Question
'N' men can complete a work together in 25 days. All of
them started the work together, but each day, two men left the work. The total time taken to complete the work is now 35 days. Which of the following statement(s) is/are true? I. 'N' is a multiple of 7 II. (2N/7) + 5 = (N/7) + 22 III. HCF of (N - 14) and (N + 21) is 7Solution
Let the efficiency of each man be '1e' unit/day. Total work = N X 25 X 1e = '25Ne' units N X 1e + (N - 2) X 1e + (N - 4) X 1e + .... + (N - 68) X 1e = 25Ne Or, N + (N - 2) + (N - 4) + .... + (N - 68) = 25N Or, 35N - 2 X (1 + 2 + 3 + .... + 34) = 25N Or, 35N - 2 X 34 X 35 ÷ 2 = 25N Or, 35N - 1190 = 25N Or, 10N = 1190 Or, 'N' = 119 Statement I: 119 is a multiple of 7. So, statement I is true. Statement II: (2N/7) + 5 = (2 X 119 ÷ 7) + 5 = 34 + 5 = 39 (N/7) + 22 = (119 ÷ 7) + 22 = 17 + 22 = 39 So, statement II is true. Statement III: N - 14 = 119 - 14 = 105 N + 21 = 119 + 21 = 140 HCF of 105 and 140 = 35 So, statement III is false. Therefore, both I and II are true.
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