The work done by 4 men in 12 days is equal to the work done by 6 women in 10 days and is also equal to the work done by 8 children in 9 days. A man, a woman and a child working together take 10 days to complete a particular job. In how many days will the same job be completed by 2 women and 5 children working together?

Options

A. 5
B. 6
C. 4
D. 7

Answer

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

### Step 1: Determine the total work done by men, women, and children.
Given:
- Work done by 4 men in 12 days = Work done by 6 women in 10 days = Work done by 8 children in 9 days.

Let's denote the total work done by 4 men in 12 days as \( W \).

The work done can be expressed as:
- Work done by 4 men in 12 days: \( W = 4 \text{ men} \times 12 \text{ days} = 48 \text{ man-days} \)
- Work done by 6 women in 10 days: \( W = 6 \text{ women} \times 10 \text{ days} = 60 \text{ woman-days} \)
- Work done by 8 children in 9 days: \( W = 8 \text{ children} \times 9 \text{ days} = 72 \text{ child-days} \)

### Step 2: Set the equations based on the work done.
From the above, we can set up the equations:
1. \( 48 \text{ man-days} = 60 \text{ woman-days} \)
2. \( 48 \text{ man-days} = 72 \text{ child-days} \)

### Step 3: Find the work rates of men, women, and children.
From the first equation:
- \( 1 \text{ man} = \frac{60}{48} \text{ women} = \frac{5}{4} \text{ women} \)
- Therefore, \( 1 \text{ woman} = \frac{4}{5} \text{ man} \)

From the second equation:
- \( 1 \text{ man} = \frac{72}{48} \text{ children} = \frac{3}{2} \text{ children} \)
- Therefore, \( 1 \text{ child} = \frac{2}{3} \text{ man} \)

### Step 4: Calculate the efficiency of each worker.
Let’s assume the work done by 1 man in 1 day is \( M \), 1 woman is \( W \), and 1 child is \( C \).

From the relationships:
- \( M = 1 \)
- \( W = \frac{4}{5}M = \frac{4}{5} \)
- \( C = \frac{2}{3}M = \frac{2}{3} \)

### Step 5: Find the combined work rate of 1 man, 1 woman, and 1 child.
The combined work rate of 1 man, 1 woman, and 1 child is:
\[ M + W + C = 1 + \frac{4}{5} + \frac{2}{3} \]
To add these fractions, we need a common denominator. The least common multiple of 1, 5, and 3 is 15.

Converting each term:
- \( M = 1 = \frac{15}{15} \)
- \( W = \frac{4}{5} = \frac{12}{15} \)
- \( C = \frac{2}{3} = \frac{10}{15} \)

Adding them together:
\[ M + W + C = \frac{15}{15} + \frac{12}{15} + \frac{10}{15} = \frac{37}{15} \]

### Step 6: Calculate the total work done in 10 days.
The total work done by 1 man, 1 woman, and 1 child in 10 days is:
\[ \text{Total Work} = 10 \times \left(\frac{37}{15}\right) = \frac{370}{15} \text{ work units} \]

### Step 7: Calculate the work done by 2 women and 5 children.
Now, we need to find out how many days it will take for 2 women and 5 children to complete the same job.

The work rate of 2 women and 5 children is:
\[ 2W + 5C = 2 \times \frac{4}{5} + 5 \times \frac{2}{3} \]
Calculating each term:
- \( 2W = \frac{8}{5} \)
- \( 5C = \frac{10}{3} \)

Finding a common denominator (15):
- \( 2W = \frac{8}{5} = \frac{24}{15} \)
- \( 5C = \frac{10}{3} = \frac{50}{15} \)

Adding them together:
\[ 2W + 5C = \frac{24}{15} + \frac{50}{15} = \frac{74}{15} \]

### Step 8: Calculate the number of days required to complete the work.
To find the number of days \( D \) required to complete \( \frac{370}{15} \) work units at a rate of \( \frac{74}{15} \):
\[ D = \frac{\text{Total Work}}{\text{Work Rate}} = \frac{\frac{370}{15}}{\frac{74}{15}} = \frac{370}{74} = 5 \]

Thus, **2 women and 5 children will complete the job in 5 days**.

### Final Answer:
**5 days**