2 men and 3 boys can do a piece of work in 10 days while 3 men and 2 boys can do the same work in 8 days. In how many days can 2 men and 1 boy do the same work ?

Options

A. `12 (1)/(2)` days
B. `11 (1)/(2)` days
C. `15 (1)/(2)` days
D. `13 (1)/(2)` days

Answer

To solve the problem step by step, we will use the concept of work done by men and boys. Let's denote the efficiency of one man as \( M \) and the efficiency of one boy as \( B \).

### Step 1: Set up the equations based on the given information

From the problem, we have two scenarios:

1. **2 men and 3 boys can complete the work in 10 days.**
\[ \text{Work} = \text{Efficiency} \times \text{Time} \]
\[ \Rightarrow 2M + 3B = \frac{W}{10} \quad \text{(1)} \]

2. **3 men and 2 boys can complete the work in 8 days.**
\[ \Rightarrow 3M + 2B = \frac{W}{8} \quad \text{(2)} \]

### Step 2: Express work in terms of efficiency

From equations (1) and (2), we can express the total work \( W \) in terms of the efficiencies:

From (1):
\[ W = 10(2M + 3B) \quad \text{(3)} \]

From (2):
\[ W = 8(3M + 2B) \quad \text{(4)} \]

### Step 3: Set the expressions for work equal to each other

Since both equations (3) and (4) represent the same work \( W \), we can set them equal to each other:
\[ 10(2M + 3B) = 8(3M + 2B) \]

### Step 4: Expand and simplify the equation

Expanding both sides:
\[ 20M + 30B = 24M + 16B \]

Now, rearranging the equation:
\[ 20M - 24M = 16B - 30B \]
\[ -4M = -14B \]

Dividing both sides by -2:
\[ 2M = 7B \]
\[ \Rightarrow \frac{M}{B} = \frac{7}{2} \quad \text{(5)} \]

### Step 5: Substitute the value of \( M \) in terms of \( B \)

From equation (5), we can express \( M \) as:
\[ M = \frac{7}{2}B \]

### Step 6: Substitute \( M \) back into one of the work equations

Let's substitute \( M \) into equation (1):
\[ 2\left(\frac{7}{2}B\right) + 3B = \frac{W}{10} \]
\[ 7B + 3B = \frac{W}{10} \]
\[ 10B = \frac{W}{10} \]
\[ W = 100B \quad \text{(6)} \]

### Step 7: Calculate the efficiency of 2 men and 1 boy

Now, we need to find the time taken by 2 men and 1 boy to complete the work:
\[ \text{Efficiency of 2 men and 1 boy} = 2M + 1B \]
Substituting \( M \):
\[ = 2\left(\frac{7}{2}B\right) + B = 7B + B = 8B \]

### Step 8: Calculate the time taken

Using the work from equation (6):
\[ \text{Time} = \frac{W}{\text{Efficiency}} = \frac{100B}{8B} = 12.5 \text{ days} \]

### Final Answer

Thus, the time taken by 2 men and 1 boy to complete the work is:
\[ \text{Time} = 12 \frac{1}{2} \text{ days} \]