The core ideas
- Ratio a : b compares two quantities of the same kind. Multiply or divide both terms by the same number to get an equal ratio.
- Proportion: a : b = c : d means a × d = b × c.
- Direct proportion: as one quantity rises, the other rises in the same ratio (cost and quantity).
- Inverse proportion: as one rises, the other falls (men and days for fixed work).
- Sharing: to divide N in the ratio a : b, the parts are N × a ÷ (a + b) and N × b ÷ (a + b).
Worked examples
Example 1. Share 1,800 in the ratio 5 : 4. The total parts are 9, each worth 200, so 1,000 and 800.
Example 2. The ratio of boys to girls is 3 : 2 in a class of 40. Boys = 24, girls = 16. If 4 more girls join, the new ratio is 24 : 20 = 6 : 5.
Example 3 (changing ratio). Two numbers are in the ratio 3 : 5. If 10 is added to each, the ratio becomes 5 : 7. Let the numbers be 3x and 5x: (3x + 10) ÷ (5x + 10) = 5 ÷ 7. Cross-multiplying: 21x + 70 = 25x + 50; 4x = 20; x = 5. The numbers are 15 and 25. Check: 25 : 35 = 5 : 7 ✓.
Example 4 (inverse proportion). 12 workers finish a job in 20 days. How many days will 15 workers take? Man-days = 240; 240 ÷ 15 = 16 days.
Example 5 (partnership). A invests 6,000 for 12 months and B invests 8,000 for 6 months. Profit is shared in the ratio of investment × time: 72,000 : 48,000 = 3 : 2. On a profit of 10,000, A gets 6,000 and B gets 4,000.
Example 6 (compound ratio). If a : b = 2 : 3 and b : c = 4 : 5, then a : b : c = 8 : 12 : 15 (make b equal: multiply the first by 4 and the second by 3).
Common traps
- Comparing unlike units (minutes with hours).
- Adding the same number to both terms and expecting the ratio to stay.
- Confusing direct and inverse proportion.
- Forgetting time in partnership problems.
Speed tips
- Use "parts": convert ratios to parts and find the value of one part.
- For inverse proportion, keep the product (man-days) constant.
- In partnership, use investment × time as the share ratio.
A practice routine
- Day 1: sharing and equal ratios, ten questions.
- Day 2: changing ratios and algebra setup.
- Day 3: proportion, direct and inverse, ten questions.
- Day 4: partnership and a timed set of 12 mixed questions at 50 seconds each.
Where it appears
Ratio and proportion appear in SSC CGL, banking, CLAT and MAT papers, and underpin time and work and profit and loss.
Practice tests below are published by institutes and educators on TestMint.
