Lesson 31 of 100 | Quantitative Aptitude / Percentages / प्रतिशत
Population, income, expenditure and price applications
Learning outcome
Apply percentages to population, income, expenditure and prices, identifying the correct reference and the assumptions that make a calculation valid.
Connect formulas to conditions
A population model applies each period's rate to that period's starting population, so repeated rates multiply factors. These rates are assumptions, not guaranteed predictions. Include any separately specified migration or other adjustments.
Savings = income − expenditure. Compare matching time periods. Changes in income, expenditure and savings use their respective original amounts as bases; all are positive here.
Do not subtract income and expenditure percentages: their bases differ. Calculate the new amounts, subtract to find savings, then compare with original savings.
For a single item with no extra charges, expenditure = unit price × quantity. If expenditure is fixed, a price multiplier requires its reciprocal quantity multiplier. A price increase of p% therefore gives new quantity = old quantity/(1 + p/100).
Fixed expenditure is essential: a buyer could otherwise change the budget rather than the purchasing quantity.
Worked example 1
A model starts with 12,000 residents and assumes 5% growth in each of two years, with no separate adjustments. Find the final population.
After the first year: 12000 × 1.05 = 12600. After the second: 12600 × 1.05 = 13230. The overall increase is 1230, giving (1230/12000) × 100% = 10.25%.
Worked example 2
Monthly income is ₹42,000 and expenditure ₹31,500. Income rises by 10% and expenditure by 12%. Find the savings change.
Original savings = 42000 − 31500 = ₹10500. New income = 42000 × 1.10 = ₹46200; new expenditure = 31500 × 1.12 = ₹35280. New savings = 46200 − 35280 = ₹10920. Savings increased by ₹420, so their percentage increase is (420/10500) × 100% = 4%.
Worked example 3
A buyer spends exactly ₹1,260 on rice. Its price rises from ₹35/kg to ₹42/kg. Find the quantity change at unchanged expenditure.
Original quantity = 1260/35 = 36 kg; new quantity = 1260/42 = 30 kg. Quantity falls by 6 kg, or (6/36) × 100% = (50/3)% ≈ 16.67%. The price increase is 20%, but the quantity decrease is not.
Common traps
Do not use income as the base for a savings change. Fixed expenditure, not price movement alone, justifies inverse proportion.
Practice questions
- A population of 8,000 grows by 5%, then falls by 2%, with no other changes. Find the final population and net percentage change.
- Monthly income is ₹32,000; expenditure is 62.5% of income. Find expenditure and savings.
- Income is ₹50,000 and expenditure ₹40,000 per month. They increase by 8% and 5% respectively. Find new savings and their percentage change.
- Exactly ₹1,440 is spent on a product whose price falls from ₹60/kg to ₹48/kg. Find both quantities and the percentage increase in quantity.
Worked answers
- The populations become 8000 × 1.05 = 8400, then 8400 × 0.98 = 8232. The net increase is (232/8000) × 100% = 2.9%.
- Expenditure = 32000 × 0.625 = ₹20000. Savings are the remainder: 32000 − 20000 = ₹12000.
- Original savings = 50000 − 40000 = ₹10000. New income = 50000 × 1.08 = ₹54000; expenditure = 40000 × 1.05 = ₹42000. New savings = 54000 − 42000 = ₹12000. The increase is (2000/10000) × 100% = 20%.
- Quantities are 1440/60 = 24 kg and 1440/48 = 30 kg. The increase is (6/24) × 100% = 25%; the unchanged budget determines this inverse relationship.