2 men and 5 boys together can finish a piece of work in 4 days, while 3 men and 6 boys can finish it in 3 days. Find the time taken by 1 man alone to finish the work and than taken by 1 boy alone.

Answer

To solve the problem, we will set up equations based on the information given and then solve for the time taken by one man and one boy to finish the work alone.

### Step 1: Define Variables
Let:
- \( X \) = time taken by 1 man to finish the work (in days)
- \( Y \) = time taken by 1 boy to finish the work (in days)

### Step 2: Set Up the Equations
From the problem, we know:
1. 2 men and 5 boys can finish the work in 4 days.
2. 3 men and 6 boys can finish the work in 3 days.

Using the work formula (Work = Rate × Time), we can express the work done in terms of \( X \) and \( Y \).

**Equation 1:**
The work done by 2 men and 5 boys in 4 days:
\[ \frac{2}{X} + \frac{5}{Y} = \frac{1}{4} \]

**Equation 2:**
The work done by 3 men and 6 boys in 3 days:
\[ \frac{3}{X} + \frac{6}{Y} = \frac{1}{3} \]

### Step 3: Clear the Fractions
To eliminate the fractions, we can multiply through by the least common multiple (LCM) of the denominators.

**For Equation 1:**
Multiply through by \( 4XY \):
\[ 4Y(2) + 4X(5) = XY \]
\[ 8Y + 20X = XY \quad \text{(Equation 1)} \]

**For Equation 2:**
Multiply through by \( 3XY \):
\[ 3Y(3) + 3X(6) = XY \]
\[ 9Y + 18X = XY \quad \text{(Equation 2)} \]

### Step 4: Rearranging the Equations
Rearranging both equations gives:
1. \( XY - 8Y - 20X = 0 \) (Equation 1)
2. \( XY - 9Y - 18X = 0 \) (Equation 2)

### Step 5: Subtract the Equations
Now, subtract Equation 1 from Equation 2:
\[ (XY - 9Y - 18X) - (XY - 8Y - 20X) = 0 \]
This simplifies to:
\[ -9Y + 8Y - 18X + 20X = 0 \]
\[ -Y + 2X = 0 \]
Thus, we have:
\[ Y = 2X \quad \text{(Equation 3)} \]

### Step 6: Substitute Back
Now substitute \( Y = 2X \) into either Equation 1 or Equation 2. Let's use Equation 1:
\[ XY - 8Y - 20X = 0 \]
Substituting \( Y \):
\[ X(2X) - 8(2X) - 20X = 0 \]
\[ 2X^2 - 16X - 20X = 0 \]
\[ 2X^2 - 36X = 0 \]
Factoring out \( 2X \):
\[ 2X(X - 18) = 0 \]
Thus, \( X = 0 \) or \( X = 18 \). Since \( X \) cannot be zero, we have:
\[ X = 18 \]

### Step 7: Find \( Y \)
Using Equation 3:
\[ Y = 2X = 2(18) = 36 \]

### Conclusion
The time taken by 1 man alone to finish the work is **18 days**, and the time taken by 1 boy alone to finish the work is **36 days**.