If 2 men and 3 boys can do a work in 10 days, 4 men can do a work in 10 days, in how many days 3 men and 3 boys can do same work?

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Define Variables
Let:
- \( X \) = daily work done by one man
- \( Y \) = daily work done by one boy

### Step 2: Set Up Equations Based on Given Information
From the problem statement:
1. **For 2 men and 3 boys working for 10 days:**
\[ \text{Total Work} = (2X + 3Y) \times 10 \]

2. **For 4 men working for 10 days:**
\[ \text{Total Work} = (4X) \times 10 \]

### Step 3: Equate the Total Work
Since both expressions represent the same total work, we can set them equal to each other:
\[ (2X + 3Y) \times 10 = (4X) \times 10 \]

### Step 4: Simplify the Equation
Dividing both sides by 10:
\[ 2X + 3Y = 4X \]

### Step 5: Rearranging the Equation
Rearranging gives:
\[ 4X - 2X = 3Y \]
\[ 2X = 3Y \]

### Step 6: Express \( X \) in Terms of \( Y \)
From \( 2X = 3Y \), we can express \( X \) in terms of \( Y \):
\[ \frac{X}{Y} = \frac{3}{2} \]
Let \( X = 3k \) and \( Y = 2k \) for some constant \( k \).

### Step 7: Calculate Total Work
Using \( X = 3k \):
\[ \text{Total Work} = 4X \times 10 = 4(3k) \times 10 = 120k \text{ units} \]

### Step 8: Calculate the Work Done by 3 Men and 3 Boys
Now, we need to find how many days 3 men and 3 boys can complete the same work:
\[ \text{Daily work of 3 men and 3 boys} = 3X + 3Y = 3(3k) + 3(2k) = 9k + 6k = 15k \]

### Step 9: Calculate the Number of Days
The number of days \( D \) required to complete the total work is given by:
\[ D = \frac{\text{Total Work}}{\text{Daily Work}} = \frac{120k}{15k} = 8 \text{ days} \]

### Final Answer
Thus, 3 men and 3 boys can complete the work in **8 days**.

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