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

Options

A. `55^(5/9)`
B. 40
C. 45
D. `44^(4/9)`

Answer

To solve the problem step by step, we will use the information provided about the average weights of boys and girls, as well as the overall average weight.

### Step 1: Define Variables
Let:
- \( x \) = number of boys in the class
- \( y \) = number of girls in the class

### Step 2: Write the Total Weights
The total weight of boys can be expressed as:
\[ \text{Total weight of boys} = 69.3x \]
The total weight of girls can be expressed as:
\[ \text{Total weight of girls} = 59.4y \]

### Step 3: Write the Overall Average Weight Equation
The average weight of all boys and girls together is given as 63.8 kg. Therefore, we can write the equation for the average weight:
\[ \frac{69.3x + 59.4y}{x + y} = 63.8 \]

### Step 4: Cross-Multiply to Eliminate the Fraction
Cross-multiplying gives us:
\[ 69.3x + 59.4y = 63.8(x + y) \]

### Step 5: Expand the Right Side
Expanding the right side:
\[ 69.3x + 59.4y = 63.8x + 63.8y \]

### Step 6: Rearrange the Equation
Rearranging the equation to isolate terms involving \( x \) and \( y \):
\[ 69.3x - 63.8x = 63.8y - 59.4y \]
This simplifies to:
\[ 5.5x = 4.4y \]

### Step 7: Express the Ratio of Boys to Girls
From the equation \( 5.5x = 4.4y \), we can express the ratio of boys to girls:
\[ \frac{x}{y} = \frac{4.4}{5.5} = \frac{44}{55} = \frac{4}{5} \]

### Step 8: Find the Total Number of Students
Let’s denote the total number of students as \( N \):
\[ N = x + y = x + \frac{5}{4}x = \frac{9}{4}x \]

### Step 9: Calculate the Percentage of Boys
The percentage of boys in the class can be calculated as:
\[ \text{Percentage of boys} = \frac{x}{x+y} \times 100 = \frac{x}{\frac{9}{4}x} \times 100 = \frac{4}{9} \times 100 \]

### Step 10: Simplify the Percentage
Calculating the percentage:
\[ \text{Percentage of boys} = \frac{400}{9} \approx 44.44\% \]

### Conclusion
Thus, the percentage of boys in the class is approximately 44.4%, which can be rounded to 45%.

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