The average weight of the students in a group 75.4 kg. Later on, four students having weight 72.9 kg, 73.8kg, 79.5 kg and 87.4 kg joined the group. As a result, the average weight of all the students in the group increased by 240 gm. What was the number of students in the group initially ?

Options

A. 46
B. 36
C. 50
D. 48

Answer

To solve the problem step by step, we can follow these calculations:

### Step 1: Understand the given information
We know:
- The initial average weight of the students in the group = 75.4 kg
- Four new students join with weights: 72.9 kg, 73.8 kg, 79.5 kg, and 87.4 kg
- The average weight increases by 240 grams (which is 0.240 kg).

### Step 2: Calculate the new average weight
The new average weight after the four students join is:
\[ 75.4 \, \text{kg} + 0.240 \, \text{kg} = 75.64 \, \text{kg} \]

### Step 3: Let the initial number of students be \( x \)
Let the initial number of students in the group be \( x \).

### Step 4: Calculate the total initial weight
The total initial weight of the students can be expressed as:
\[ \text{Total initial weight} = 75.4 \times x \]

### Step 5: Calculate the total weight of the new students
Now, we calculate the total weight of the four new students:
\[ 72.9 + 73.8 + 79.5 + 87.4 = 313.6 \, \text{kg} \]

### Step 6: Calculate the new total weight
The new total weight after the four students join is:
\[ \text{New total weight} = 75.4x + 313.6 \]

### Step 7: Calculate the new total number of students
The new total number of students is:
\[ \text{New number of students} = x + 4 \]

### Step 8: Set up the equation for the new average
The new average weight can be expressed as:
\[ \text{New average} = \frac{\text{New total weight}}{\text{New number of students}} \]
Substituting the values we have:
\[ 75.64 = \frac{75.4x + 313.6}{x + 4} \]

### Step 9: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 75.64(x + 4) = 75.4x + 313.6 \]

### Step 10: Expand and simplify
Expanding the left side:
\[ 75.64x + 302.56 = 75.4x + 313.6 \]

### Step 11: Rearranging the equation
Rearranging to isolate \( x \):
\[ 75.64x - 75.4x = 313.6 - 302.56 \]
\[ 0.24x = 11.04 \]

### Step 12: Solve for \( x \)
Dividing both sides by 0.24:
\[ x = \frac{11.04}{0.24} = 46 \]

### Conclusion
The initial number of students in the group was **46**.

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