The price of sugar is increased by 25%. The percentage of reduction in…

When the price of a commodity increases by r%, the percentage reduction in consumption needed to keep total expenditure (price times consumption) unchanged is given by the reverse-percentage relation: reduction % = [r / (100 + r)] × 100. This holds because expenditure is a product of two factors; when one factor rises by a given percentage, the other must fall by a smaller, compensating percentage — not by the same percentage — to keep the product constant.

1. Let the original price be P and the original consumption be C, so the original expenditure is P × C.

2. The price rises by 25%, so the new price is P + 25% of P = 1.25P.

3. For expenditure to remain unchanged, the new consumption C' must satisfy 1.25P × C' = P × C.

4. Solving for C' gives C' = C / 1.25 = 0.8C.

5. The reduction in consumption is C - 0.8C = 0.2C, which is 20% of the original consumption.

6. Equivalently, using the reverse-percentage formula directly: reduction % = [25 / (100 + 25)] × 100 = (25 / 125) × 100 = 20%.

Cross-check by substitution: new expenditure = 1.25P × 0.8C = 1.00PC, which equals the original expenditure PC. Since the two match, the 20% reduction is confirmed.