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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 Only I 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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