The average weight of some children in a group was 48.5kg .When 6 students having average weight 50.4kg left the group and 4 students having average weight 56.55 kg joined the group, the average weight of the children in the group increased by 0.8 kg. The number of children, initially, in the group was?

Options

A. A. 32
B. B. 30
C. C. 28
D. D. 26

Answer

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

### Step 1: Define Variables
Let the initial number of children in the group be \( n \).

### Step 2: Calculate Initial Total Weight
The average weight of the children in the group is given as 48.5 kg. Therefore, the total weight of the children initially can be calculated as:
\[ \text{Total initial weight} = 48.5 \times n \]

### Step 3: Calculate Weight of Students Leaving
When 6 students leave the group, each having an average weight of 50.4 kg, the total weight of these students is:
\[ \text{Weight of students leaving} = 6 \times 50.4 = 302.4 \text{ kg} \]

### Step 4: Calculate Weight of Students Joining
When 4 students join the group, each having an average weight of 56.55 kg, the total weight of these students is:
\[ \text{Weight of students joining} = 4 \times 56.55 = 226.2 \text{ kg} \]

### Step 5: Calculate New Total Weight
After the changes, the new total weight of the group becomes:
\[ \text{New total weight} = (48.5n - 302.4 + 226.2) \]
\[ = 48.5n - 76.2 \]

### Step 6: Calculate New Number of Students
After the changes, the new number of students in the group is:
\[ \text{New number of students} = n - 6 + 4 = n - 2 \]

### Step 7: Calculate New Average Weight
The average weight of the children in the group increased by 0.8 kg, so the new average weight is:
\[ \text{New average weight} = 48.5 + 0.8 = 49.3 \text{ kg} \]

### Step 8: Set Up the Equation
Now we can set up the equation based on the new average weight:
\[ \frac{48.5n - 76.2}{n - 2} = 49.3 \]

### Step 9: Cross Multiply to Solve for \( n \)
Cross multiplying gives:
\[ 48.5n - 76.2 = 49.3(n - 2) \]
Expanding the right side:
\[ 48.5n - 76.2 = 49.3n - 98.6 \]

### Step 10: Rearranging the Equation
Rearranging the equation to isolate \( n \):
\[ 48.5n - 49.3n = -98.6 + 76.2 \]
\[ -0.8n = -22.4 \]

### Step 11: Solve for \( n \)
Dividing both sides by -0.8:
\[ n = \frac{22.4}{0.8} = 28 \]

### Conclusion
The initial number of children in the group was \( n = 28 \).

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