Ratios:
simplify them and split any amount

What a ratio is and how to write it

A ratio compares quantities of the same kind. If a class has 18 girls and 12 boys, the ratio of girls to boys is 18 : 12, which simplifies to 3 : 2.

Order matters. The ratio of boys to girls is 2 : 3, which is a different statement. A ratio can also be written as a fraction (3/2) or in words (3 to 2).

Ratios do not carry units once the quantities are in the same unit. 50 cm to 2 m must be converted first: 50 : 200 = 1 : 4.

You will meet ratios in cooking, mixing paint or fertiliser, map scales, screen shapes, speeds, financial comparisons and sharing money. The two skills that cover almost every case are simplifying a ratio and dividing an amount in a ratio. Both are shown below.

How to simplify a ratio

If you cannot see the greatest common factor, divide both parts by any common factor and repeat until no factor remains. 90 : 120 halves to 45 : 60, then divides by 15 to 3 : 4.

Ratios involving decimals or fractions are easiest if you multiply through first. 0.5 : 2.5 becomes 5 : 25 when multiplied by 10, which simplifies to 1 : 5.

Three-part ratios work the same way. For 24 : 36 : 60, the greatest common factor of all three is 12, so the simplest form is 2 : 3 : 5. Test every part, because a factor that fits two of them may not fit the third.

How to divide an amount in a given ratio

1. 1Add the parts of the ratio to get the total number of shares.
2. 2Divide the amount by the total shares to find the value of one share.
3. 3Multiply the value of one share by each part.

This is the method used for splitting a bill, a profit between partners or an inheritance in agreed proportions. Always add the shares back together as a check.

Rounding can spoil an exact split. If ₹1,00,000 is shared 1 : 1 : 1, each share is ₹33,333.33, so decide who receives the leftover paisa before you commit. Where the shares are exact, as above, no rounding is needed.

Scaling a ratio, finding the total and combining ratios

Multiplying every part by the same number gives an equivalent ratio. A paint mix of 3 parts blue to 4 parts white is the same shade at 6 : 8 or 15 : 20. If you have 15 tins of blue, you need 15 ÷ 3 × 4 = 20 tins of white.

You can also work backwards from one known share. Two partners split profit 3 : 2 and the second partner receives ₹60,000. Two shares are worth ₹60,000, so one share is ₹30,000 and the total profit is 5 × ₹30,000 = ₹1,50,000. The first partner receives ₹90,000.

To combine two ratios that share a term, make the shared term equal. If A : B is 2 : 3 and B : C is 4 : 5, scale the first by 4 and the second by 3. That gives A : B = 8 : 12 and B : C = 12 : 15, so A : B : C is 8 : 12 : 15.

Make sure the terms genuinely refer to the same thing before combining. A ratio of boys to girls cannot be combined with a ratio of girls to teachers unless the girls are the same group.

Aspect ratios from common pixel sizes

Use the aspect ratio calculator on this page to reduce any width and height to this simplest form. If a design must fit a 4 : 5 slot and your image is 3 : 2, you will need to crop it.

To crop that 3000 × 2000 photo to 4 : 5, keep the full height of 2000 px and set the width to 2000 × 4 ÷ 5 = 1,600 px. You lose 1,400 px of width, so decide in advance which side of the frame you can spare.

Convert a ratio to a fraction or percentage

A part-to-part ratio becomes fractions of the whole by dividing each part by the total. In a 3 : 2 mixture, the first part is 3/5 = 60% and the second is 2/5 = 40%.

Be careful to distinguish part-to-part from part-to-whole. A 3 : 1 mixture of milk to water is 3/4 milk, which is 75%, and 1/4 water. It is not 1/3 water, because the whole is four parts.

Common ratio mistakes

• Writing the ratio in the wrong order. 2 : 3 and 3 : 2 describe different things.
• Comparing different units without converting, such as 50 cm : 2 m.
• Dividing the total by one part instead of by the sum of all parts.
• Reading 3 : 1 as 1/3 when it means 3/4 and 1/4 of the whole.
• Stopping at 6 : 8 when the simplest form is 3 : 4.
• Treating the ratio as a difference. A 3 : 4 split of ₹70,000 gives ₹30,000 and ₹40,000, a gap of ₹10,000, not ₹1.

Whenever a ratio question includes a total, write the total next to the sum of the parts. If the amount does not divide evenly by the number of shares, recheck the question before rounding, because most textbook and exam ratios are built to divide exactly.

When a ratio is stated as two equal ratios and one number is missing, you have a proportion problem, which is covered in the proportion guide. A related set of conversions is in the fraction and decimal guides.