A shopkeeper has 50 kg of rice. Part is sold at 20% profit and the rest at 40% profit. If the overall gain is 25%, find the quantities sold at 20% and at 40% profit.

Introduction / Context:
We are given two profit rates on parts of a single stock and an overall profit rate. This is a weighted average situation. Let x be the amount sold at 20% profit; the remainder is sold at 40% profit. Use the average profit equation to find x.

Given Data / Assumptions:

• Total quantity = 50 kg.
• Profit rates = 20% on x kg; 40% on (50 − x) kg.
• Overall profit = 25% on the whole.

Concept / Approach:
Average profit% = (20*x + 40*(50 − x)) / 50. Set equal to 25 and solve for x. The quantities are x at 20% and (50 − x) at 40%.

Step-by-Step Solution:
(20x + 40(50 − x)) / 50 = 25. 20x + 2000 − 40x = 1250 ⇒ −20x = −750 ⇒ x = 37.5 kg. Hence, at 20%: 37.5 kg; at 40%: 12.5 kg.

Verification / Alternative check:
Weighted profit = (20*37.5 + 40*12.5)/50 = (750 + 500)/50 = 1250/50 = 25% overall, as required.

Why Other Options Are Wrong:
12.5 & 37.5 reverse the roles; other pairs do not satisfy the weighted average equation at all.

Common Pitfalls:
Averaging 20 and 40 directly to 30% and assuming equal split—this ignores the required 25% overall and the unknown split.

Final Answer:
37.5 kgs and 12.5 kgs