In this guide
A good way to think about ratios is in parts. If money is shared between Asha and Bilal in the ratio 3 : 5, imagine the whole amount split into 8 equal parts: Asha gets 3 of them and Bilal gets 5. Almost every ratio question becomes easy once you ask, "How many parts, and what is one part worth?"
Basics
- A ratio a : b compares two quantities of the same kind: a/b.
- Ratios can be simplified like fractions: 24 : 36 = 2 : 3.
- Multiplying or dividing both terms by the same number doesn't change a ratio.
Dividing a quantity
Worked example: Divide ₹4,800 between A and B in the ratio 5 : 7.
Total parts = 12; one part = 400. A gets 5 × 400 = ₹2,000; B gets ₹2,800.
Comparing ratios
To compare 3 : 5 and 5 : 8, compare the fractions 3/5 = 0.6 and 5/8 = 0.625. So 5 : 8 is larger. Or cross-multiply: 3 × 8 = 24 < 5 × 5 = 25.
Combining ratios
If A : B = 2 : 3 and B : C = 4 : 5, find A : B : C.
Make B the same in both: multiply the first ratio by 4 and the second by 3.
A : B = 8 : 12 and B : C = 12 : 15, so A : B : C = 8 : 12 : 15.
Changing ratios
Worked example: The ratio of boys to girls in a class is 4 : 5. If 8 more boys join, the ratio becomes 1 : 1. How many girls are there?
Let boys = 4x and girls = 5x. Then 4x + 8 = 5x, so x = 8. Girls = 40.
Proportion
Four numbers a, b, c, d are in proportion if a : b = c : d, that is, ad = bc.
| Term | Formula |
|---|---|
| Fourth proportional to a, b, c | d = bc/a |
| Third proportional to a, b | c = b²/a |
| Mean proportional between a and b | √(ab) |
Worked example: The mean proportional between 9 and 16 = √144 = 12.
Direct and inverse proportion
- Direct: more workers, more output (at a constant rate).
- Inverse: more workers, fewer days to finish the same job.
Partnership
Profits are shared in the ratio of capital × time.
Worked example: A invests ₹40,000 for 12 months and B invests ₹60,000 for 6 months. Profit is ₹21,000. Find each share.
A : B = 40,000 × 12 : 60,000 × 6 = 4,80,000 : 3,60,000 = 4 : 3.
A gets 4/7 × 21,000 = ₹12,000; B gets ₹9,000.
Worked example (changing capital): A starts with ₹30,000. After 4 months, A withdraws ₹10,000. B invests ₹25,000 for the full year. Find the ratio of profits.
A = 30,000 × 4 + 20,000 × 8 = 1,20,000 + 1,60,000 = 2,80,000.
B = 25,000 × 12 = 3,00,000.
Ratio = 14 : 15.
Ratio in age problems
Worked example: The ages of a father and son are in the ratio 7 : 2. After 10 years, the ratio will be 9 : 4. Find their present ages.
7x + 10 : 2x + 10 = 9 : 4 → 4(7x + 10) = 9(2x + 10) → 28x + 40 = 18x + 90 → x = 5.
Father = 35, son = 10.
Common traps
| Trap | Correct approach |
|---|---|
| Adding ratios directly | Convert to a common term first |
| Forgetting time in partnership | Multiply capital by months |
| Treating a ratio as actual numbers | Use "x" as the value of one part |
Practice
- Divide ₹3,600 in the ratio 2 : 3 : 4.
- If A : B = 3 : 4 and B : C = 6 : 7, find A : B : C.
- Find the fourth proportional to 4, 9 and 12.
- A and B invest ₹50,000 and ₹75,000. A invests for 12 months and B for 8 months. Find the ratio of their profits.
- Two numbers are in the ratio 3 : 5. If 9 is added to each, the ratio becomes 3 : 4. Find the numbers.
- Find the mean proportional between 0.25 and 16.
Answers: 1. ₹800, ₹1,200, ₹1,600. 2. 9 : 12 : 14. 3. 27. 4. 1 : 1. 5. 9 and 15. 6. 2.
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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