12 men, 8 women and 6 boys are assigned to do a piece of work. In how many…

2023

12 men, 8 women and 6 boys are assigned to do a piece of work. In how many days can they complete the work?

I. 2 men, 3 women and 5 boys can complete the work in 67 days.

II. 18 men, 12 women and 9 boys can complete the work in 24 days.

  1. A.

    Statement i alone is sufficient but statement ii alone is not sufficient.

  2. B.

    Either statement i alone or statement ii alone is sufficient.

  3. C.

    Both statements i and ii together are sufficient but neither of statements alone is sufficient.

  4. D.

    Statement ii alone is sufficient but statement i alone is not sufficient.

Show answer & explanation

Correct answer: D

Concept: In a data-sufficiency work-rate problem, a statement's men-women-boys combination is “alone sufficient” for a target group only when its counts are a constant multiple of the target's counts — the statement's work-rate equation then scales directly to the target. If a statement's counts are not a multiple of the target's, one equation cannot pin down the three unknown individual rates (men, women, boys), so that statement alone is insufficient.

Application:

  1. Target group: 12 men + 8 women + 6 boys — ratio 12:8:6 = 6:4:3.

  2. Statement II: 18 men + 12 women + 9 boys finish the work in 24 days, so rate(18M+12W+9B) = 1/24 per day. Its ratio 18:12:9 = 6:4:3 is identical to the target ratio, scaled by 1.5 (18÷1.5=12, 12÷1.5=8, 9÷1.5=6).

  3. Since work rate scales linearly with the group's composition, rate(12M+8W+6B) = rate(18M+12W+9B) ÷ 1.5 = (1/24) ÷ 1.5 = 1/36 — so the target group finishes in 36 days using Statement II alone.

  4. Statement I: 2 men + 3 women + 5 boys finish the work in 67 days — ratio 2:3:5. Checking against 6:4:3: 12÷2=6, 8÷3≈2.67, 6÷5=1.2 — the three scale factors differ, so Statement I's single equation cannot isolate the target group's rate.

Cross-check: (18÷1.5)M + (12÷1.5)W + (9÷1.5)B = 12M + 8W + 6B, and (1/24)÷1.5 = 1/36 — this independently confirms the target group finishes in 36 days from Statement II alone, with no dependence on Statement I.

Result: Statement II alone is sufficient, but Statement I alone is not sufficient.

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