A group of boys has an average weight of 36 kg. One boy weighing 42 kg leaves the group and another boy weighing 30 kg joins the group. If the average now becomes 35.7 kg, then how many boys are there in the group?

Options

A. 30
B. 32
C. 40
D. 56

Answer

To solve the problem step by step, we will follow the logical reasoning based on the information given in the question.

### Step 1: Understand the initial conditions
We know that the average weight of the boys in the group is 36 kg. Let's denote the number of boys in the group as \( n \).

### Step 2: Calculate the total initial weight of the boys
The total weight of the boys can be calculated using the formula for average:
\[ \text{Total Weight} = \text{Average Weight} \times \text{Number of Boys} \]
Thus, the total initial weight of the boys is:
\[ \text{Total Initial Weight} = 36 \times n \]

### Step 3: Account for the boy leaving and the new boy joining
When a boy weighing 42 kg leaves the group, the total weight decreases by 42 kg. When a boy weighing 30 kg joins, the total weight increases by 30 kg. Therefore, the change in total weight can be calculated as:
\[ \text{Change in Total Weight} = -42 + 30 = -12 \text{ kg} \]
So, the new total weight becomes:
\[ \text{Total New Weight} = 36n - 12 \]

### Step 4: Calculate the new average weight
After the changes, the average weight of the boys is now 35.7 kg. The number of boys in the group after one leaves and one joins is \( n \) (the same number since one boy left and one joined). Thus, we can express the new average as:
\[ \text{New Average} = \frac{\text{Total New Weight}}{n} \]
Substituting the values we have:
\[ 35.7 = \frac{36n - 12}{n} \]

### Step 5: Solve for \( n \)
To eliminate the fraction, we can multiply both sides by \( n \):
\[ 35.7n = 36n - 12 \]
Now, rearranging the equation gives:
\[ 36n - 35.7n = 12 \]
\[ 0.3n = 12 \]
Dividing both sides by 0.3:
\[ n = \frac{12}{0.3} = 40 \]

### Conclusion
The number of boys in the group is \( n = 40 \).