A ratio compares two quantities and a proportion says two ratios are equal. Between them they cover recipes, maps, speeds, mixing, sharing and scaling.
Ratios and proportions are how mathematics compares quantities without needing to know their exact sizes. A recipe, a map, a paint mix, a speed and a currency exchange are all ratio problems in disguise. This guide builds from the simplest ratio through sharing, direct and inverse proportion, and three-term ratios, with a worked example and a real-life use for every type.
A ratio compares two (or more) quantities of the same kind. It is written with a colon, a : b, and read “a to b”. A ratio tells you how many times bigger one quantity is than another, or how the quantities are shared.
A fruit bowl has 4 apples and 6 oranges. The ratio of apples to oranges is 4 : 6. Order matters: the ratio of oranges to apples is 6 : 4, which is a different statement.
When the two quantities have different units (for example 150 km in 3 hours), the comparison is called a rate, such as 50 km per hour. Ratios compare like with like, and rates compare unlike things.
Just like fractions, ratios simplify by dividing every term by their common factors, ideally the HCF. A ratio is in its simplest form when the terms share no factor except 1.
(a) 12 : 18. HCF(12, 18) = 6. Dividing both by 6 gives 2 : 3.
(b) Three terms: 24 : 36 : 60. HCF = 12, so the simplest form is 2 : 3 : 5.
To compare ratios easily, it helps to make one term equal to 1. Divide both terms by the first: 4 : 10 = 1 : 2.5. Or by the second: 5 : 2 = 2.5 : 1.
Multiplying or dividing every term 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 same comparison. A table of equivalent ratios is a good way to solve many problems: write the ratio, then keep multiplying until you reach the number you need.
This distinction trips up more students than any other ratio idea. A ratio can compare one part to another part, or one part to the whole total, and the two give different numbers.
A class has 12 boys and 18 girls (30 students in all). 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 as a part-to-whole fraction.
The fix: boys : girls = 2 : 3 means that for every 2 boys there are 3 girls, which makes 5 parts in total. Boys are 2 out of those 5 parts, so boys are 2/5 of the class (40%), not 2/3. The 3 in a part-to-part ratio refers to the other part, not to the whole.
Ratios, fractions and percentages are linked. A ratio of 3 : 7 splits the whole into 10 parts, so the two parts are 3/10 = 30% and 7/10 = 70% of the whole. (See Percentages.)
To share a total in a ratio: add up the parts, find the value of one part, then multiply back out.
Divide Rs 60 between two people in the ratio 2 : 3. Total parts = 2 + 3 = 5. One part = 60 ÷ 5 = Rs 12. Shares: 2 × 12 = Rs 24 and 3 × 12 = Rs 36. Check: 24 + 36 = 60. ✓
Divide 180 sweets among three children in the ratio 2 : 3 : 4. Total parts = 9. One part = 180 ÷ 9 = 20. Shares: 40, 60 and 80 sweets. Check: 40 + 60 + 80 = 180. ✓
Two people share money in the ratio 7 : 4, and one gets Rs 3,600 more than the other. The difference is 7 − 4 = 3 parts, so 3 parts = Rs 3,600 and 1 part = Rs 1,200. The shares are 7 × 1,200 = Rs 8,400 and 4 × 1,200 = Rs 4,800.
Money is shared in the ratio 3 : 5, and the larger share is Rs 4,000. The larger share is 5 parts, so 1 part = Rs 800. The smaller share is 3 × 800 = Rs 2,400, and the total is 8 × 800 = Rs 6,400.
In a class, boys : girls = 3 : 2, and there are 24 boys. One part = 24 ÷ 3 = 8, so there are 2 × 8 = 16 girls. If 8 more girls join, there are 24 girls and the ratio becomes 24 : 24 = 1 : 1.
A : B = 2 : 3 and B : C = 4 : 5. Find A : B : C. Make the B terms equal using the LCM of 3 and 4, which is 12. A : B = 8 : 12 and B : C = 12 : 15. So A : B : C = 8 : 12 : 15.
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A proportion is a statement that two ratios are equal: a : b = c : d. When four numbers are in proportion, cross-multiplying gives a × d = b × c. That is the basis for finding 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. Their ratio stays constant, and a graph of one against the other is a straight line through the origin (0, 0).
5 pens cost Rs 20. How much do 8 pens cost? One pen costs 20 ÷ 5 = Rs 4. So 8 pens cost 4 × 8 = Rs 32.
12 eggs cost Rs 240. How much do 18 eggs cost? One egg costs 240 ÷ 12 = Rs 20, so 18 eggs cost 18 × 20 = Rs 360.
Two quantities are in inverse proportion when one increases as the other decreases in such a way that their product stays constant. A graph of one against the other is a curve, not a straight line.
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.
(a) 2 identical pipes fill a tank in 6 hours. 3 pipes take 12 ÷ 3 = 4 hours, because 2 × 6 = 12 pipe-hours of work.
(b) A car covers 300 km at 60 km/h in 5 hours. At 75 km/h it takes 300 ÷ 75 = 4 hours.
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 is the signal for inverse proportion. In direct proportion both quantities move the same way (more pens, more cost). In inverse proportion they move in opposite ways (more workers, fewer days). A quick test: does doubling one double the other (direct), or halve it (inverse)?
Not every “more gives more” situation is direct proportion.
The fix: 2 km costs 100 + 50 × 2 = Rs 200, and 4 km costs 100 + 50 × 4 = Rs 300. Doubling the distance did not double the cost, because of the fixed Rs 100 charge. Direct proportion needs the ratio to be the same every time, and here 200 ÷ 2 = 100 per km but 300 ÷ 4 = 75 per km. The graph would not pass through (0, 0).
Three numbers a, b, c are in continued proportion when a : b = b : c. The middle term b is called the mean proportional, and it equals √(a × c). (See Squares, Cubes & Roots.)
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 the HCF |
| Splitting a total by a ratio | Add the parts, find one part, scale up |
| Both quantities increase together | Direct proportion (unitary method) |
| One 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) |
| Two linked ratios sharing a term | Make the shared term equal using the LCM |
Map scales use direct proportion: at a scale of 1 cm : 5 km, a 3.5 cm map distance represents 3.5 × 5 = 17.5 km. Speed and time for a fixed distance are in inverse proportion: a car covering 300 km at 60 km/h takes 5 hours, but at 75 km/h takes only 300 ÷ 75 = 4 hours. Mixing paint in a red : blue ratio of 3 : 5 means that if 15 L of blue is used, the red needed is 15 × 3/5 = 9 L.
On a map with scale 1 : 50,000, 4.5 cm represents 4.5 × 50,000 = 225,000 cm. Since 100,000 cm = 1 km, that is 2.25 km.
(a) A mortar mix uses cement : sand = 1 : 4. If 5 bags of cement are used, the sand needed is 5 × 4 = 20 bags.
(b) Rice and water are cooked in the ratio 1 : 2. For 3 cups of rice, use 3 × 2 = 6 cups of water.
The Golden Ratio, written φ (phi), is about 1.618 : 1. It is linked to the Fibonacci numbers: the ratio of each Fibonacci number to the one before it (8/5 = 1.6, 13/8 = 1.625, 21/13 ≈ 1.615) gets closer and closer to φ. Some plants show Fibonacci-type spiral counts. But many popular claims about the golden ratio, such as that it appears in hurricanes, in every shell, or was used deliberately in ancient buildings, are exaggerated or unproven. Check a claim before you repeat it.
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1. HCF = 6, so 18 : 24 = 3 : 4.
2. 4 + 5 = 9 parts. One part = 900 ÷ 9 = 100. Shares: Rs 400 and Rs 500.
3. One pencil costs 60 ÷ 4 = Rs 15. Seven cost 7 × 15 = Rs 105.
1. HCF = 12, so the ratio is 2 : 3 : 5.
2. 3 + 5 + 4 = 12 parts. One part = 6,000 ÷ 12 = 500. Shares: Rs 1,500, Rs 2,500 and Rs 2,000.
3. Cross-multiply: 12x = 9 × 8 = 72, so x = 6.
4. Inverse proportion: total = 6 × 10 = 60 worker-days. Workers needed = 60 ÷ 4 = 15.
5. 7.5 × 50,000 = 375,000 cm = 3,750 m = 3.75 km.
Bar models are the most useful tool in this topic. Ask learners to draw the total as a bar and cut it into the right number of equal parts before they calculate anything. Practise deciding between direct and inverse proportion by asking “if one doubles, what happens to the other?”, and by using real contexts such as recipes, journeys and workers. Pay attention to units, and ask learners to check that their shares add up to the total. Recipes at home are a natural place to practise scaling.
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 of differently sized packages. They are also the direct foundation for percentages, since a percentage is simply a ratio to 100.
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