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 do a work in ‘n’ days and ‘n’ women can do the same work in ‘2m’ days. ‘x’ men and ‘1.5y’ women together can complete the same work in 20 days. 5 men and (y+1) women started working together and they did work only for D days and remaining work is completed by a woman in (25/3)D days. Find the value of ‘D’.

  1. A.

    33

  2. B.

    18

  3. C.

    21

  4. D.

    24

  5. E.

    27

Attempted by 2 students.

Show answer & explanation

Correct answer: E

Concept

Work problems reduce to a single idea: efficiency (work done per day) is additive. If a group finishes a job in T days, its combined daily efficiency is 1/T of the job; conversely, summing the per-day efficiencies of all workers and dividing the total work by that sum gives the time. Ratios of efficiencies translate directly into ratios of daily output, and unequal worker types (men vs women) are reconciled by expressing everyone in one common unit before adding.

Application — Step 1: efficiencies of A, B, C, D

  1. A, B, C, D together finish work X in 7 days, so their combined efficiency is 1/7 of X per day.

  2. D alone did ¼ of X. Since the group ran for 7 days, D's daily efficiency = (¼)/7 = 1/28.

  3. Then A + B + C = 1/7 − 1/28 = 4/28 − 1/28 = 3/28 per day.

  4. Given A : B = 3 : 4 and B is 100% more efficient than C (so B = 2C, i.e. C = B/2): take A = 3k, B = 4k, C = 2k. Then 9k = 3/28, giving k = 1/84.

  5. Hence A = 3/84 = 1/28, B = 4/84 = 1/21, C = 2/84 = 1/42, and D = 1/28.

Application — Step 2: find x and y

  1. A + B = 1/28 + 1/21 = 3/84 + 4/84 = 7/84 = 1/12, so x = 12 days.

  2. A + D = 1/28 + 1/28 = 2/28 = 1/14, so y = 14 days.

Application — Step 3: convert men and women to one unit

‘m’ men finish a job in ‘n’ days and ‘n’ women finish the same job in ‘2m’ days. Equating total work: m·n·(man) = n·2m·(woman), which gives man = 2 × woman. Take 1 woman = 1 unit/day, so 1 man = 2 units/day.

Application — Step 4: total work from the 20-day scenario

  1. ‘x’ men = 12 men and ‘1.5y’ women = 1.5 × 14 = 21 women complete the job in 20 days.

  2. Daily output = 12 × 2 + 21 × 1 = 24 + 21 = 45 units/day.

  3. Total work = 45 × 20 = 900 units.

Application — Step 5: solve for D

  1. 5 men and (y + 1) = 15 women work for D days. Their daily output = 5 × 2 + 15 × 1 = 25 units/day, so they complete 25D units.

  2. The remaining work is finished by one woman (1 unit/day) in (25/3)·D days, i.e. (25/3)·D units.

  3. Work balance: 900 − 25D = (25/3)·D.

  4. So 900 = 25D + (25/3)D = (100/3)·D, giving D = 900 × 3/100 = 27.

Cross-check

With D = 27: the team does 25 × 27 = 675 units; the lone woman does (25/3) × 27 = 225 units; 675 + 225 = 900 units = total work. The two parts add back to the whole, so D = 27 is consistent.

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