A ratio compares two quantities. A proportion says two ratios are equal. Between them, they cover recipes, maps, speeds, mixing, sharing, and scaling — this guide covers every major question type.
Ratios and proportions are how math compares quantities without needing to know their exact sizes. This guide builds from the simplest ratio all the way through direct proportion, inverse proportion, and three-term ratios — with a worked example for every type.
A ratio compares two quantities of the same kind, showing how many times one contains the other. It's written with a colon: a : b, read "a to b."
A fruit bowl has 4 apples and 6 oranges. The ratio of apples to oranges is 4 : 6.
Just like fractions, ratios simplify by dividing both terms by their HCF.
12 : 18 — HCF(12,18) = 6. Dividing both terms by 6 gives 2 : 3, the simplest form.
Multiplying or dividing both terms of a ratio by the same number never changes what it represents — exactly like equivalent fractions.
2 : 3 = 4 : 6 = 6 : 9 = 20 : 30 — all of these describe the exact same comparison.
This distinction trips up more students than any other ratio concept: a ratio can compare one part to another part, or one part to the entire total — and they give completely different numbers.
A class has 12 boys and 18 girls (30 students total). Boys to girls (part-to-part) = 12:18 = 2:3. Boys to the whole class (part-to-whole) = 12:30 = 2:5.
A part-to-part ratio is not the same fraction as a part-to-whole comparison.
The fix: boys:girls = 2:3 means for every 2 boys there are 3 girls — 5 parts total. Boys make up 2 out of those 5 parts of the whole class, so boys are 2/5 of the class, not 2/3. The "3" in a part-to-part ratio refers to the other part, not the whole.
To split a total amount according to a ratio, add up the ratio's parts to find the total number of "shares," find the value of one share, then multiply back out.
Divide $60 between two people in the ratio 2:3. Total parts = 2+3 = 5. One part = 60÷5 = $12. Shares: 2×12=$24 and 3×12=$36. Check: 24+36=60. ✓
Divide 180 sweets among three people in the ratio 2:3:4. Total parts = 2+3+4 = 9. One part = 180÷9 = 20. Shares: 40, 60, and 80 sweets. Check: 40+60+80=180. ✓
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A proportion is a statement that two ratios are equal: a:b = c:d. Whenever four numbers are in proportion, cross-multiplying gives a×d = b×c — the basis for solving for any missing value.
Solve x:5 = 12:15. Cross-multiply: x×15 = 5×12 = 60. So x = 60÷15 = 4. Check: 4:5 and 12:15 both simplify to 4:5. ✓
Two quantities are in direct proportion when they increase or decrease together, at the same rate — double one, and the other doubles too.
5 pens cost $20. How much do 8 pens cost? Using the unitary method: 1 pen costs 20÷5=$4. So 8 pens cost 4×8=$32.
Two quantities are in inverse proportion when one increases as the other decreases, in a way that keeps their product constant.
4 workers build a wall in 12 days. How many days would 6 workers take? The total work stays constant: 4×12=48 "worker-days." With 6 workers: 48÷6=8 days.
Assuming every scaling problem is direct proportion is one of the most common ratio mistakes.
The fix: more workers means less time needed, not more — that's the signal for inverse proportion. Direct proportion problems have both quantities moving the same direction (more pens, more cost); inverse proportion problems have them moving in opposite directions (more workers, less time).
Three numbers a, b, c are in continued proportion when a:b = b:c. The middle term, b, is called the mean proportional, calculated as the square root of a×c.
Find the mean proportional between 4 and 16. b = √(4×16) = √64 = 8. Check: 4:8 = 8:16, since both simplify to 1:2. ✓
| Situation | Method |
|---|---|
| Comparing two quantities directly | Simplify the ratio using HCF |
| Splitting a total by a ratio | Add the parts, find one share, scale up |
| Both quantities increase together | Direct proportion (unitary method) |
| One quantity increases as the other decreases | Inverse proportion (keep the product constant) |
| Solving for an unknown in a:b = c:d | Cross-multiplication |
| Finding the middle term of a:b = b:c | Mean proportional = √(a×c) |
Map scales use direct proportion: at a scale of 1cm:5km, a 3.5cm map distance represents 3.5×5=17.5km in real life. Speed and time for a fixed distance are inversely proportional: a car covering 300km at 60km/h takes 5 hours, but at 75km/h takes only 300÷75=4 hours. Mixing paint in a red:blue ratio of 3:5 — if 15L of blue is used, red needed = 15×3/5=9L.
The Golden Ratio (approximately 1.618:1) appears throughout nature — in flower petal arrangements, spiral shells, and hurricane shapes — and has been used deliberately in art and architecture for thousands of years for its pleasing proportions.
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Ratio and proportion are used constantly outside the classroom — scaling a recipe, reading a map, mixing chemicals or paint safely, converting currencies, and comparing prices between differently sized packages. They're also the direct foundation for percentages, our next topic, since a percentage is simply a ratio to 100.
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