In this guide
Ratio questions and age problems look different but use the same tool: let one part be x and translate the words into an equation. Age problems have one extra fact that makes them easier: everyone ages at the same rate, so the difference between two people's ages never changes.
Ratio basics
- A ratio 3 : 5 means the quantities are 3x and 5x for some x.
- Dividing an amount in the ratio a : b gives a/(a + b) and b/(a + b) of it.
Worked example: Divide ₹2,700 between P and Q in the ratio 4 : 5.
Total parts = 9; one part = 300. P = ₹1,200, Q = ₹1,500.
Combining ratios
If A : B = 3 : 4 and B : C = 6 : 5, make B equal: A : B = 9 : 12 and B : C = 12 : 10.
A : B : C = 9 : 12 : 10.
Coins
Worked example: A bag has ₹1, 50-paise and 25-paise coins in the ratio 2 : 3 : 4. The total value is ₹135. How many 50-paise coins are there?
Values: 2x × ₹1 + 3x × ₹0.50 + 4x × ₹0.25 = 2x + 1.5x + x = 4.5x = 135 → x = 30.
Fifty-paise coins = 3x = 90.
Income and expenditure
Worked example: The incomes of A and B are in the ratio 5 : 4, and their expenditures in the ratio 3 : 2. If each saves ₹4,000, find their incomes.
Income 5x and 4x; expenditure 3y and 2y.
5x − 3y = 4,000 and 4x − 2y = 4,000.
From the second, y = 2x − 2,000. Substitute: 5x − 6x + 6,000 = 4,000 → x = 2,000.
Incomes: ₹10,000 and ₹8,000.
Problems on ages
Worked example: The present ages of a mother and her daughter are in the ratio 7 : 2. After 5 years, the ratio will be 8 : 3. Find their present ages.
(7x + 5)/(2x + 5) = 8/3 → 21x + 15 = 16x + 40 → 5x = 25 → x = 5.
Present ages: 35 and 10. Check: in 5 years they will be 40 and 15, and 40 : 15 = 8 : 3. ✓
Worked example: The ages of two siblings are in the ratio 3 : 5. Eight years ago the ratio was 1 : 3. Find their present ages.
(3x − 8)/(5x − 8) = 1/3 → 9x − 24 = 5x − 8 → x = 4.
Present ages: 12 and 20.
Using options
Age questions are often fastest by testing the options: check which pair fits both conditions.
Worked example: The sum of the ages of a father and son is 50. Five years ago, the father was 7 times as old as the son. Find their present ages.
Option (40, 10): five years ago, 35 and 5 → 35 = 7 × 5 ✓. Answer: 40 and 10.
Practice
- Divide ₹3,600 among A, B and C in the ratio 2 : 3 : 7.
- If A : B = 2 : 3 and B : C = 4 : 5, find A : B : C.
- The ratio of the ages of two people is 4 : 7. The sum of their ages is 66. Find the elder's age.
- Six years ago, the ratio of the ages of X and Y was 3 : 4. The sum of their present ages is 54. Find X's present age.
- The incomes of two people are in the ratio 3 : 2 and their expenditures in the ratio 5 : 3. If each saves ₹2,000, find their incomes.
Answers: 1. ₹600, ₹900, ₹2,100. 2. 8 : 12 : 15. 3. 42. 4. 24 (six years ago: 18 and 24, sum 42 + 12 = 54). 5. ₹12,000 and ₹8,000.
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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