Question

A work was completed by three persons of equal ability, first one doing \( m \) hours for \( m \) days, second one doing \( n \) hours for \( n \) days (where \( m \) and \( n \) are integers) and third one doing 16 hours for 16 days. The work could have been completed in 29 days by the third person alone with his respective working hours. If all of them do the work together with their respective working hours, then they can complete it in about:

• A. 12 days
• B. 13 days
• C. 14 days
• D. 15 days

Hint:

Ensure integer solutions for rates or times when using squared terms, as these often reflect real, practical constraints in work-related problems.

Answer 1

The Correct Option is B

Given that the total work \( W \) could be completed by the third person alone in 29 days, working 16 hours a day, we first calculate the total amount of work in terms of "work hours." \[ W = 16 \text{ hours/day} \times 29 \text{ days} = 464 \text{ work hours} \] Step 1: Establish the total work and the contribution of each person. Each person's contribution to the work is given by:
- Third person: \( 16 \times 16 = 256 \text{ work hours} \) Thus, the work done by the first two people together must be: \[ 464 \text{ total work} - 256 \text{ by third person} = 208 \text{ work hours} \]

Step 2: Analyzing the work by first and second persons. We know: \[ m^2 + n^2 = 208 \] From the image, it's suggested that \( m = 12 \) and \( n = 8 \) are solutions to this equation because \( 12^2 + 8^2 = 144 + 64 = 208 \).

Step 3: Calculate the total daily work output when all work together. The daily work output if they all worked together would be: \[ m + n + 16 = 12 + 8 + 16 = 36 \text{ work hours/day} \]

Step 4: Calculate the number of days required to complete the work together. The time required to complete 464 work hours if they all work together is: \[ \text{Time taken} = \frac{464 \text{ work hours}}{36 \text{ work hours/day}} \approx 12.89 \text{ days} \] Rounding up since we can't work a fraction of a day in this context, it is approximately 13 days.