The average of the weights of boys in a class is 69.3 kg and that of girls in the same class is 59.4 kg. If the average of the weights of all the boys and girls in the class is 63.8 kg. then the percentage of the number of girls in the class is:

Options

A. `44(5)/(9)%`
B. `45(5)/(9)%`
C. `55(5)/(9)%`
D. `54(5)/(9)%`

Answer

To solve the problem step by step, we will use the allegation method to find the percentage of girls in the class.

### Step-by-Step Solution:

1. **Identify the given averages**:
- Average weight of boys (A_b) = 69.3 kg
- Average weight of girls (A_g) = 59.4 kg
- Average weight of the entire class (A_c) = 63.8 kg

2. **Set up the allegation method**:
- We will use the formula for the allegation method which helps in finding the ratio of two groups based on their averages.
- The formula states that the difference between the average of one group and the overall average is compared to the difference between the other group and the overall average.

3. **Calculate the differences**:
- Difference between the average weight of boys and the overall average:
\[ D_b = A_b - A_c = 69.3 - 63.8 = 5.5 \text{ kg} \]
- Difference between the average weight of girls and the overall average:
\[ D_g = A_c - A_g = 63.8 - 59.4 = 4.4 \text{ kg} \]

4. **Set up the ratio of boys to girls**:
- The ratio of the number of boys (B) to the number of girls (G) can be expressed as:
\[ \frac{B}{G} = \frac{D_g}{D_b} = \frac{4.4}{5.5} \]

5. **Simplify the ratio**:
- To simplify \( \frac{4.4}{5.5} \), we can multiply both the numerator and the denominator by 10 to eliminate the decimal:
\[ \frac{44}{55} \]
- This simplifies to:
\[ \frac{4}{5} \]

6. **Express the number of boys and girls in terms of a common variable**:
- Let the number of boys be \( 4k \) and the number of girls be \( 5k \).

7. **Calculate the total number of students**:
- Total number of students = Number of boys + Number of girls:
\[ \text{Total} = 4k + 5k = 9k \]

8. **Calculate the percentage of girls in the class**:
- The percentage of girls is given by:
\[ \text{Percentage of girls} = \left( \frac{\text{Number of girls}}{\text{Total number of students}} \right) \times 100 \]
- Substituting the values:
\[ \text{Percentage of girls} = \left( \frac{5k}{9k} \right) \times 100 = \frac{500}{9} \approx 55.56\% \]

9. **Final answer**:
- The percentage of girls in the class is \( \frac{500}{9} \) or approximately \( 55 \frac{5}{9} \% \).