14. 10 women can complete a piece of work in 8 days and 10 children take 12 days to complete the same work. How many days will 6 women and 3 children working together take to complete the same piece of work? \( \begin{array}{lllll}\text { 1) } 2 \text { days } & \text { 2) } 6 \text { days } & \text { 3) } 10 \text { days } & \text { 4) } 12 \text { days } & \text { 5) } 15 \text { days }\end{array} \)

To solve this problem, we first need to determine the work rate of women and children. 1. **Women’s Work Rate**: If 10 women can complete the work in 8 days, then the work done by 10 women in one day is \( \frac{1}{8} \). Thus, the work done by one woman in one day is \( \frac{1}{8 \times 10} = \frac{1}{80} \). 2. **Children’s Work Rate**: If 10 children can complete the work in 12 days, then the work done by 10 children in one day is \( \frac{1}{12} \). So, the work done by one child in one day is \( \frac{1}{12 \times 10} = \frac{1}{120} \). Now, let's calculate the combined work rate of 6 women and 3 children: - Work rate of 6 women: \( 6 \text{ women} = 6 \times \frac{1}{80} = \frac{6}{80} = \frac{3}{40} \) - Work rate of 3 children: \( 3 \text{ children} = 3 \times \frac{1}{120} = \frac{3}{120} = \frac{1}{40} \) Now, add the two work rates together: \[ \text{Total work rate} = \frac{3}{40} + \frac{1}{40} = \frac{4}{40} = \frac{1}{10} \] This means together, 6 women and 3 children can complete \( \frac{1}{10} \) of the work in one day. To find out how many days it will take to complete the full work, we take the reciprocal of their total work rate. Thus, it will take \( 10 \) days. So the answer is **3) 10 days**.