To finish a piece of work, P takes five times as long as Q, and six times as…

2025

To finish a piece of work, P takes five times as long as Q, and six times as long as R. Working together, they can finish the work in 4 days. In how many days can Q alone complete the work?

  1. A.

    48/5

  2. B.

    8

  3. C.

    48

  4. D.

    12/5

Show answer & explanation

Correct answer: A

Work-rate principle: if a person completes a task alone in D days, their work rate is 1/D of the task per day. When several people work together, their combined work rate is the sum of their individual rates, and the time they take together equals 1 divided by that combined rate.

  1. Let the number of days Q takes to finish the work alone be x.

  2. Since P takes five times as long as Q, P's time alone is 5x days.

  3. Since P takes six times as long as R, R's time alone is P's time divided by 6, i.e. 5x/6 days.

  4. Individual work rates: P's rate = 1/(5x), Q's rate = 1/x, R's rate = 6/(5x).

  5. Combined rate = 1/(5x) + 1/x + 6/(5x) = (1 + 5 + 6)/(5x) = 12/(5x).

  6. Since together they finish the work in 4 days, the combined rate must equal 1/4.

  7. Setting 12/(5x) = 1/4 gives 5x = 48, so x = 48/5.

Cross-check: with x = 48/5, Q's rate = 5/48, P's time = 5x = 48 days (rate 1/48), and R's time = 5x/6 = 8 days (rate 6/48). Combined rate = 5/48 + 1/48 + 6/48 = 12/48 = 1/4, matching the given combined time of 4 days.

Therefore, Q alone can complete the work in 48/5 days (9.6 days).

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