so the simplest form is 7:5.

Simplify 96:27. The factors of 27 are 1, 3, 9 and 27. Of these, only 1 and 3 go into 96, so the HCF is 3. 96 ÷ 3 = 32 and 27 ÷ 3 = 9, so the simplest form is 32:9.

Simplify the three-part ratio 18:24:30. The method is the same with three or more parts, as long as the factor divides every part exactly. The largest number that goes into 18, 24 and 30 is 6. 18 ÷ 6 = 3, 24 ÷ 6 = 4 and 30 ÷ 6 = 5, so the simplest form is 3:4:5.

A workplace example. An office has 36 laptops and 84 desktop computers. What is the ratio of laptops to desktops in its simplest form? Both numbers are in the 12 times table: 36 ÷ 12 = 3 and 84 ÷ 12 = 7. The ratio is 3:7.

How to simplify ratios with decimals

A simplified ratio should be written in whole numbers, so clear the decimals first. Multiply every part by 10, 100 or 1,000, whichever removes the decimal point from all of them, then simplify as normal.

Simplify 1.2:0.8. Multiply both parts by 10 to get 12:8. The HCF of 12 and 8 is 4, so the simplest form is 3:2.

Simplify 0.25:1.5. Multiply both parts by 100 to get 25:150. The HCF is 25, so the simplest form is 1:6.

How to simplify ratios with fractions

The idea is the same: turn the fractions into whole numbers, then simplify. There are four steps.

1. Turn any mixed numbers into improper fractions. Multiply the whole number by the denominator and add the numerator.
2. Rewrite both fractions over the lowest common denominator. Multiply the top and bottom of each fraction by the same number so their values do not change.
3. Drop the denominators and write the numerators as a ratio. Once both fractions share a denominator, the ratio of the fractions is the ratio of their numerators.
4. Simplify the whole-number ratio by dividing by the HCF.

Simplify 1/2:3/8. The lowest common denominator is 8, and 1/2 = 4/8. The ratio is 4/8:3/8, which gives 4:3. Nothing divides both 4 and 3, so the answer is 4:3.

Simplify 12/17:15/68. Since 17 × 4 = 68, the lowest common denominator is 68. Multiply the first fraction top and bottom by 4: 12/17 = 48/68. The ratio is 48/68:15/68, which gives 48:15. The HCF of 48 and 15 is 3, so the answer is 16:5.

Simplify 1 5/12:7/30.

1. Convert the mixed number: 1 × 12 = 12, and 12 + 5 = 17, so 1 5/12 = 17/12.
2. The lowest common denominator of 12 and 30 is 60. 17/12 = 85/60 (multiply by 5) and 7/30 = 14/60 (multiply by 2).
3. Drop the denominators: 85:14.
4. 85 and 14 share no factor other than 1, so the answer is 85:14.

Simplify 3 2/9:5/18. Convert the mixed number: 3 × 9 = 27, and 27 + 2 = 29, so 3 2/9 = 29/9. The lowest common denominator is 18, and 29/9 = 58/18. The ratio is 58/18:5/18, which gives 58:5. That will not simplify further, so the answer is 58:5.

A shortcut that gives the same result: multiply every part of the ratio by the lowest common denominator. For 1/2:3/4, multiplying both parts by 4 gives 2:3 straight away.

How to simplify ratios with different units

A ratio only works when both parts are in the same unit. Convert first, simplify second, and leave the unit off the final answer.

Simplify 45 minutes to 2 hours. 2 hours is 120 minutes, so the ratio is 45:120. The HCF is 15, so the simplest form is 3:8.

Simplify 750g to 3kg. 3kg is 3,000g, so the ratio is 750:3,000. The HCF is 750, so the simplest form is 1:4.

Writing a ratio in the form 1:n

Some questions ask for a ratio with 1 as the first part, because it makes comparisons easier. Divide both parts by the first number and accept that the second part may be a decimal. For 5:12, 5 ÷ 5 = 1 and 12 ÷ 5 = 2.4, so the ratio is 1:2.4. This is not the simplest whole-number form, but it is what the question asks for.

Tips for simplifying ratios faster

• Use your times tables first. If both numbers are in the 7 times table, as with 21:14, divide by 7 straight away to get 3:2.
• Strip shared zeros. For 3,000:6,000, divide by 1,000 to get 3:6, then keep going to 1:2.
• Simplify in stages if you need to. With 88:44 you might divide by 11 to get 8:4, then by 4 to reach 2:1. You end up in the same place.
• Watch for primes. If one part is a prime number, such as 5, 7 or 11, the ratio only simplifies if the other parts are multiples of it. 5:15 becomes 1:3, but 5:12 cannot be reduced.
• Keep the order. “The ratio of A to B” means A comes first. Swapping the parts gives a different ratio.

Common mistakes

• Stopping too early because you divided by a common factor rather than the highest one.
• Dividing only one part of the ratio.
• Leaving a decimal or fraction in an answer that should be in whole numbers.
• Simplifying two different units without converting them first.

Once you can simplify a ratio quickly, the next steps are sharing an amount in a ratio and converting a ratio into a fraction. To practise all of it against the clock, try a timed numerical reasoning practice test.