How Ratios Work: Simplifying, Comparing and Sharing
A ratio compares quantities of the same kind. If a drink mix uses 2 parts concentrate to 5 parts water, the ratio is 2:5, read "2 to 5". Ratios appear in recipes, mixtures, maps, gear systems, class sizes and money splits, and the same few skills solve nearly all of them.
Below you will see how to write and simplify a ratio, why part-to-part and part-to-whole ratios are easy to mix up, and how to divide a total in a given ratio. The examples use whole numbers you can check by hand.
Reading and writing a ratio
A ratio can be written with a colon (3:5), with the word "to" (3 to 5), or as a fraction (3/5). Order matters: 3:5 is not the same as 5:3. The first number belongs to the first item named, so a class with 12 girls and 15 boys has a girls-to-boys ratio of 12:15 and a boys-to-girls ratio of 15:12.
A ratio has no units of its own, but both quantities must be measured in the same unit. A ratio of 30 cm to 2 m is not 30:2. Convert 2 m to 200 cm first, giving 30:200, which simplifies to 3:20.
Simplifying a ratio
Like fractions, ratios can be reduced by dividing every term by the same number. The fastest way is to divide by the greatest common divisor (GCD) of all the terms.
Worked example: simplify 24:36. The GCD of 24 and 36 is 12. Dividing both terms by 12 gives 2:3. Another: 18:30:12 has a GCD of 6, so it simplifies to 3:5:2.
Ratios involving decimals are cleared first by multiplying every term by a power of 10. So 1.5:4.5 becomes 15:45 after multiplying by 10, then 1:3 after dividing by 15. Two ratios are equivalent if they simplify to the same form: 2:3, 4:6 and 30:45 all describe the same comparison.
Part-to-part versus part-to-whole
Suppose a bag holds 3 red counters for every 5 blue ones. The part-to-part ratio of red to blue is 3:5. But red counters make up 3 out of 3 + 5 = 8 counters in total, so the part-to-whole fraction is 3/8, or 37.5%.
This distinction causes the most common ratio error: treating 3:5 as if it meant 3/5. The fraction 3/5 would mean 3 red out of every 5 counters in all, which is a different mix. To get a fraction of the whole from a part-to-part ratio, always add the parts first.
| Description | Ratio or fraction | Meaning |
|---|---|---|
| Red to blue | 3:5 | 3 red for every 5 blue |
| Red to total | 3:8 | 3 red for every 8 counters |
| Share that is red | 3/8 = 37.5% | Red as a fraction of the whole bag |
Sharing an amount in a given ratio
To split a total, add the parts to count the shares, divide the total by that number to find the value of one part, then multiply.
Worked example: three people share a 360 prize in the ratio 2:3:4. There are 2 + 3 + 4 = 9 parts. One part is 360 ÷ 9 = 40. The shares are 2 × 40 = 80, 3 × 40 = 120 and 4 × 40 = 160. Check: 80 + 120 + 160 = 360.
The same steps solve mixing problems. A cleaning concentrate is diluted 1:4 (one part concentrate to four parts water). For 2 litres of finished solution, there are 5 parts, each 0.4 litres, so you need 0.4 litres of concentrate and 1.6 litres of water.
Finding a missing amount
If you know one quantity and the ratio, scale the ratio until the known term matches. A paint mix uses red to white in the ratio 3:7. If you have 21 tins of white, the multiplier is 21 ÷ 7 = 3, so you need 3 × 3 = 9 tins of red.
- Write the ratio with its labels so you never swap the order.
- Find the multiplier by dividing the known amount by its ratio term.
- Multiply every other term by the same multiplier.
- Check that the finished quantities simplify back to the original ratio.
For quick checks, the ratio calculator simplifies and scales ratios, and the proportion calculator solves for a missing term. To reduce terms by hand, the GCD calculator finds the divisor.
- Ratio Calculator: Fill in three of the four numbers in a:b = c:d, leave one empty and read off the missing value. The guide covers simplifying and combining ratios, odds and screen aspect ratios.
- Proportion Calculator: Enter three of the four numbers in two equal fractions and leave one box empty. The calculator solves for the missing term and the guide explains the cross-multiplication rule behind it.
- Greatest Common Divisor (GCD) Calculator: Paste or type two or more numbers and get their greatest common divisor in one click. The guide shows three ways to find it by hand and what the result is good for.
This article is general information, not professional advice. See our Disclaimer and Editorial Policy.
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