Allocation of Rs. 675 among 75 students (boys and girls): each boy receives Rs. 20 and each girl receives Rs. 5. Determine the exact counts of boys and girls in the group.

Introduction / Context:
This is a classic two-variable allocation problem that tests linear equations and weighted sums. We are given a total headcount and a total amount distributed with different rates for boys and girls. The goal is to determine how many boys and how many girls are present when each boy gets Rs. 20 and each girl gets Rs. 5.

Given Data / Assumptions:

• Total students = 75.
• Amount distributed = Rs. 675.
• Each boy receives Rs. 20.
• Each girl receives Rs. 5.

Concept / Approach:
Let b = number of boys and g = number of girls. Then b + g = 75. The money equation is 20*b + 5*g = 675. Solving this simultaneous system yields the unique pair (b, g) that satisfies both headcount and payment constraints.

Step-by-Step Solution:
b + g = 75 … (1) 20b + 5g = 675 … (2) Divide (2) by 5 ⇒ 4b + g = 135 … (3) Subtract (1) from (3): (4b + g) − (b + g) = 135 − 75 ⇒ 3b = 60 ⇒ b = 20. Then g = 75 − 20 = 55.

Verification / Alternative check:
Payment check: 20 boys * Rs. 20 = Rs. 400; 55 girls * Rs. 5 = Rs. 275; total = Rs. 675, which matches the given sum.

Why Other Options Are Wrong:
15, 60 → money becomes 15*20 + 60*5 = 300 + 300 = 600 (too low). 25, 50 → 25*20 + 50*5 = 500 + 250 = 750 (too high). 30, 45 → 30*20 + 45*5 = 600 + 225 = 825 (too high).

Common Pitfalls:
Mixing the rates or forgetting to apply both the headcount and money equations simultaneously. Also, do not round—this is an exact integer solution.

Final Answer:
20 , 55