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Working alone, A can complete a task in 27 hours and B can complete the same task in 54 hours. A, B, and C together finish the task in 12 hours and receive Rs 4,320 as total wages. What is the share of C in these wages?

Difficulty: Medium

Correct Answer: 1440

Explanation:


Introduction / Context:

This question combines time and work with wage distribution based on individual contributions. It assesses the ability to find individual work rates, use combined rate information to determine the rate of a third worker, and then divide total payment in proportion to work done. Such problems appear frequently in aptitude tests where profit sharing and wage sharing are tied to work rates.


Given Data / Assumptions:

  • A alone can complete the task in 27 hours.
  • B alone can complete the task in 54 hours.
  • A, B, and C together complete the task in 12 hours.
  • Total wages paid for the completed task are Rs 4,320.
  • Wages are shared in proportion to the amount of work each person actually does.


Concept / Approach:

The key steps are to convert the given completion times into hourly work rates, then use the combined completion time with three workers to calculate C's rate. Once we know each worker's rate, we compute how much of the total task each worker completes during the 12 hours. The fraction of the total work done by C is then used to find C's share of the total wages, since payment is proportional to contribution.


Step-by-Step Solution:

Let total work = 1 unit.Rate of A per hour = 1/27.Rate of B per hour = 1/54.A, B, and C together finish in 12 hours, so combined rate = 1/12 per hour.Let rate of C per hour be r_c. Then 1/27 + 1/54 + r_c = 1/12.Compute 1/27 + 1/54: common denominator is 54, so 1/27 = 2/54, and 2/54 + 1/54 = 3/54 = 1/18.So 1/18 + r_c = 1/12, hence r_c = 1/12 - 1/18.Compute r_c: 1/12 - 1/18 = (3 - 2) / 36 = 1/36.In 12 hours, total work done by A = 12 * 1/27 = 12/27 = 4/9 of the job.Work done by B = 12 * 1/54 = 12/54 = 2/9 of the job.Work done by C = 12 * 1/36 = 12/36 = 1/3 of the job.Expressed with a common denominator 9, these are 4/9, 2/9, and 3/9 respectively, so the work ratio is A : B : C = 4 : 2 : 3.Total parts = 4 + 2 + 3 = 9 parts, so each part corresponds to 4320 / 9 = Rs 480.C gets 3 parts, so C's share = 3 * 480 = Rs 1,440.


Verification / Alternative Check:

The shares of A and B would then be 4 * 480 = Rs 1,920 and 2 * 480 = Rs 960. Summing all three gives 1,920 + 960 + 1,440 = Rs 4,320, which matches the total wages. Also, using C's rate of 1/36 with the others gives a combined rate of 1/18 + 1/36 = 3/36 = 1/12, so they finish in 12 hours, consistent with the problem statement.


Why Other Options Are Wrong:

Options like Rs 960 or Rs 1,280 assume a smaller share for C than is justified by the work ratio of 4 : 2 : 3. Option Rs 1,920 would overpay C and underpay A. Only Rs 1,440 respects the proportional distribution based on the calculated fractions of work completed by each worker.


Common Pitfalls:

Students may forget to multiply the rates by 12 hours to find the exact amount of work done by each worker. Others may incorrectly assume equal sharing or use only time ratios without converting them to actual work contributions. Keeping the focus on work rates and actual fractions of the total job prevents such errors.


Final Answer:

C's correct share of the wages is Rs 1,440.

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