# Ratio and Proportion Questions with Answers and Formulas

> Ratio and proportion formulas, chained ratios, partnership profit sharing and mixture problems, with solved questions and explanations.

Source: https://www.sansal.in/aptitude/ratio-and-proportion · Last updated: 2026-09-19 · How often it appears: High. Shows up directly and inside mixture, partnership and age questions.

Ratios describe relative size without fixing actual values, which is exactly why they appear everywhere in aptitude papers. The technique that solves most of them is introducing a common multiplier k and letting the algebra do the rest.

## Concepts

### Introduce the multiplier k

If two quantities are in the ratio 3:5, write them as 3k and 5k. Every condition in the question then becomes an equation in k. This one habit turns most ratio questions into single-variable algebra.

### Chaining two ratios needs a common term

Given a:b = 2:3 and b:c = 4:5, you cannot just write 2:3:5 — b means 3 in one ratio and 4 in the other. Scale both so b matches: multiply the first by 4 and the second by 3, giving a:b:c = 8:12:15.

### Partnership profit follows capital × time

Profit is shared in proportion to the product of the money invested and how long it stayed invested. Someone who invests half as much for twice as long gets the same share.

## Formulas

| Formula | Rule | Note |
| --- | --- | --- |
| Proportion | If a:b = c:d then a × d = b × c |  |
| Dividing an amount in a ratio | Share = (that part / sum of parts) × Total |  |
| Chaining ratios | a:b = m:n and b:c = p:q → a:b:c = mp : np : nq |  |
| Partnership share | Profit share ∝ Capital × Time |  |
| Mean proportional between a and b | √(ab) |  |

## Worked examples

### Example 1

**Question:** Divide ₹600 between two people in the ratio 2:3.

**Solution:**

1. Sum of the ratio parts = 2 + 3 = 5
2. First share = (2/5) × 600 = 240
3. Second share = (3/5) × 600 = 360

**Answer:** ₹240 and ₹360

**Check:** They add back to ₹600, and 240:360 simplifies to 2:3.

### Example 2

**Question:** If a:b = 2:3 and b:c = 4:5, find a:b:c.

**Solution:**

1. b is 3 in the first ratio and 4 in the second — make them match at 12.
2. Multiply the first ratio by 4: a:b = 8:12
3. Multiply the second by 3: b:c = 12:15
4. Now b agrees, so chain them.

**Answer:** a:b:c = 8:12:15

### Example 3

**Question:** A invests ₹5,000 for 12 months and B invests ₹6,000 for 10 months. How should a profit of ₹2,200 be split?

**Solution:**

1. Profit follows capital × time, not capital alone.
2. A's weight = 5000 × 12 = 60,000
3. B's weight = 6000 × 10 = 60,000
4. The ratio is 60,000 : 60,000 = 1:1

**Answer:** ₹1,100 each

**Check:** B put in more money but for less time, and the two effects cancel exactly.

## Practice questions

1. ₹1,200 divided in the ratio 1:2:3 gives shares of:
   - A. 200, 400, 600
   - B. 100, 400, 700
   - C. 300, 400, 500
   - D. 150, 450, 600
   - **Answer:** A. 200, 400, 600 — Parts total 6. Shares = 1200 × (1/6, 2/6, 3/6) = 200, 400, 600.

2. If a:b = 3:4 and b:c = 8:9, then a:b:c is:
   - A. 3:4:9
   - B. 6:8:9
   - C. 3:8:9
   - D. 6:8:12
   - **Answer:** B. 6:8:9 — Make b common at 8: multiply the first ratio by 2 → 6:8. Second is already 8:9. So 6:8:9.

3. Two numbers are in the ratio 3:5 and their sum is 64. The numbers are:
   - A. 21 and 43
   - B. 24 and 40
   - C. 27 and 37
   - D. 30 and 34
   - **Answer:** B. 24 and 40 — 3k + 5k = 64 → 8k = 64 → k = 8. Numbers are 24 and 40.

4. A mixture of 40 litres has milk and water in the ratio 3:1. Water to be added to make it 1:1 is:
   - A. 10 litres
   - B. 15 litres
   - C. 20 litres
   - D. 25 litres
   - **Answer:** C. 20 litres — Milk = 30 L, water = 10 L. For 1:1 the water must reach 30 L, so add 20 L.

5. The mean proportional between 4 and 9 is:
   - A. 5
   - B. 6
   - C. 6.5
   - D. 7
   - **Answer:** B. 6 — √(4 × 9) = √36 = 6.

6. If 3 pens cost ₹45, then 7 pens cost:
   - A. ₹95
   - B. ₹100
   - C. ₹105
   - D. ₹110
   - **Answer:** C. ₹105 — One pen = ₹15. Seven pens = ₹105.

## Where marks get lost

- Chaining two ratios without first making the common term match.
- Sharing partnership profit by capital alone, ignoring the time invested.
- Treating ratio parts as actual quantities instead of introducing k.
- In mixture problems, changing the total when only one component was added.

## Related topics

- [Percentages](https://www.sansal.in/aptitude/percentages)
- [Averages](https://www.sansal.in/aptitude/averages)
- [Profit & Loss](https://www.sansal.in/aptitude/profit-and-loss)

Practise these on Sansal: https://www.sansal.in/practice/aptitude