In this guide
A ratio compares two quantities of the same kind. If a class has 20 boys and 30 girls, the ratio of boys to girls is 20 : 30, which simplifies to 2 : 3. For every 2 boys there are 3 girls.
Ratio is worth mastering early because several other chapters rest on it: ages, partnership, mixtures, and the sharing of wages in time and work. Almost every ratio question is solved with one idea, the value of one part. Find it, and the rest is multiplication.
Rules to know first
- Order matters. Boys : girls = 2 : 3 is not the same as girls : boys, which is 3 : 2.
- Same units. 50 paise : ₹2 must become 50 paise : 200 paise = 1 : 4.
- A ratio is not an actual amount. 2 : 3 might mean 2 and 3, or 20 and 30, or 200 and 300.
Simplifying a ratio
Divide both terms by their HCF: 45 : 60 → divide by 15 → 3 : 4.
If the terms are fractions, multiply every term by the LCM of the denominators. For 1/2 : 1/3, multiply by 6 to get 3 : 2.
Dividing an amount in a ratio
- Add the terms of the ratio to get the total parts.
- Divide the amount by the total parts to get one part.
- Multiply one part by each term.
If you are given a difference instead of a total, subtract the terms instead of adding them. The rest is the same.
Combining two ratios
If A : B and B : C are given, make the B terms equal. Multiply each ratio so that B becomes the LCM of its two values.
Proportion
Four numbers a, b, c, d are in proportion (written a : b :: c : d) when a/b = c/d. The key property is:
Product of the outer terms = product of the middle terms, that is, a × d = b × c.
| Term | Meaning | Example |
|---|---|---|
| Fourth proportional of a, b, c | x such that a : b = c : x | 4, 6, 10 → x = 6 × 10 ÷ 4 = 15 |
| Mean proportional of a and b | x such that a : x = x : b, so x = √(ab) | 4 and 9 → √36 = 6 |
| Third proportional of a and b | x such that a : b = b : x, so x = b² ÷ a | 4 and 6 → 36 ÷ 4 = 9 |
Direct and inverse proportion
| Type | What happens | Example | Method |
|---|---|---|---|
| Direct | Both rise together | More pens, more cost | Find the cost of one, then multiply |
| Inverse | One rises, the other falls | More workers, fewer days | Product stays the same |
If 5 pens cost ₹40, one pen costs ₹8 and 8 pens cost ₹64. If 6 workers take 10 days, 12 workers take 6 × 10 ÷ 12 = 5 days.
Solved examples
Example 1. Simplify 1/2 : 2/3 : 3/4.
- LCM of 2, 3 and 4 is 12.
- Multiply each term by 12: 6 : 8 : 9.
- Answer: 6 : 8 : 9.
Example 2. Divide ₹3,600 among A, B and C in the ratio 2 : 3 : 4.
- Total parts = 2 + 3 + 4 = 9.
- One part = 3,600 ÷ 9 = ₹400.
- Shares: ₹800, ₹1,200 and ₹1,600. Check: they add to ₹3,600.
Example 3. Two numbers are in the ratio 5 : 8, and their difference is 36. Find them.
- Difference in parts = 8 − 5 = 3.
- 3 parts = 36, so one part = 12.
- Numbers: 5 × 12 = 60 and 8 × 12 = 96.
Example 4. If A : B = 2 : 3 and B : C = 4 : 5, find A : B : C.
- B is 3 in the first ratio and 4 in the second. LCM of 3 and 4 is 12.
- Multiply the first ratio by 4: A : B = 8 : 12.
- Multiply the second by 3: B : C = 12 : 15.
- So A : B : C = 8 : 12 : 15.
Example 5. Two numbers are in the ratio 3 : 5. If 10 is added to each, the ratio becomes 5 : 7. Find the numbers.
- Let the numbers be 3x and 5x.
- (3x + 10) ÷ (5x + 10) = 5/7.
- Cross-multiply: 7(3x + 10) = 5(5x + 10), so 21x + 70 = 25x + 50.
- 4x = 20, so x = 5. Numbers: 15 and 25.
- Check: 25 : 35 = 5 : 7.
Example 6. A bag has ₹1, 50 paise and 25 paise coins in the ratio 1 : 2 : 4 by number. The total value is ₹60. How many coins of each kind are there?
- Let the numbers be x, 2x and 4x.
- Value in rupees: 1 × x + 0.5 × 2x + 0.25 × 4x = x + x + x = 3x.
- 3x = 60, so x = 20.
- Coins: 20 one-rupee, 40 fifty-paise and 80 twenty-five-paise.
Common mistakes
| Mistake | Correct way |
|---|---|
| Comparing ₹1 with 50 paise as 1 : 50 | Same units: 100 : 50 = 2 : 1 |
| Reversing the order of the ratio | Keep the order the question gives |
| Combining A : B and B : C without equalising B | Make B the same in both first |
| Counting coins instead of their value | Convert each kind to its rupee value |
| Treating an inverse case as direct | More workers means fewer days |
Practice set
- Simplify 72 : 96.
- Divide ₹2,400 in the ratio 3 : 5.
- The ratio of boys to girls in a school is 7 : 5. There are 480 students. How many are boys?
- Two numbers are in the ratio 4 : 7 and their sum is 132. Find them.
- If A : B = 3 : 4 and B : C = 6 : 7, find A : B : C.
- Find the fourth proportional to 3, 5 and 12.
- Find the mean proportional of 9 and 16.
- Two numbers are in the ratio 2 : 3. If 5 is added to each, the ratio becomes 3 : 4. Find the numbers.
Answers:
- HCF 24: 3 : 4.
- 8 parts, one part ₹300: ₹900 and ₹1,500.
- 12 parts, one part 40: boys = 280.
- 11 parts = 132, one part 12: 48 and 84.
- LCM of 4 and 6 is 12: A : B = 9 : 12, B : C = 12 : 14, so 9 : 12 : 14.
- 3 : 5 = 12 : x, so x = 5 × 12 ÷ 3 = 20.
- √(9 × 16) = √144 = 12.
- (2x + 5) ÷ (3x + 5) = 3/4 gives 8x + 20 = 9x + 15, so x = 5: 10 and 15. Check: 15 : 20 = 3 : 4.
What to do next
- Solve 10 "divide an amount" questions until each takes under 30 seconds.
- Practise five combine-the-ratio questions a day for a week.
- Use ratios in ages and mixture and partnership.
- Revise percentage: a ratio of 1 : 4 means the first is 25% of the second.
A note on dates and numbers. Exam patterns, vacancies and schedules change from year to year. Always confirm the current details in the latest notification on the Staff Selection Commission website .
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