How to Simplify a Ratio and Solve Proportions

A ratio compares two quantities. Writing 24 : 36 says that for every 24 of the first there are 36 of the second. Simplifying a ratio and solving for a missing part are the two skills that cover most problems.

Simplifying a ratio

Divide both numbers by their greatest common divisor, the largest number that divides both exactly.

For 24 : 36 the greatest common divisor is 12. Dividing both by 12 gives 2 : 3.

If a ratio includes decimals, multiply both sides by 10, 100 or 1,000 until they are whole numbers, then simplify. 1.5 : 4.5 becomes 15 : 45, then 1 : 3.

Solving a proportion

A proportion says two ratios are equal: A : B = C : X. Cross-multiply to solve for X.

A × X = B × C     so     X = (B × C) ÷ A

A recipe uses flour and sugar in the ratio 2 : 3. You have 250 g of flour. How much sugar?

X = (3 × 250) ÷ 2 = 375 g of sugar.

Scaling a map

A map scale of 1 : 50,000 means 1 cm on the map is 50,000 cm on the ground, which is 500 m. A road measured as 4.2 cm on the map is 4.2 × 50,000 = 210,000 cm, or 2.1 km.

Sharing in a ratio

To split 90 in the ratio 2 : 3, add the parts (2 + 3 = 5), find one part (90 ÷ 5 = 18), then multiply: 2 × 18 = 36 and 3 × 18 = 54.

Common mistakes

• Mixing units. Convert both quantities to the same unit before forming a ratio.
• Simplifying only one side.
• Reversing the order. 2 : 3 and 3 : 2 are different ratios.

The ratio calculator simplifies ratios and solves for the missing value in a proportion.

Try it yourself

1. Simplify 18 : 48.
2. A paint mix uses 3 parts concentrate to 5 parts water. How much water goes with 12 litres of concentrate?
3. Share 120 in the ratio 3 : 7.

Answers: (1) 3 : 8; (2) 20 litres; (3) 36 and 84.