In a class of 92 students, the ratio of the number of boys to that of girls is 12:11. The average score of the girls, in a test, is `33 (1)/(3) %` more than that of boys. If the average score of all students is 80, then what is the average score of girls?

Options

A. 88
B. 84
C. 92
D. 96

Answer

To solve the problem step by step, we will follow the logical sequence based on the information provided.

### Step 1: Determine the number of boys and girls in the class
Given the total number of students is 92 and the ratio of boys to girls is 12:11.

Let the number of boys be \( 12x \) and the number of girls be \( 11x \).

The total number of students can be expressed as:
\[ 12x + 11x = 92 \]
\[ 23x = 92 \]
\[ x = \frac{92}{23} = 4 \]

Now, substituting \( x \) back to find the number of boys and girls:
- Number of boys = \( 12x = 12 \times 4 = 48 \)
- Number of girls = \( 11x = 11 \times 4 = 44 \)

### Step 2: Set up the average scores
Let the average score of boys be \( S_b \) and the average score of girls be \( S_g \).

According to the problem, the average score of the girls is \( 33 \frac{1}{3} \% \) more than that of the boys. This can be expressed as:
\[ S_g = S_b + \frac{1}{3} S_b = \frac{4}{3} S_b \]

### Step 3: Calculate the total average score
The average score of all students is given as 80. The total score of all students can be calculated as:
\[ \text{Total score} = \text{Average score} \times \text{Total number of students} = 80 \times 92 = 7360 \]

### Step 4: Express total score in terms of boys and girls
The total score can also be expressed as the sum of the scores of boys and girls:
\[ \text{Total score} = \text{Total score of boys} + \text{Total score of girls} \]
\[ = (S_b \times 48) + (S_g \times 44) \]

Substituting \( S_g = \frac{4}{3} S_b \):
\[ = (S_b \times 48) + \left(\frac{4}{3} S_b \times 44\right) \]
\[ = 48 S_b + \frac{176}{3} S_b \]
To combine these, convert \( 48 S_b \) to a fraction:
\[ = \frac{144}{3} S_b + \frac{176}{3} S_b = \frac{320}{3} S_b \]

### Step 5: Set up the equation
Now we have:
\[ \frac{320}{3} S_b = 7360 \]
Multiplying both sides by 3:
\[ 320 S_b = 22080 \]
Dividing by 320:
\[ S_b = \frac{22080}{320} = 69 \]

### Step 6: Calculate the average score of girls
Now that we have \( S_b \), we can find \( S_g \):
\[ S_g = \frac{4}{3} S_b = \frac{4}{3} \times 69 = 92 \]

### Final Answer
The average score of the girls is \( \boxed{92} \).

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