In a class of 45 students, the ratio of the number of boys and girls is 4:5. The average score of boys in mathematics is 78. If the average score of all the students in the class in mathematics is 76, then what is the average score of girls in mathematics?

Options

A. 73.6
B. 71.8
C. 72.4
D. 74.4

Answer

To solve the problem step by step, we will follow the given information systematically.

### Step 1: Determine the number of boys and girls
Given the ratio of boys to girls is 4:5, we can let the number of boys be \(4A\) and the number of girls be \(5A\).

Since the total number of students is 45, we can set up the equation:
\[ 4A + 5A = 45 \]
This simplifies to:
\[ 9A = 45 \]
Now, solving for \(A\):
\[ A = \frac{45}{9} = 5 \]

### Step 2: Calculate the number of boys and girls
Now we can find the number of boys and girls:
- Number of boys = \(4A = 4 \times 5 = 20\)
- Number of girls = \(5A = 5 \times 5 = 25\)

### Step 3: Calculate the total score of boys
The average score of boys in mathematics is 78. Therefore, the total score of boys can be calculated as:
\[ \text{Total score of boys} = \text{Number of boys} \times \text{Average score of boys} = 20 \times 78 = 1560 \]

### Step 4: Calculate the total score of the class
The average score of all students in the class is 76. Therefore, the total score of all students can be calculated as:
\[ \text{Total score of class} = \text{Total number of students} \times \text{Average score of class} = 45 \times 76 = 3420 \]

### Step 5: Calculate the total score of girls
Now we can find the total score of girls by subtracting the total score of boys from the total score of the class:
\[ \text{Total score of girls} = \text{Total score of class} - \text{Total score of boys} = 3420 - 1560 = 1860 \]

### Step 6: Calculate the average score of girls
Finally, we can find the average score of girls by dividing the total score of girls by the number of girls:
\[ \text{Average score of girls} = \frac{\text{Total score of girls}}{\text{Number of girls}} = \frac{1860}{25} = 74.4 \]

### Final Answer
The average score of girls in mathematics is **74.4**.

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