`10` women can complete a work in 7 days and 10 children take 14 days to complete the same work. How many days will 5 women and 10 children take to complete the same work ?

Options

A. `3`
B. `5`
C. `7`
D. Cannot be determined

Answer

To solve the problem step by step, let's break it down:

### Step 1: Calculate the work done by 10 women in one day
10 women can complete the work in 7 days. Therefore, the work done by 10 women in one day is:
\[ \text{Work done by 10 women in 1 day} = \frac{1}{7} \text{ (of the total work)} \]
Thus, the work done by 1 woman in one day is:
\[ \text{Work done by 1 woman in 1 day} = \frac{1}{7 \times 10} = \frac{1}{70} \text{ (of the total work)} \]

### Step 2: Calculate the work done by 10 children in one day
10 children can complete the work in 14 days. Therefore, the work done by 10 children in one day is:
\[ \text{Work done by 10 children in 1 day} = \frac{1}{14} \text{ (of the total work)} \]
Thus, the work done by 1 child in one day is:
\[ \text{Work done by 1 child in 1 day} = \frac{1}{14 \times 10} = \frac{1}{140} \text{ (of the total work)} \]

### Step 3: Calculate the work done by 5 women in one day
Now, we calculate the work done by 5 women in one day:
\[ \text{Work done by 5 women in 1 day} = 5 \times \frac{1}{70} = \frac{5}{70} = \frac{1}{14} \text{ (of the total work)} \]

### Step 4: Calculate the work done by 10 children in one day
From Step 2, we know that the work done by 10 children in one day is:
\[ \text{Work done by 10 children in 1 day} = \frac{1}{14} \text{ (of the total work)} \]

### Step 5: Calculate the total work done by 5 women and 10 children in one day
Now, we add the work done by 5 women and 10 children together:
\[ \text{Total work done in 1 day} = \text{Work done by 5 women} + \text{Work done by 10 children} \]
\[ \text{Total work done in 1 day} = \frac{1}{14} + \frac{1}{14} = \frac{2}{14} = \frac{1}{7} \text{ (of the total work)} \]

### Step 6: Calculate the number of days to complete the work
If 5 women and 10 children together can complete \(\frac{1}{7}\) of the work in one day, then the total number of days required to complete the entire work is:
\[ \text{Total days} = \frac{1}{\text{Total work done in 1 day}} = \frac{1}{\frac{1}{7}} = 7 \text{ days} \]

### Final Answer
Thus, 5 women and 10 children will take **7 days** to complete the same work.