Directions: Read the information carefully and answer the questions. A, B, C…

2023

Directions: Read the information carefully and answer the questions. A, B, C and D together can do a work ‘X’ in seven days and D did ¼th of the work ‘X’. The ratio of efficiency of A to that of B is 3 : 4, while B is 100% more efficient than C. A and B together can complete work ‘X’ in ‘x’ days, while A and D can do the same work together in ‘y’ days.

Two persons P and Q together can complete another work ‘Y’ in (x+12) days, while Q and R together can complete the same work in (y − 2) days. If P, Q and R together can complete the work ‘Y’ in 6x/(y−7) days, then find by what percent the efficiency of ‘Q’ is more or less than the efficiency of ‘P’.

  1. A.

    80%

  2. B.

    120%

  3. C.

    100%

  4. D.

    150%

  5. E.

    60%

Show answer & explanation

Correct answer: C

Concept

Work-rate principle: if a worker (or group) finishes a job in t days, its one-day efficiency is 1/t of the job, and efficiencies of people working together simply add. To isolate one person's efficiency from combined rates, subtract: a worker's rate equals the whole group's rate minus the rate of the rest. Two efficiencies in ratio a:b means one is (a−b)/b, i.e. 100×(a−b)/b percent, of the difference relative to the smaller.

Application — Work X (find x and y)

  1. Let A = 3k and B = 4k (ratio 3:4). “B is 100% more efficient than C” means B = 2C, so C = 2k.

  2. D alone does ¼ of X in 7 days, so D’s rate = (1/4)÷7 = 1/28 of X per day.

  3. All four finish X in 7 days, so their combined rate = 1/7. Hence A+B+C = 1/7 − 1/28 = 3/28.

  4. A+B+C = 9k = 3/28, so k = 1/84. Then A = 3/84 = 1/28, B = 4/84 = 1/21, C = 2/84 = 1/42, D = 1/28.

  5. x = time for A+B: A+B = 1/28 + 1/21 = 1/12, so x = 12 days.

  6. y = time for A+D: A+D = 1/28 + 1/28 = 1/14, so y = 14 days.

Application — Work Y (compare Q and P)

  1. P+Q finish Y in x+12 = 24 days → rate 1/24. Q+R finish in y−2 = 12 days → rate 1/12.

  2. P+Q+R finish in 6x/(y−7) = 72/7 days → rate 7/72.

  3. P = (P+Q+R) − (Q+R) = 7/72 − 1/12 = 7/72 − 6/72 = 1/72.

  4. R = (P+Q+R) − (P+Q) = 7/72 − 1/24 = 7/72 − 3/72 = 4/72.

  5. Q = (P+Q+R) − P − R = 7/72 − 1/72 − 4/72 = 2/72.

Cross-check and result

Q = 2/72 and P = 1/72, so Q is exactly double P. Percent more = 100×(2−1)/1 = 100%. Verify: P+Q = 1/72+2/72 = 3/72 = 1/24 ✓ and Q+R = 2/72+4/72 = 6/72 = 1/12 ✓, both matching the given times. So Q is 100% more efficient than P.

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