Ratio-of-Ages Problems

🎯 Learning Objective

Solve age problems where present or changed ages are given as ratios, using a common multiplier.

💡 Concept

• If two ages are in ratio a : b, write them as ax and bx — one unknown multiplier x for both.
• For a future/past condition, add or subtract the SAME number of years from each part.
• The second ratio gives an equation; cross-multiply and solve for x.
• Once x is known, each present age = (its ratio part) × x.
• The multiplier x is a real number of years worth of scaling — it must be positive.

🧮 Key Formulas

Ages in ratio a : b → write as ax and bx

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Present age = ratio-part × x

✏️ Easy Example

Q. The ages of A and B are in the ratio 5 : 7 and their sum is 48 years. Find their ages.

1. Let ages be 5x and 7x
2. 5x + 7x = 48 → 12x = 48
3. x = 4
4. A = 5×4 = 20, B = 7×4 = 28

Answer: A = 20 years, B = 28 years

🇮🇳 Real-Life Example

Selectors compare candidate experience like 'seniors to juniors 3 : 2' — the same single-multiplier trick that cracks age ratios also splits any real-world ratio into actual numbers.

📝 Exam-Level Example

Q. The present ages of A and B are in the ratio 4 : 5. After 5 years, the ratio becomes 5 : 6. Find their present ages.

1. Present ages = 4x and 5x
2. After 5 years: (4x + 5) / (5x + 5) = 5/6
3. 6(4x + 5) = 5(5x + 5)
4. 24x + 30 = 25x + 25
5. x = 5 → A = 20, B = 25

Answer: A = 20 years, B = 25 years

📝 Exam-Level Example

Q. The ratio of the ages of a father and son is 7 : 3 and the product of their ages is 756. What will be the ratio of their ages after 6 years?

1. Ages = 7x and 3x. Product = 7x × 3x = 21x² = 756
2. x² = 36 → x = 6
3. Father = 42, Son = 18
4. After 6 years: 48 and 24 → ratio 48 : 24 = 2 : 1

Answer: 2 : 1

🪄 Memory Trick

One ratio needs one variable x; a second condition (sum, product, or a later ratio) gives the equation to pin down x.

⚠️ Common Mistakes

• ❌ Using two different variables for a single ratio (only one x is needed)
• ❌ Adding years to only the numerator or only the denominator
• ❌ Forgetting to multiply x back to get the actual ages

🏆 Exam Tips

• ✅ Ratio 5 : 7 always means 5x : 7x — never 5 and 7 directly
• ✅ For a 'product of ages' clue, expect a quadratic like 21x² = 756

📌 Summary

• Ratio a : b → ages ax and bx
• Apply the same +t or −t to both parts
• Second condition → equation for x
• Actual age = ratio-part × x