To simplify a ratio, divide every part of it by their highest common factor, the largest whole number that goes into all of them exactly. For 12:15 the highest common factor is 3, so 12 ÷ 3 = 4 and 15 ÷ 3 = 5, and the ratio in its simplest form is 4:5.
A ratio is fully simplified when the only whole number that divides every part exactly is 1. Simplifying never changes what a ratio means, only how it looks: 12:15 and 4:5 describe the same proportion.
This page covers reducing a ratio to its lowest terms, including ratios that contain decimals, fractions or two different units. If you have a total to share out rather than a ratio to tidy up, see how to divide an amount into a given ratio.
What is a ratio?
A ratio compares two or more quantities and is written with a colon between the numbers, such as 1:2. A ratio of 1:2 could show that for every 1 apple in a fruit basket there are 2 bananas.
Ratios taken from real figures are often too large to read at a glance. Comparing a city of 10,000,000 people with a city of 500,000 gives a raw ratio of 10,000,000:500,000, which tells you far less than the simplified 20:1, meaning that for every 20 people in the first city there is 1 person in the second.
Simplified ratios are quicker to compare, which is why they run through the data interpretation questions in numerical reasoning tests.
How to simplify a ratio, step by step
Step 1: Find the highest common factor (HCF) of the numbers in the ratio
A factor is a whole number that divides into your number exactly, leaving no remainder. The factors of 12 are 1, 2, 3, 4, 6 and 12. The factors of 15 are 1, 3, 5 and 15.
The highest common factor, or HCF, is the largest factor the numbers share. Here 12 and 15 share only 1 and 3, so the HCF is 3.
Step 2: Divide every part of the ratio by the HCF
Divide each number in the ratio by the HCF, and by nothing else, because dividing one part on its own would change what the ratio means.
12 ÷ 3 = 4
15 ÷ 3 = 5
So 12:15 simplifies to 4:5.
Step 3: Check the ratio cannot be simplified any further
Look for another whole number, larger than 1, that divides every part of your new ratio exactly. For 4:5 there is none, so 4:5 is the simplest form.
If you do find one, you did not start with the highest common factor. Divide again and repeat until 1 is the only number left that divides them all.
Worked examples
Example 1: simplify 700:500
Start by writing out the factors of both numbers.
The factors of 700 are 1, 2, 4, 5, 7, 10, 14, 20, 25, 28, 35, 50, 70, 100, 140, 175, 350 and 700.
The factors of 500 are 1, 2, 4, 5, 10, 20, 25, 50, 100, 125, 250 and 500.
They share 1, 2, 4, 5, 10, 20, 25, 50 and 100, so the HCF is 100.
700 ÷ 100 = 7 and 500 ÷ 100 = 5, so the simplest form is 7:5.
Example 2: simplify 96:27
The factors of 96 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48 and 96.
The factors of 27 are 1, 3, 9 and 27.
The only factors they share are 1 and 3, so the HCF is 3.
96 ÷ 3 = 32 and 27 ÷ 3 = 9, so the simplest form is 32:9.
Example 3: simplify the three-part ratio 18:24:30
The method is the same with three or more parts, as long as the factor you use divides every part exactly.
The factors of 18 are 1, 2, 3, 6, 9 and 18. The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. The factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30. The largest factor all three share is 6.
18 ÷ 6 = 3, 24 ÷ 6 = 4 and 30 ÷ 6 = 5, so the simplest form is 3:4:5.
Simplifying ratios that contain 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 is enough to remove the decimal point from all of them, then simplify as normal.
To simplify 1.2:0.8, multiply both parts by 10 to get 12:8. The HCF of 12 and 8 is 4, so 12 ÷ 4 = 3 and 8 ÷ 4 = 2, giving 3:2.
To simplify 0.25:1.5, multiply both parts by 100 to get 25:150. The HCF is 25, so 25 ÷ 25 = 1 and 150 ÷ 25 = 6, giving 1:6.
Simplifying ratios that contain fractions
Fractions come out the same way once you have turned them into whole numbers. Multiply every part by the lowest common denominator, then simplify what is left.
For 1/2 : 3/4, the lowest common denominator is 4. Multiplying both parts by 4 gives 2:3, which will not reduce any further. There is more detail in our guide to simplifying ratios with fractions.
Simplifying ratios with different units
A ratio only means something when both sides are measured in the same unit, so convert first, simplify second, and leave the unit off the final answer.
To simplify 45 minutes to 2 hours, convert the 2 hours into 120 minutes, which gives 45:120. The HCF of 45 and 120 is 15, so 45 ÷ 15 = 3 and 120 ÷ 15 = 8, giving 3:8.
To simplify 750g to 3kg, convert the 3kg into 3,000g, which gives 750:3,000. The HCF is 750, so the ratio simplifies to 1:4.
Writing a ratio in the form 1:n
Comparison questions sometimes ask for a ratio in the form 1:n, where the first part is reduced to 1. Divide both parts by the first number and accept that the second part may not come out as a whole number.
For 5:12, divide both parts by 5: 5 ÷ 5 = 1 and 12 ÷ 5 = 2.4, so the ratio is 1:2.4. That is not the simplest whole-number form, because the HCF of 5 and 12 is already 1, but it makes the comparison easier to read.
Tips for simplifying ratios faster
Use your times tables before you write anything out. If you can see that 21 and 14 are both in the 7 times table, divide by 7 straight away to get 3:2, then decide whether it will go any further.
Strip the zeros when both parts end in one. For 70:50, divide by 10 to get 7:5. For 3,000:6,000, divide by 1,000 to get 3:6, then keep going, because 3:6 still divides by 3 to give 1:2.
Write the factors out in full when the numbers are awkward. For 60:40, list them: 60 has 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60, while 40 has 1, 2, 4, 5, 8, 10, 20 and 40. The largest number in both lists is 20, so 60:40 simplifies to 3:2. Underlining the factors that appear in both lists makes the HCF hard to miss.
Simplifying in stages is fine. With 88:44 you might spot 11 first, giving 8:4, then divide by 4 to reach 2:1. You end up in the same place as spotting the HCF of 44 in one go.
Watch what a prime number does. A prime number is divisible only by itself and 1, and the first five are 2, 3, 5, 7 and 11. If one part of your ratio is prime, the ratio will only simplify when the other parts are multiples of that prime: 5:15 becomes 1:3, while 5:12 cannot be reduced at all.
Common mistakes to avoid
- Stopping too early, because you divided by a common factor rather than the highest one.
- Dividing only one part of the ratio, which changes the proportion completely.
- Leaving a decimal or a fraction in an answer that is supposed to be in its simplest form.
- Simplifying two different units without converting them to the same unit first.
Practice ratio questions under test conditions
Simplifying only helps in an assessment if it is quick, so practise it against the clock. Ratios run through numerical reasoning tests, and basic numeracy tests will drill the underlying arithmetic if factors and times tables are what slow you down.
