UPSC CSE Prelims 2020

Correct Option (A)

Let G represent the number of girls and B represent the number of boys. The overall average marks are calculated as the total marks obtained by all students divided by the total number of students.

Based on the provided data for English marks:

• Average marks for girls in English = 9
• Average marks for boys in English = 8
• Overall average marks in English = 8.8

The total marks in English can be expressed as $9G+8B$. The total number of students is $G+B$. Therefore, the equation for the overall average in English is:

$$\frac{\text{9G + 8B}}{\text{G + B}}=8.8$$

Now, consider the data for Hindi marks:

• Average marks for girls in Hindi = 8
• Average marks for boys in Hindi = 7
• Overall average marks in Hindi = X

The total marks in Hindi can be expressed as $8G+7B$. The total number of students is $G+B$. Therefore, the equation for the overall average in Hindi is:

$$X=\frac{\text{8G + 7B}}{\text{G + B}}$$

To determine the value of X, we manipulate the equation derived from the English average data:

$$\frac{\text{9G + 8B}}{\text{G + B}}=8.8$$

This expression can be rewritten by separating the terms to align with the structure of X:

$$\Rightarrow \frac{\text{8G + 7B + G + B}}{\text{G + B}}=8.8$$

Separating the fraction into two components:

$$\Rightarrow \frac{\text{8G + 7B}}{\text{G + B}}+\frac{\text{G + B}}{\text{G + B}}=8.8$$

Simplifying the second term, as $\frac{\text{G + B}}{\text{G + B}}=1$:

$$\Rightarrow \frac{\text{8G + 7B}}{\text{G + B}}+1=8.8$$

Subtracting 1 from both sides of the equation:

$$\Rightarrow \frac{\text{8G + 7B}}{\text{G + B}}=8.8-1$$

$$\Rightarrow \frac{\text{8G + 7B}}{\text{G + B}}=7.8$$

Since X is defined as $\frac{\text{8G + 7B}}{\text{G + B}}$, the value of X is 7.8. This method demonstrates the consistent application of average calculation across both subjects to find the unknown average.

Incorrect Options:

Options (B) 7.6, (C) 7.4, and (D) 7.2 are incorrect. These values do not align with the result derived from the consistent application of the average calculation formula using the provided data for both English and Hindi marks. The unique solution obtained through algebraic manipulation is 7.8.