Question

24 men can complete a piece of work in 16 days. 16 women can complete the same work in 12 days. 8 men and 8 women started working on the task. After 6 days, x men were added to the group such that the work was finished in 4 more days. Find the value of x.

• A. 24
• B. 32
• C. 36
• D. 40
• E. 48

Hint:

In work problems, ensure that the rate of work is consistently applied across different groups and time frames. A careful breakdown of the work done in intervals can help solve for unknowns.

Answer 1

The Correct Option is C

Step 1: Calculate total work.
Total work is calculated in 'men-days'. 24 men complete the work in 16 days. Therefore: \[ \text{Total Work} = 24 \times 16 = 384 \text{ men-days.} \] Step 2: Determine work rates.
Rate of work for 1 man per day: \[ \text{Rate of 1 man} = \frac{1}{384} \text{ work per day.} \] Rate of work for 1 woman per day: \[ \text{Rate of 1 woman} = \frac{1}{192} \text{ work per day.} \quad (\text{since 16 women complete the work in 12 days}) \] Step 3: Work done in the first 6 days.
In the first 6 days, 8 men and 8 women worked. Their combined work rate per day is: \[ 8 \times \frac{1}{384} + 8 \times \frac{1}{192} = \frac{8}{384} + \frac{8}{192} = \frac{1}{48} + \frac{1}{24} = \frac{3}{48} = \frac{1}{16} \text{ work per day.} \] Work done in 6 days: \[ 6 \times \frac{1}{16} = \frac{6}{16} = \frac{3}{8}. \] Step 4: Remaining work and new group.
Remaining work: \[ 1 - \frac{3}{8} = \frac{5}{8}. \] Remaining work must be completed in 4 days. The new group's daily work rate is: \[ \frac{5}{8} \div 4 = \frac{5}{32}. \] The new group's work rate equation is: \[ 8 \times \frac{1}{384} + 8 \times \frac{1}{192} + x \times \frac{1}{384} = \frac{1}{48} + \frac{1}{24} + \frac{x}{384} = \frac{5}{32}. \] Simplifying: \[ \frac{1}{48} + \frac{1}{24} = \frac{1}{16}, \] \[ \frac{1}{16} + \frac{x}{384} = \frac{5}{32}, \] \[ \frac{x}{384} = \frac{5}{32} - \frac{1}{16} = \frac{5}{32} - \frac{2}{32} = \frac{3}{32}, \] \[ x = 36. \] Step 5: Conclusion.
The value of \( x \) is 36.