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 need to set up equations based on the information provided. Let's break it down step by step.

### Step 1: Define Variables
Let:
- \( X \) = the number of days one man alone can finish the work.
- \( Y \) = the number of days one boy alone can finish the work.

### Step 2: Express Work Done in One Day
The work done by one man in one day is \( \frac{1}{X} \) and the work done by one boy in one day is \( \frac{1}{Y} \).

### Step 3: Set Up the First Equation
According to the problem, 2 men and 5 boys can finish the work in 4 days. Therefore, their combined work in one day is:
\[ 2 \times \frac{1}{X} + 5 \times \frac{1}{Y} = \frac{1}{4} \]
This simplifies to:
\[ \frac{2}{X} + \frac{5}{Y} = \frac{1}{4} \tag{1} \]

### Step 4: Set Up the Second Equation
Similarly, for 3 men and 6 boys who can finish the work in 3 days, we have:
\[ 3 \times \frac{1}{X} + 6 \times \frac{1}{Y} = \frac{1}{3} \]
This simplifies to:
\[ \frac{3}{X} + \frac{6}{Y} = \frac{1}{3} \tag{2} \]

### Step 5: Clear the Fractions
To eliminate the fractions, we can multiply both sides of equation (1) by \( 4XY \):
\[ 4Y \cdot 2 + 4X \cdot 5 = XY \]
This gives us:
\[ 8Y + 20X = XY \tag{3} \]

Now, multiply both sides of equation (2) by \( 3XY \):
\[ 3Y \cdot 3 + 3X \cdot 6 = XY \]
This gives us:
\[ 9Y + 18X = XY \tag{4} \]

### Step 6: Rearranging the Equations
From equations (3) and (4), we can rearrange them:
\[ XY - 8Y - 20X = 0 \tag{5} \]
\[ XY - 9Y - 18X = 0 \tag{6} \]

### Step 7: Subtract the Equations
Now, subtract equation (6) from equation (5):
\[ (XY - 8Y - 20X) - (XY - 9Y - 18X) = 0 \]
This simplifies to:
\[ Y - 2X = 0 \]
Thus, we have:
\[ Y = 2X \tag{7} \]

### Step 8: Substitute Back
Substituting \( Y = 2X \) into equation (3):
\[ 8(2X) + 20X = X(2X) \]
This simplifies to:
\[ 16X + 20X = 2X^2 \]
\[ 36X = 2X^2 \]
Dividing both sides by \( 2X \) (assuming \( X \neq 0 \)):
\[ 18 = X \]
So, \( X = 18 \).

### Step 9: Find Y
Using equation (7):
\[ Y = 2X = 2 \times 18 = 36 \]

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