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

Answer

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

### Step 1: Define Variables
Let the work done by one man in one day be \( m \) and the work done by one boy in one day be \( b \).

### Step 2: Set Up Equations
From the first statement, "2 men and 3 boys can do a work in 10 days," we can express this as:
\[ (2m + 3b) \times 10 = \text{Total Work} \quad \text{(Equation 1)} \]

From the second statement, "3 men and 2 boys can do a work in 8 days," we can express this as:
\[ (3m + 2b) \times 8 = \text{Total Work} \quad \text{(Equation 2)} \]

### Step 3: Equate Total Work
Since both equations represent the same total work, we can set them equal to each other:
\[ (2m + 3b) \times 10 = (3m + 2b) \times 8 \]

### Step 4: Expand and Simplify
Expanding both sides gives:
\[ 20m + 30b = 24m + 16b \]

Now, rearranging the equation to isolate \( m \) and \( b \):
\[ 20m - 24m + 30b - 16b = 0 \]
\[ -4m + 14b = 0 \]
\[ 4m = 14b \]
\[ m = \frac{14}{4}b = \frac{7}{2}b \]

### Step 5: Substitute for Men in Terms of Boys
Now, we can express the work done by 2 men in terms of boys:
\[ 2m = 2 \times \frac{7}{2}b = 7b \]

### Step 6: Calculate Total Work for 2 Men and 1 Boy
Now, we need to find out how much work 2 men and 1 boy can do in one day:
\[ 2m + 1b = 7b + 1b = 8b \]

### Step 7: Find the Number of Days to Complete the Work
We know from Equation 1 that the total work can be expressed as:
\[ \text{Total Work} = (2m + 3b) \times 10 = (7b + 3b) \times 10 = 10b \times 10 = 100b \]

Now, if 2 men and 1 boy can do \( 8b \) work in one day, the number of days \( x \) required to complete the total work is:
\[ 8b \times x = 100b \]
Dividing both sides by \( 8b \):
\[ x = \frac{100b}{8b} = \frac{100}{8} = 12.5 \]

### Final Answer
Thus, 2 men and 1 boy can complete the work in **12.5 days**.

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