6 children and 2 men complete a certain piece of work in 6 days. Each child takes twice the time taken by a man to finish the work. In how many days will 5 men finish the same work?

Options

A. 6
B. 8
C. 9
D. 15

Answer

To solve the problem step by step, we will first determine the work done by the children and men, and then find out how many days it will take for 5 men to complete the same work.

### Step 1: Define the Work Rate of Men and Children
Let the time taken by one man to complete the work be \( t \). Then, the time taken by one child to complete the work is \( 2t \) (since each child takes twice the time taken by a man).

The work done by one man in one day is \( \frac{1}{t} \) and the work done by one child in one day is \( \frac{1}{2t} \).

### Step 2: Calculate the Total Work Done by 6 Children and 2 Men
Now, we have 6 children and 2 men working together. The total work done by them in one day can be calculated as follows:

- Work done by 6 children in one day:
\[ 6 \times \frac{1}{2t} = \frac{6}{2t} = \frac{3}{t} \]

- Work done by 2 men in one day:
\[ 2 \times \frac{1}{t} = \frac{2}{t} \]

- Total work done by 6 children and 2 men in one day:
\[ \text{Total work} = \frac{3}{t} + \frac{2}{t} = \frac{5}{t} \]

### Step 3: Calculate the Total Work in 6 Days
Since they complete the work in 6 days, the total work can be calculated as:
\[ \text{Total Work} = \text{Work done in one day} \times \text{Number of days} = \frac{5}{t} \times 6 = \frac{30}{t} \]

### Step 4: Find the Work Done by 5 Men
Now, we need to find out how many days it will take for 5 men to complete the same total work of \( \frac{30}{t} \).

The work done by 5 men in one day is:
\[ 5 \times \frac{1}{t} = \frac{5}{t} \]

### Step 5: Calculate the Number of Days for 5 Men to Complete the Work
Let \( d \) be the number of days required for 5 men to complete the work. The equation is:
\[ \text{Total Work} = \text{Work done by 5 men in one day} \times d \]
Substituting the values we have:
\[ \frac{30}{t} = \frac{5}{t} \times d \]

### Step 6: Solve for \( d \)
To find \( d \), we can simplify the equation:
\[ d = \frac{30}{5} = 6 \]

Thus, it will take 5 men **6 days** to finish the same work.

### Summary of Steps:
1. Define work rates for men and children based on their time to complete the work.
2. Calculate the total work done by 6 children and 2 men in one day.
3. Calculate the total work done in 6 days.
4. Find the work done by 5 men in one day.
5. Set up the equation to find the number of days \( d \) for 5 men to complete the work.
6. Solve for \( d \).