6 women alone can complete a piece of work in 10 days, whereas 10 children alone take 15 days to complete the same piece of work. How many days will 6 women and 10 children together take to complete the piece of work ?

Options

A. 7
B. 8
C. 6
D. 4

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Determine the work done by women and children separately.

1. **Work done by 6 women in 10 days:**
- If 6 women can complete the work in 10 days, then the total work can be represented as:
\[ \text{Total Work} = \text{Number of Women} \times \text{Days} = 6 \times 10 = 60 \text{ woman-days} \]

2. **Work done by 10 children in 15 days:**
- If 10 children can complete the same work in 15 days, then the total work can also be represented as:
\[ \text{Total Work} = \text{Number of Children} \times \text{Days} = 10 \times 15 = 150 \text{ child-days} \]

### Step 2: Find the efficiency of women and children.

1. **Efficiency of 1 woman:**
- Since 6 women can complete the work in 10 days, the efficiency of 1 woman can be calculated as:
\[ \text{Efficiency of 1 woman} = \frac{60 \text{ woman-days}}{6} = 10 \text{ days} \]
- Therefore, the work done by 1 woman in 1 day is:
\[ \text{Efficiency of 1 woman} = \frac{1}{10} \text{ of the work} \]

2. **Efficiency of 1 child:**
- Since 10 children can complete the work in 15 days, the efficiency of 1 child can be calculated as:
\[ \text{Efficiency of 1 child} = \frac{150 \text{ child-days}}{10} = 15 \text{ days} \]
- Therefore, the work done by 1 child in 1 day is:
\[ \text{Efficiency of 1 child} = \frac{1}{15} \text{ of the work} \]

### Step 3: Calculate the combined efficiency of 6 women and 10 children.

1. **Combined efficiency:**
- The total efficiency of 6 women is:
\[ \text{Efficiency of 6 women} = 6 \times \frac{1}{10} = \frac{6}{10} = \frac{3}{5} \text{ of the work per day} \]
- The total efficiency of 10 children is:
\[ \text{Efficiency of 10 children} = 10 \times \frac{1}{15} = \frac{10}{15} = \frac{2}{3} \text{ of the work per day} \]

2. **Total combined efficiency:**
- Now, we add the efficiencies of women and children:
\[ \text{Total Efficiency} = \frac{3}{5} + \frac{2}{3} \]
- To add these fractions, we need a common denominator:
\[ \text{Common denominator} = 15 \]
- Convert the fractions:
\[ \frac{3}{5} = \frac{9}{15}, \quad \frac{2}{3} = \frac{10}{15} \]
- Now, add them:
\[ \text{Total Efficiency} = \frac{9}{15} + \frac{10}{15} = \frac{19}{15} \text{ of the work per day} \]

### Step 4: Calculate the time taken to complete the work together.

1. **Time taken to complete the work:**
- The total work is 1 (the whole work), and the combined efficiency is \(\frac{19}{15}\). Therefore, the time taken (T) is:
\[ T = \frac{\text{Total Work}}{\text{Total Efficiency}} = \frac{1}{\frac{19}{15}} = \frac{15}{19} \text{ days} \]

### Final Answer:
The time taken by 6 women and 10 children together to complete the work is approximately **0.789 days** or **15/19 days**.