The price of petrol increases by 25%. By what percentage must a customer…

Concept — A bill is the product of two factors: Bill = Price × Consumption. If the price is multiplied by (1 + r) and the consumption by (1 − x), the bill is unchanged only when (1 + r)(1 − x) = 1, which rearranges to x = r ÷ (1 + r). The required cut is therefore measured against the raised price, not against the old one, so it is always smaller than the rise itself.

Application — Apply that identity to a rise of 25%, i.e. r = 0.25.

1. Let the original price be P per litre and the original consumption be C litres, so the original bill is P × C.

2. After the 25% rise the new price is 1.25P per litre.

3. Let the consumption fall by a fraction x, so the new consumption is C(1 − x) and the new bill is 1.25P × C(1 − x).

4. The bill must not alter, so 1.25P × C(1 − x) = P × C; dividing both sides by P × C leaves 1.25(1 − x) = 1.

5. Hence 1 − x = 1 ÷ 1.25 = 0.8, so x = 0.2 — a reduction of 20%.

Cross-check — Put P = ₹100 per litre and C = 100 litres, so the original bill is ₹10,000. At the new price of ₹125 per litre that same ₹10,000 buys 10,000 ÷ 125 = 80 litres, which is 20 litres below 100 litres — a 20% cut, exactly as the algebra says.

Contrast — Each candidate reduction applied to a 100-litre purchase, valued at the new price of ₹125 per litre:

Only the 20% cut brings the bill back to ₹10,000. Subtracting the 25% rise straight from the quantity over-corrects, because the rise is measured on the old price while the cut has to be measured on the new one. General rule: a price rise of r% needs a consumption cut of 100r ÷ (100 + r) per cent — here 100 × 25 ÷ 125 = 20.