3 men and 4 boys can do a piece of work in 14 days, while 4 men and 6 boys can do it in 10 days. How long would it take 1 boy to finish the work?

Answer

To solve the problem step by step, we will denote the work done by one man as \(M\) and the work done by one boy as \(B\).

### Step 1: Set up the equations based on the given information.
From the first statement, we know that:
- 3 men and 4 boys can complete the work in 14 days.
This can be expressed as:
\[ (3M + 4B) \times 14 = 1 \quad \text{(1)} \]
From the second statement, we know that:
- 4 men and 6 boys can complete the work in 10 days.
This can be expressed as:
\[ (4M + 6B) \times 10 = 1 \quad \text{(2)} \]

### Step 2: Simplify the equations.
From equation (1):
\[ 3M + 4B = \frac{1}{14} \quad \text{(3)} \]
From equation (2):
\[ 4M + 6B = \frac{1}{10} \quad \text{(4)} \]

### Step 3: Eliminate one variable.
To eliminate \(M\) or \(B\), we can multiply equation (3) by 10 and equation (4) by 14:
\[ 10(3M + 4B) = 10 \times \frac{1}{14} \implies 30M + 40B = \frac{10}{14} = \frac{5}{7} \quad \text{(5)} \]
\[ 14(4M + 6B) = 14 \times \frac{1}{10} \implies 56M + 84B = \frac{14}{10} = \frac{7}{5} \quad \text{(6)} \]

### Step 4: Set up a new equation.
Now we have two new equations (5) and (6):
1. \(30M + 40B = \frac{5}{7}\)
2. \(56M + 84B = \frac{7}{5}\)

### Step 5: Solve the equations.
We can multiply equation (5) by 7 and equation (6) by 5 to eliminate the fractions:
\[ 7(30M + 40B) = 7 \times \frac{5}{7} \implies 210M + 280B = 5 \quad \text{(7)} \]
\[ 5(56M + 84B) = 5 \times \frac{7}{5} \implies 280M + 420B = 7 \quad \text{(8)} \]

### Step 6: Subtract the equations.
Now, we can subtract equation (7) from equation (8):
\[ (280M + 420B) - (210M + 280B) = 7 - 5 \]
This simplifies to:
\[ 70M + 140B = 2 \]
Dividing through by 2 gives:
\[ 35M + 70B = 1 \quad \text{(9)} \]

### Step 7: Express \(M\) in terms of \(B\).
From equation (9), we can express \(M\) in terms of \(B\):
\[ M = \frac{1 - 70B}{35} \]

### Step 8: Substitute back to find \(B\).
Now, we can substitute \(M\) back into either equation (3) or (4) to find \(B\). Let's substitute into equation (3):
\[ 3\left(\frac{1 - 70B}{35}\right) + 4B = \frac{1}{14} \]
Multiplying through by 35 to eliminate the fraction:
\[ 3(1 - 70B) + 140B = \frac{35}{14} \]
This simplifies to:
\[ 3 - 210B + 140B = 2.5 \]
Combining like terms gives:
\[ 3 - 70B = 2.5 \]
So, solving for \(B\):
\[ 70B = 3 - 2.5 = 0.5 \implies B = \frac{0.5}{70} = \frac{1}{140} \]

### Step 9: Find the time taken by 1 boy to finish the work.
Since \(B\) represents the work done by one boy in one day, to find how long it takes one boy to finish the work, we take the reciprocal:
\[ \text{Time taken by 1 boy} = \frac{1}{B} = 140 \text{ days} \]

### Final Answer:
It would take 1 boy 140 days to finish the work.