A grocer has 18 kg of rice, a part of which he sells at 8% profit and the rest…

Concept: When two quantities of the same item are sold at different profit percentages and the combined lot shows one overall profit percentage, the two quantities split in the ratio of how far each individual profit rate is from the overall rate (Rule of Alligation): Quantity at lower rate : Quantity at higher rate = (higher rate − overall rate) : (overall rate − lower rate). Equivalently, with cost price taken as a fixed value per unit, the weighted profit of the two parts must add up to the overall profit on the whole lot.

Application:

1. Let the quantity sold at 26% profit be x kg; then the quantity sold at 8% profit is (18 − x) kg, since the total stock is 18 kg.

2. Take the cost price as ₹1 per kg (profit % is scale-invariant, so any convenient cost price works). The total profit on the whole 18 kg at the overall 24% rate is 18 × 0.24 = ₹4.32.

3. The same total profit, built from the two parts, is 0.08 × (18 − x) + 0.26 × x.

4. Equate the two expressions: 0.08(18 − x) + 0.26x = 4.32 → 1.44 − 0.08x + 0.26x = 4.32 → 1.44 + 0.18x = 4.32.

5. Solve: 0.18x = 2.88 → x = 16.

Cross-check: By the alligation rule: Quantity at 8% : Quantity at 26% = (26 − 24) : (24 − 8) = 2 : 16 = 1 : 8. Dividing 18 kg in the ratio 1 : 8 gives 2 kg (at 8% profit) and 16 kg (at 26% profit) — the same result as the equation method.

Result: 16 kg of rice was sold at 26% profit.