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 these calculations:

### Step 1: Determine the number of boys and girls in the class.
The ratio of boys to girls is given as 4:5. Let the number of boys be \(4x\) and the number of girls be \(5x\). The total number of students is 45.

\[ 4x + 5x = 45 \]

### Step 2: Solve for \(x\).
Combining the terms gives us:

\[ 9x = 45 \]

Now, divide both sides by 9:

\[ x = 5 \]

### Step 3: Calculate the number of boys and girls.
Now substitute \(x\) back into the expressions for boys and girls:

- Number of boys = \(4x = 4 \times 5 = 20\)
- Number of girls = \(5x = 5 \times 5 = 25\)

### Step 4: Calculate the total score of boys.
The average score of boys is given as 78. 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 5: Calculate the total score of the class.
The average score of all students in the class is given as 76. The total score of the class 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 6: Calculate the total score of girls.
To find the total score of girls, we subtract 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 7: Calculate the average score of girls.
Now, 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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