The monthly incomes of A, B and C are in the ratio 2 : 9 : 3 and their expenses are in the ratio 3 : 9 : 5. If A’s saving is half of his total income, then savings of A, B and C are, respectively, in the ratio of:

Correct — C. Incomes and expenses need two different constants — one ratio may not be scaled by the other's multiplier.

Incomes: 2k, 9k, 3k. Expenses: 3m, 9m, 5m.
A saves half his income: 2k − 3m = k → 3m = k, so m = k⁄3.

Saving of A = k
Saving of B = 9k − 9m = 9k − 3k = 6k
Saving of C = 3k − 5m = 3k − 5k⁄3 = 4k⁄3

k : 6k : 4k⁄3, every term ×3 → 3 : 18 : 4 → option (c).

Why the others are wrong

• (a)
  3 : 9 : 2 — A saves k and B saves 6k, so A : B must be 1 : 6. This option makes it 1 : 3, which means B's expenses have been over-subtracted.
• (b)
  2 : 5 : 2 — It gives A and C the same saving, but A saves k against C's 4k⁄3. A : B is also 2 : 5 here instead of the required 1 : 6.
• (d)
  1 : 2 : 1 — Once m = k⁄3 is fixed, B saves six times what A saves, not twice. This option also equates A's and C's savings, which the 4k⁄3 rules out.

Two ratios in one question need two independent constants. Incomes are 2k, 9k, 3k and expenses 3m, 9m, 5m — writing both with the same k is the error the question is built to punish.

The extra condition supplies the link. A saving half his income becomes 2k − 3m = k, which pins m = k⁄3 and lets every expense be rewritten in k.

Saving = income − expenditure throughout, so all three savings then land in one unknown and the ratio survives.

Key facts

• Savings = income − expenditure, so both must be expressed in a common unknown before a savings ratio exists.
• Two independent ratios in one question require two independent constants.
• Multiplying every term of a ratio by the same number leaves the ratio unchanged, which turns 1 : 6 : 4⁄3 into 3 : 18 : 4.

Common traps

• Using one constant for both ratios, which turns A's saving into 2k − 3k and makes it negative.
• Multiplying only the fractional term when clearing 4k⁄3, instead of scaling all three parts.
• Reading a saving of half the income as a saving of half the expenditure.

No directly related past PYQ was found.