15. 2 men and 3 women working together can finish a piece of work in 10 days, while 4 men working together can do it in 10 days. In how many days will 3 men and 3 women working together finish the job? \( \begin{array}{lllll}\text { 1) } 8 \text { days } & \text { 2) } 10 \text { days } & \text { 3) } 11 \text { days } & \text { 4) } 12 \text { days } & \text { 5) } 15 \text { days }\end{array} \)

To solve this problem, let’s assign a variable for the work rate of men and women. Let the work rate of one man be \( M \) and the work rate of one woman be \( W \). From the information given, the two men and three women together finish the work in 10 days. This means: \[ (2M + 3W) \times 10 = 1 \quad \text{(one complete work)} \] which simplifies to: \[ 2M + 3W = \frac{1}{10} \quad \text{(1)} \] Similarly, for 4 men: \[ (4M) \times 10 = 1 \quad \text{(one complete work)} \] which simplifies to: \[ 4M = \frac{1}{10} \quad \text{(2)} \] Thus: \[ M = \frac{1}{40} \quad \text{(3)} \] Now, substitute \( M \) from equation (3) into equation (1): \[ 2\left(\frac{1}{40}\right) + 3W = \frac{1}{10} \] This gives: \[ \frac{1}{20} + 3W = \frac{1}{10} \] Subtracting \( \frac{1}{20} \) from both sides: \[ 3W = \frac{1}{10} - \frac{1}{20} = \frac{2 - 1}{20} = \frac{1}{20} \] Thus: \[ W = \frac{1}{60} \quad \text{(4)} \] Now, we need to find out how many days it will take for 3 men and 3 women to complete the job. The combined work rate is: \[ 3M + 3W = 3\left(\frac{1}{40}\right) + 3\left(\frac{1}{60}\right) \] Calculating this: \[ = \frac{3}{40} + \frac{3}{60} \] To add these fractions, the least common multiple of 40 and 60 is 120. Therefore, we convert: \[ \frac{3}{40} = \frac{9}{120} \quad \text{and} \quad \frac{3}{60} = \frac{6}{120} \] Now, adding these: \[ 3M + 3W = \frac{9}{120} + \frac{6}{120} = \frac{15}{120} = \frac{1}{8} \] Thus, the group of 3 men and 3 women can complete \(\frac{1}{8}\) of the work in one day. Therefore, they will finish the job in: \[ \text{Days} = \frac{1}{\frac{1}{8}} = 8 \text{ days.} \] So the answer is \( \textbf{1) 8 days} \).