In aid of charity, every student in a class contributes as many rupees as the number of students in that class. With the additional contribution of 2…

Correct Option (2)

Let the number of students in the class be $x$x. According to the problem statement, each student contributes an amount in rupees equal to the number of students. Therefore, the total contribution from all students, excluding any additional amount, is $x \times x = x^2$x×x=x2 rupees.

An additional contribution of 2 rupees was made by one student. The total collection is stated to be 443 rupees.

Thus, the total collection can be represented by the equation:

$$x^2 + 2 = 443$$x2+2=443

To determine the sum contributed solely by the students based on their number, subtract the additional contribution from the total collection:

$$x^2 = 443 - 2$$x2=443−2

$$x^2 = 441$$x2=441

To find the number of students, take the square root of 441:

$$x = \sqrt{441}$$x=441​

$$x = \pm 21$$x=±21

Since the number of students cannot be a negative value, the positive root is considered.

Therefore, the number of students in the class is 21.

Incorrect Options:

The fundamental equation derived from the problem statement is $x^2 = 441$x2=441, where $x$x represents the number of students. Let's evaluate the other options:

• If the number of students were 12 (Option 1), the total contribution based on the number of students would be $12^2 = 144$122=144. Adding the additional 2 rupees, the total collection would be $144 + 2 = 146$144+2=146, which does not match the given total of 443 rupees.
• If the number of students were 43 (Option 3), the total contribution based on the number of students would be $43^2 = 1849$432=1849. Adding the additional 2 rupees, the total collection would be $1849 + 2 = 1851$1849+2=1851, which does not match the given total of 443 rupees.
• If the number of students were 45 (Option 4), the total contribution based on the number of students would be $45^2 = 2025$452=2025. Adding the additional 2 rupees, the total collection would be $2025 + 2 = 2027$2025+2=2027, which does not match the given total of 443 rupees.

Only $x = 21$x=21 satisfies the condition $x^2 = 441$x2=441, resulting in a total collection of $21^2 + 2 = 441 + 2 = 443$212+2=441+2=443 rupees, consistent with the problem statement.