6 men and 8 boys can do a piece of work in 7 days, while 8 men and 12 boys do the same work in 5 days. How long would it take one boy to finish the same work?

Answer

To solve the problem, we need to find out how long it would take one boy to finish the same work. We can break down the solution step by step.

### Step 1: Define Variables
Let:
- \( M \) = work done by one man in one day
- \( B \) = work done by one boy in one day

### Step 2: Set Up the First Equation
From the first scenario, we have:
- 6 men and 8 boys complete the work in 7 days.

The total work done can be expressed as:
\[ 6 \times 7M + 8 \times 7B = 1 \quad \text{(1)} \]
This simplifies to:
\[ 42M + 56B = 1 \]

### Step 3: Set Up the Second Equation
From the second scenario, we have:
- 8 men and 12 boys complete the work in 5 days.

The total work done can be expressed as:
\[ 8 \times 5M + 12 \times 5B = 1 \quad \text{(2)} \]
This simplifies to:
\[ 40M + 60B = 1 \]

### Step 4: Equate the Two Equations
Now we have two equations:
1. \( 42M + 56B = 1 \)
2. \( 40M + 60B = 1 \)

Since both equations equal 1, we can set them equal to each other:
\[ 42M + 56B = 40M + 60B \]

### Step 5: Rearrange the Equation
Rearranging gives:
\[ 42M - 40M = 60B - 56B \]
This simplifies to:
\[ 2M = 4B \]
Thus, we can express \( M \) in terms of \( B \):
\[ M = 2B \quad \text{(3)} \]

### Step 6: Substitute \( M \) into One of the Equations
Now, substitute \( M = 2B \) into the first equation:
\[ 42(2B) + 56B = 1 \]
This simplifies to:
\[ 84B + 56B = 1 \]
Combining the terms gives:
\[ 140B = 1 \]

### Step 7: Solve for \( B \)
Now, solve for \( B \):
\[ B = \frac{1}{140} \]
This means that one boy can complete \( \frac{1}{140} \) of the work in one day.

### Step 8: Find the Time for One Boy to Complete the Work
To find out how long it would take one boy to finish the work alone, we take the reciprocal of \( B \):
\[ \text{Time taken by one boy} = \frac{1}{B} = 140 \text{ days} \]

### Final Answer
Therefore, it would take one boy **140 days** to finish the same work alone.

---