Problems on Ages
Subject: Mental Ability | Frequency: 1-2 questions per APPSC paper | Time: 60-90 sec/question
Introduction
Age problems use algebraic equations to find ages based on past, present, or future conditions. The cross-multiplication technique for ratio-based problems covers 80%+ of exam questions.
Core Method
- Let present age = x (use ONE variable)
- Age after n years = x + n; Age n years ago = x − n
- If ratio is p:q, ages are px and qx
- Form equations, solve, verify
Key Formulas
| Concept | Formula |
|---|---|
| Age after n years | x + n |
| Age n years ago | x − n |
| Ratio a:b | Ages = ax, bx |
| Sum + Difference | Older = (S+D)/2, Younger = (S−D)/2 |
| Age difference never changes | Critical insight |
Worked Examples — Easy
Q1: Father is 3× son's age. After 12 years, twice son's age. Find ages. Son = x, Father = 3x. 3x+12 = 2(x+12) → x = 12. Son=12, Father=36
Q2: A+B = 50, A = B+10. → B=20, A=30. A=30, B=20
Q3: Ratio 5:4. After 8 years, 6:5. → (5x+8)/(4x+8) = 6/5 → x=8. A=40, B=32
Worked Examples — Medium
Q4: 5 years ago ratio 3:1. Sum = 50. → A−5 = 3(B−5), A+B=50 → B=15, A=35
Q5: Ratio 3:5. After 9 years, 3:4. → 12x+36 = 15x+27 → x=3. A = 9 years
Worked Examples — Hard
Q6: Man 24 years older than son. In 2 years, twice son's age. Son+24+2 = 2(Son+2) → Son = 22 years
Q7: A = 2×(B's age when A was B's current age). Sum = 63. 3a = 4b, a+b = 63 → b=27, a=36. Verify: When A was 27, B was 18. 2×18=36 ✓
Shortcuts & Tricks
| Shortcut | Details |
|---|---|
| One variable only | Express all ages in terms of x |
| Ratio → multiply by k | a:b means ak and bk |
| Both age equally | Everyone ages by exactly n years |
| Cross multiply ratios | (a+n)/(b+n) = p/q → q(a+n) = p(b+n) |
| Difference is constant | Age gap between two people never changes |
| Sum+Difference shortcut | Older = (S+D)/2 |
Common Mistakes
- Forgetting both people age equally
- Wrong time direction ("5 years ago" = subtract)
- Ratio confusion (which person is numerator?)
- Not verifying (negative/non-integer ages = wrong)
- Multiple variables when one suffices
Practice Questions
- Father 4× daughter's age. In 6 years, 2.5× → Daughter = 18, Father = 72? Let d, 4d. 4d+6 = 2.5(d+6) → 1.5d = 9 → d=6. D=6, F=24
- A:B = 7:5. Difference = 8. → 2x=8, x=4 → A=28, B=20
- 6 years ago, A was 3× B. A+B = 54 now. → A = 39, B = 15
- Ratio 4:3. After 6 years, 5:4. → x=6. A=24, B=18
- Mother is 25 years older than daughter. In 5 years, twice daughter's age. → D+25+5 = 2(D+5) → D = 20