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    Question

    ‘N’ men can complete a work together in 24 days. All

    of them started the work together, but each day, four men left the work. The total time taken to complete the work is now 32 days. Which of the following statement(s) is/are true?  I. ‘N’ is a multiple of 4  II. 0.5N + 20 = 0.7N – 22  III. HCF of (N – 35) and (N + 25) is 40  
    A Only II Correct Answer Incorrect Answer
    B None of the these Correct Answer Incorrect Answer
    C Both I and II Correct Answer Incorrect Answer
    D Only III Correct Answer Incorrect Answer
    E Both I and III Correct Answer Incorrect Answer

    Solution

    Let the efficiency of each man be 1e unit/day. 
    Total work = N × 24 × 1e = 24Ne units 
    Now, 
    N × 1e + (N – 4) × 1e + (N – 8) × 1e + … + (N – 124) × 1e = 24Ne 

    Or,  N + (N – 4) + (N – 8) + … + (N – 124) = 24N 

    Or,  32N – 4 × (1 + 2 + 3 + … + 31) = 24N 
    Or,  8N = 4 × (31 × 32 ÷ 2) 
    Or,  8N = 4 × 496 = 1984 
    Or,  N = 248 
    Statement I: 
    248 is a multiple of 4. 
    So, Statement I is true. 
    Statement II: 
    0.5N + 20 = 0.5 × 248 + 20 = 124 + 20 = 144 
    0.7N – 22 = 0.7 × 248 – 22 = 173.6 – 22 = 151.6 (not equal) 
    So, Statement II is false. 
    Statement III: 
    N – 35 = 248 – 35 = 213 
    N + 25 = 248 + 25 = 273 
    HCF of 213 and 273 = 3 
    So, Statement III is false. 
    Therefore, only Statement I is true.

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