Three partners A, B and C entered into a business. A invested one-third of the capital for one-third duration. B invested one-fourth of the capital for one-fourth duration. C invested the remaining capital for the whole duration. Out of a profit of ₹ 17,000, how much profit will C get?

Why the correct option is correct: In business partnerships, profit is distributed in proportion to the product of the capital invested and the time period for which it is invested (Profit ∝ Capital × Time). This concept is a core component of the UPSC Civil Services Aptitude Test (CSAT) Paper-II quantitative syllabus.

Let the total capital be 1 unit and the total duration be 1 unit.

• Partner A: Invests ⅓ capital for ⅓ time. Product = ⅓ × ⅓ = ⅑.
• Partner B: Invests ¼ capital for ¼ time. Product = ¼ × ¼ = 1/16.
• Partner C: Invests the remaining capital for the whole duration. Remaining capital = 1 - (⅓ + ¼) = 1 - 7/12 = 5/12. Duration = 1. Product = 5/12 × 1 = 5/12.

The ratio of profit shares for A : B : C is ⅑ : 1/16 : 5/12. To simplify, multiply by the least common multiple (LCM) of the denominators (9, 16, and 12), which is 144.

• A's share = 144 × (⅑) = 16
• B's share = 144 × (1/16) = 9
• C's share = 144 × (5/12) = 60

The simplified profit ratio is 16 : 9 : 60. The total number of ratio parts is 16 + 9 + 60 = 85. Given the total profit is ₹ 17,000, the value of one part is 17,000 / 85 = ₹ 200. Therefore, C's profit share is 60 × 200 = ₹ 12,000. Thus, Option A is correct.

Why the incorrect options are wrong:

• Option B (₹ 10,000): Incorrect; this figure typically results from a fundamental error, such as failing to multiply the capital by the respective time duration.
• Option C (₹ 12,500): Incorrect; this arises if a candidate mistakenly calculates C's investment fraction incorrectly without correctly subtracting A and B's shares.
• Option D (₹ 10,750): Incorrect; this is an arbitrary distractor value resulting from arithmetic errors when distributing the ₹ 17,000 among miscalculated ratio parts.

Takeaway: In partnership problems, always rely on the rule: Profit Share = Capital × Time. Calculate the remaining fractions systematically before establishing the final simplified ratio using the LCM.