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.

‘m’ men can complete a work in (x+9) day, while ‘y’ men can complete the same work in ‘n’ days. If ‘y – 7’ men can complete the same work in (n+12) days, then find in how many days (n–m) men can complete the same work?

  1. A.

    21 days

  2. B.

    42 days

  3. C.

    35 days

  4. D.

    14 days

  5. E.

    84 days

Show answer & explanation

Correct answer: B

Concept

Work problems use two ideas. (1) Efficiency-share: when several workers finish a job together, the time each pair or group takes equals the total work divided by the sum of their per-day efficiencies, where efficiencies are fixed by the given ratios. (2) Man-days constancy: for one fixed job, (number of workers) × (days) is a constant, so workers and days are inversely proportional.

Application – finding x and y

  1. Fix efficiencies from the ratios. A : B = 3 : 4, and “B is 100% more efficient than C” means B = 2C, so C = 2 when B = 4. Hence A : B : C = 3 : 4 : 2.

  2. D's share: A, B, C, D finish X in 7 days and D alone does ¼ of X, so A, B, C together do ¾ of X in those same 7 days.

  3. Convert to per-day rates. Take 1 unit = X/84 per day; then A = 3 units = X/28, B = 4 units = X/21, C = 2 units = X/42 per day. Together A + B + C = 9 units = 9X/84 = 3X/28 per day, finishing ¾ of X in 7 days as required (3X/28 × 7 = 3X/4). D must finish ¼ of X in 7 days, so D = X/28 per day.

  4. x = time for A and B together = X ÷ (X/28 + X/21) = X ÷ (3X/84 + 4X/84) = X ÷ (7X/84) = 12 days.

  5. y = time for A and D together = X ÷ (X/28 + X/28) = X ÷ (2X/28) = X ÷ (X/14) = 14 days.

Application – the men–days part

  1. Substitute x = 12 and y = 14. So m men finish in x + 9 = 21 days; y = 14 men finish in n days; y − 7 = 7 men finish in n + 12 days.

  2. Apply man-days constancy to the 14-men and 7-men statements: 14 × n = 7 × (n + 12). Then 14n = 7n + 84, giving 7n = 84, so n = 12.

  3. Total work = 14 × n = 14 × 12 = 168 man-days.

  4. From the m-men statement: m × 21 = 168, so m = 8.

  5. Required crew = n − m = 12 − 8 = 4 men.

  6. Days for that crew = total work ÷ workers = 168 ÷ 4 = 42 days.

Cross-check

Confirm the man-days match for every crew: 14 × 12 = 168, 7 × 24 = 168, 8 × 21 = 168, and 4 × 42 = 168. All four agree, so 42 days is consistent.

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