Ratios and Proportions: Simplify, Solve, Share

A ratio says how much of one thing there is for every amount of another. A class with 12 boys and 18 girls has a boys-to-girls ratio of 12 : 18, read "12 to 18".

• Order matters: girls to boys is 18 : 12.
• Part to part vs part to whole: boys to girls is 12 : 18, but boys to the whole class is 12 : 30.
• A ratio can have more than two parts: cement, sand and gravel in 1 : 2 : 3.

Divide every part by the greatest common factor, exactly like simplifying a fraction:

Two things to fix first:

• Decimals: multiply every part by 10 or 100 until they're whole. 1.5 : 2.25 → 150 : 225 → 2 : 3.
• Different units: convert to the same unit. 50 cm : 2 m → 50 : 200 → 1 : 4.

Not sure of the GCF? The GCF guide shows three ways to find it.

A unit ratio has 1 as one of its parts. Divide every part by that part:

Unit ratios make comparisons easy. Is 5 : 8 or 7 : 11 the bigger share for the second part? 1 : 1.6 vs 1 : 1.571…, so 5 : 8 gives the second part slightly more. Map scales such as 1 : 50,000 are unit ratios too.

A proportion says two ratios are equal, with one number missing. Solve 3 : 4 = 15 : x two ways:

• Scale factor: 3 became 15, which is × 5. Do the same to the other part: x = 4 × 5 = 20. Quickest when the factor is a whole number.
• Cross-multiply: write 3/4 = 15/x. The cross products are equal: 3 × x = 4 × 15 = 60, so x = 60 ÷ 3 = 20. Works every time.

On a basic calculator, multiply the two numbers that sit diagonally from each other, then divide by the third: 4 × 15 ÷ 3 = 20.

To split $100 between two people in the ratio 2 : 3:

1. Add the parts: 2 + 3 = 5 parts.
2. Find one part: $100 ÷ 5 = $20.
3. Multiply: 2 × $20 = $40 and 3 × $20 = $60.

Always check: the shares add up to $100, and 40 : 60 simplifies back to 2 : 3. With three people in 1 : 2 : 3, there are 6 parts, so $600 becomes $100, $200 and $300.

• Recipes: a recipe for 4 uses 300 g of rice. For 6 people, set up 4 : 300 = 6 : x. Then x = 300 × 6 ÷ 4 = 450 g. The "unitary method" does the same thing in two steps: 300 ÷ 4 = 75 g per person, then × 6.
• Maps: on a 1 : 50,000 map, 4.5 cm stands for 4.5 × 50,000 = 225,000 cm. That's 2,250 m, or 2.25 km.
• Screens: a 16 : 9 screen 1,920 pixels wide is 1,920 ÷ 16 × 9 = 1,080 pixels tall.

• Treating a ratio as a fraction of the whole: in a 2 : 3 split, the first share is 2/5 of the total, not 2/3.
• Swapping the order: "boys to girls" and "girls to boys" are different ratios.
• Adding instead of multiplying: 3 : 4 = 15 : x is not 4 + 12 = 16. A ratio scales by multiplying.
• Mixing units: convert minutes and hours, or cm and m, before simplifying.

Check your answers with the Ratio Calculator.

1. Simplify 24 : 36
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   2 : 3 (GCF 12)
2. Simplify 1.2 : 3
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   2 : 5. Multiply by 10 to get 12 : 30, then divide by 6.
3. Simplify 45 : 30 : 15
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   3 : 2 : 1 (GCF 15)
4. Solve 5 : 8 = x : 40
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   x = 25. 8 × 5 = 40, so x = 5 × 5.
5. Solve 7 : x = 21 : 30
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   x = 10. 7 × 30 = 210, and 210 ÷ 21 = 10.
6. Share $240 in the ratio 3 : 5
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   $90 and $150. 8 parts, so one part is $30.
7. Share 84 in the ratio 1 : 2 : 4
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   12, 24 and 48. 7 parts, so one part is 12.
8. 250 g of flour makes 10 cookies. How much for 24 cookies?
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   600 g. 250 ÷ 10 = 25 g per cookie, × 24.
9. On a 1 : 25,000 map, two towns are 8 cm apart. How far apart are they?
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   2 km. 8 × 25,000 = 200,000 cm = 2,000 m.