Ratio and proportion
A ratio compares two quantities of the same kind; a proportion says two ratios are equal, so the product of extremes equals the product of means.
Key terms
| Term | Meaning | Example |
|---|---|---|
| Ratio | Comparison of like quantities by division | 30 girls : 20 boys = 3 : 2 |
| Proportion | Two equal ratios, a : b :: c : d | 2 : 3 :: 8 : 12 |
| Extremes and means | a and d are extremes, b and c are means; ad = bc | 2 × 12 = 3 × 8 |
| Fourth proportional to a, b, c | d = bc ÷ a | to 4, 6, 10 → 15 |
| Third proportional to a, b | b² ÷ a | to 4, 6 → 9 |
| Mean proportional of a, b | √(ab) | of 4 and 9 → 6 |
| Compounded ratio | Multiply the ratios | (2 : 3) and (4 : 5) → 8 : 15 |
| Duplicate ratio | a² : b² | of 2 : 3 → 4 : 9 |
| Direct proportion | Both rise together | more rice, more cost |
| Inverse proportion | One rises as the other falls | more workers, fewer days |
Worked example 1: combining ratios and sharing
If A : B = 2 : 3 and B : C = 4 : 5, find A : B : C, and divide ₹7,000 among A, B and C in this ratio.
- Standard method: make the B terms equal. B is 3 in one ratio and 4 in the other; LCM 12. A : B = 8 : 12 and B : C = 12 : 15, so A : B : C = 8 : 12 : 15.
- Shortcut (cross-multiplying): A = 2 × 4 = 8; B = 3 × 4 = 12; C = 3 × 5 = 15.
- Sharing: 8 + 12 + 15 = 35 parts; one part = 7,000 ÷ 35 = ₹200. Shares: ₹1,600, ₹2,400 and ₹3,000.
- Check: 8 : 12 = 2 : 3, 12 : 15 = 4 : 5, and 1,600 + 2,400 + 3,000 = 7,000.
A : B : C = 8 : 12 : 15; shares ₹1,600, ₹2,400 and ₹3,000.
Worked example 2: a changing ratio
In a hostel, the ratio of boys to girls is 5 : 3. When 20 more girls join, the ratio becomes 5 : 4. How many boys are there?
- Standard method: let boys = 5k and girls = 3k. Then 5k : (3k + 20) = 5 : 4, so 20k = 15k + 100 and k = 20. Boys = 100.
- Shortcut: the number of boys does not change, and it is 5 parts in both ratios, so the parts are the same size. Girls go from 3 parts to 4 parts, so 1 part = 20. Boys = 5 × 20 = 100.
- Check: 100 boys and 60 girls; with 80 girls, 100 : 80 = 5 : 4.
100 boys.
Worked example 3: scaling a recipe (direct proportion)
A recipe for 6 people uses 450 g of rice. How much rice is needed for 20 people?
- Standard method (unitary): rice per person = 450 ÷ 6 = 75 g. For 20 people: 75 × 20 = 1,500 g.
- Shortcut (proportion): 6 : 20 :: 450 : x, so x = 450 × 20 ÷ 6 = 1,500 g.
- Check: 1,500 ÷ 20 = 75 g per person, the same as in the original recipe.
1,500 g (1.5 kg).
Worked example 4: making numbers proportional
What number must be added to each of 6, 15, 20 and 43 so that they become proportional?
- Standard method: (6 + x) : (15 + x) :: (20 + x) : (43 + x), so (6 + x)(43 + x) = (15 + x)(20 + x). Then 258 + 49x = 300 + 35x, so 14x = 42 and x = 3.
- Shortcut: x = (bc − ad) ÷ (a + d − b − c) = (15 × 20 − 6 × 43) ÷ (6 + 43 − 15 − 20) = 42 ÷ 14 = 3.
- Check: 9 : 18 = 1 : 2 and 23 : 46 = 1 : 2.
3.
Every bank, SSC, railway and state PSC paper uses ratio, both directly (sharing money, combining ratios, changing ratios) and inside ages, partnership, mixtures and time-work. Placement tests like the 'what must be added to make them proportional' type.
2 marks: “Find the mean proportional between 9 and 16.” √(9 × 16) = √144 = 12.
5 marks: “If A : B = 2 : 3 and B : C = 4 : 5, find A : B : C and divide ₹7,000 in that ratio.” 8 : 12 : 15; one part ₹200; shares ₹1,600, ₹2,400 and ₹3,000.
10 marks: “(a) Boys and girls are 5 : 3; 20 more girls join and the ratio becomes 5 : 4: find the boys. (b) A recipe for 6 uses 450 g of rice: find the rice for 20. (c) Find the number to be added to 6, 15, 20 and 43 to make them proportional.” (a) 100. (b) 1,500 g. (c) 3. Show the ratio working and a check for each.
Ratio = comparison; proportion = equal ratios. In a : b :: c : d, extremes × extremes = means × means (ad = bc). Combine ratios by making the common term equal.