A and B invested in a business in the ratio 5:6. The profit ratio was 10:9. If B invested his money for 12 months, for how many months did A invest?

Correct — B. Profit is shared in the ratio of capital × time.

A: capital × time = 5 × t = 5t
B: capital × time = 6 × 12 = 72
Set the ratio: 5t : 72 = 10 : 9

Cross-multiply: 5t × 9 = 72 × 10, so 45t = 720
Divide: t = 720 ÷ 45 = 16 months → option (b)

Check: 5 × 16 = 80, and 80 : 72 = 10 : 9

Why the others are wrong

• (a)
  15 months — 15 months gives 5 × 15 = 75 against 72, a profit ratio of 75 : 72 = 25 : 24, not 10 : 9.
• (c)
  17 months — 17 months gives 5 × 17 = 85 against 72, and 85 : 72 does not reduce to 10 : 9 (that would need 80).
• (d)
  18 months — 18 months gives 5 × 18 = 90 against 72, a profit ratio of 90 : 72 = 5 : 4, more than 10 : 9.

In a partnership, each partner's profit share is proportional to capital × time. Knowing any two of the three ratios (capital, time, profit) fixes the third.

Here A put in less money (5 against 6) yet took the larger profit (10 against 9), so A must have stayed longer than B's 12 months. Term by term, time ratio = (10 ÷ 5) : (9 ÷ 6) = 2 : 1.5 = 4 : 3, and 4⁄3 × 12 = 16.

Key facts

• Profit ratio = (capital of A × time of A) : (capital of B × time of B).
• Time ratio = profit ratio ÷ capital ratio, term by term: (10 ÷ 5) : (9 ÷ 6) = 4 : 3.
• 'Invested in the ratio 5:6' describes the capitals, not the profits.

Common traps

• Equating the capital ratio 5 : 6 to the profit ratio 10 : 9 and forgetting that time multiplies capital.
• Attaching the 12 months to A instead of B: 5 × 12 : 6 × t = 10 : 9 gives t = 9, which is not an option.

No directly related past PYQ was found.