3.7 Effect of Percentage Change on Product / Expenditure

Expenditure = Price × Quantity (consumption). Area = Length × Breadth. Total wage bill = Number of workers × Wage per worker. All of these are products of two factors, so the successive percentage change formula applies directly to find the net change in the product when both factors change.

A very common exam question flips this around: given that price changes by r%, by what percentage must consumption change so that expenditure remains constant (net change = 0)? Setting the successive-change formula to zero and solving for b in terms of a gives two clean results:

If price increases by r%, consumption must be decreased by: (r)/(100+r) × 100 %

If price decreases by r%, consumption must be increased by: (r)/(100-r) × 100 %

Worked Example 1: The price of sugar increases by 25%. By what percentage must a family reduce its consumption so that its expenditure on sugar remains unchanged? Required reduction = [25/(100+25)] × 100 = 25/125 × 100 = 20%.

Worked Example 2: The price of petrol falls by 20%. By what percentage can a person increase his consumption while keeping his expenditure the same? Required increase = [20/(100−20)] × 100 = 20/80 × 100 = 25%. Notice the asymmetry: a 20% price fall permits a 25% consumption rise (not 20%) to balance out — this asymmetry is exactly why “increase by x% then decrease by x%” never nets to zero.

Worked Example 3: Due to a price rise, a family reduces its consumption of rice by 10%, while the price itself rose by 10%. What is the net percentage change in the family’s total expenditure on rice? a = +10 (price), b = −10 (quantity). Net change = 10 − 10 + (10 × −10)/100 = 0 − 1 = −1%. Expenditure falls by 1%.

Worked Example 4: The price of a commodity increases by 30%. A household decides it can only afford to increase its total spending on the commodity by 4%. By what percentage must it reduce consumption? Let price change a = +30. Let expenditure’s net change be +4 (this is the target net %, not one of the two “input” percentages). We need to find b such that: 30 + b + (30b)/100 = 4. 30 + b(1 + 0.30) = 4 → b × 1.30 = 4 − 30 = −26 → b = −26/1.30 = −20%. So consumption must be reduced by 20%. This variant — where the final net change is given and one of the two input changes must be found — is a favourite twist in RRB NTPC papers; always isolate b algebraically from the successive-change formula rather than guessing.

Worked Example 5: The price of a commodity rises by 25%. A family decides that it can afford to let its total expenditure on the commodity rise by only 5%. By what percentage must it reduce its consumption? Let a = +25 (price) and the target net change = +5. We need b such that: 25 + b + (25b)/100 = 5. b(1 + 0.25) = 5 − 25 → b × 1.25 = −20 → b = −20/1.25 = −16%. (Check: 1.25 × 0.84 = 1.05, i.e., a net +5% change. Confirmed — consumption must be reduced by 16%.)

Worked Example 6: The price of petrol falls by 10%, and as a result a family’s total expenditure on petrol decreases by only 1% (since the family also increases its consumption somewhat). Find the percentage increase in consumption. Let a = −10 (price) and the target net change = −1. We need b such that: −10 + b + (−10b)/100 = −1. b(1 − 0.10) = −1 + 10 → b × 0.90 = 9 → b = 9/0.90 = 10%. (Check: price factor 0.90, consumption factor 1.10, product = 0.90 × 1.10 = 0.99, i.e., a net −1% change. Confirmed — consumption must increase by 10%.)