12 men take 18 days to complete a job whereas 12 women in 18 days can complete 3/4 part of the same job. How many days will 10 men and 8 women together take to complete the same job?

Question

Accepted Answer

To determine how many days 10 men and 8 women will take to complete the job, we need to first understand the work rates of the men and women separately.

1. **Work Rate of Men:**
   - 12 men complete the job in 18 days.
   - Therefore, the work rate of 12 men is 1 job per 18 days.
   - The work rate of 1 man is $ \frac{1}{12 \times 18} = \frac{1}{216} $ of the job per day.

2. **Work Rate of Women:**
   - 12 women complete $ \frac{3}{4} $ of the job in 18 days.
   - Therefore, the work rate of 12 women is $ \frac{3}{4} $ job per 18 days.
   - The work rate of 1 woman is $ \frac{3}{4} \times \frac{1}{12 \times 18} = \frac{1}{288} $ of the job per day.

3. **Combined Work Rate of 10 Men and 8 Women:**
   - The work rate of 10 men is $ 10 \times \frac{1}{216} = \frac{10}{216} = \frac{5}{108} $ of the job per day.
   - The work rate of 8 women is $ 8 \times \frac{1}{288} = \frac{8}{288} = \frac{1}{36} $ of the job per day.
   - Combined work rate of 10 men and 8 women is $ \frac{5}{108} + \frac{1}{36} $.

4. **Calculating the Combined Work Rate:**
   - Convert $ \frac{1}{36} $ to have a common denominator with $ \frac{5}{108} $:
     $$
     \frac{1}{36} = \frac{3}{108}
     $$
   - Combined work rate is $ \frac{5}{108} + \frac{3}{108} = \frac{8}{108} = \frac{2}{27} $ of the job per day.

5. **Calculating the Number of Days:**
   - To complete 1 job, the number of days required is the reciprocal of the combined work rate:
     $$
     \text{Days} = \frac{1}{\frac{2}{27}} = \frac{27}{2} = 13 \frac{1}{2}
     $$

Therefore, 10 men and 8 women together will take $ 13 \frac{1}{2} $ days to complete the job, which corresponds to Option 3.

**Explanation of Incorrect Options:**

- **Option 1 (6 days):** This option is incorrect because the combined work rate of 10 men and 8 women is not sufficient to complete the job in just 6 days.
  
- **Option 2 (7 $\frac{1}{2}$ days):** This option is incorrect because the calculation shows that the combined work rate results in a longer duration than 7 $\frac{1}{2}$ days.

- **Option 4 (14 days):** This option is incorrect because the calculated time of $ 13 \frac{1}{2} $ days is slightly less than 14 days, making this option inaccurate.

Thus, the correct answer is indeed Option 3, $ 13 \frac{1}{2} $ days.