Profit and loss, interest and discount
Profit and loss are percentages of cost price; discount is a percentage of marked price. Simple interest grows evenly; compound interest grows on interest too.
Worked example: profit and loss
By selling an article for ₹720, a man loses 10%. At what price should he sell it to gain 15%?
- A 10% loss means SP = 90% of CP.
- CP = 720 ÷ 0.90 = ₹800.
- For a 15% gain, SP = 115% of 800 = ₹920.
- One-line method: 720 × 115/90 = 720 × 23/18 = 40 × 23 = ₹920.
₹920. Shortcut: new SP = old SP × (100 + new gain%) ÷ (100 − old loss%).
Worked example: successive discounts
An item is marked at ₹2,500. Successive discounts of 20% and 10% are given. Find the selling price and the single equivalent discount.
- After 20%: 2,500 × 0.80 = ₹2,000.
- After a further 10%: 2,000 × 0.90 = ₹1,800.
- Single equivalent discount = 20 + 10 − (20 × 10)/100 = 28%.
- Check: 2,500 × 0.72 = ₹1,800.
SP ₹1,800; single equivalent discount 28%. Shortcut: discounts a% and b% equal a + b − ab/100, always less than a + b.
Worked example: simple vs compound interest
Find the simple and compound interest on ₹10,000 at 10% per annum for 2 years, compounded yearly.
- SI = P × R × T ÷ 100 = 10,000 × 10 × 2 ÷ 100 = ₹2,000.
- Compound amount = 10,000 × (1.10)² = 10,000 × 1.21 = ₹12,100.
- CI = 12,100 − 10,000 = ₹2,100.
- Check the gap: CI − SI for 2 years = P(R/100)² = 10,000 × 0.01 = ₹100.
SI ₹2,000; CI ₹2,100. Shortcut: for 2 years, CI − SI = P(R/100)²; for 3 years it is P(R/100)²(3 + R/100).
Formula sheet
| Item | Formula |
|---|---|
| Profit % and loss % | Profit or loss ÷ CP × 100 |
| SP from CP | CP × (100 + gain%) ÷ 100, or CP × (100 − loss%) ÷ 100 |
| Discount | Discount % = discount ÷ MP × 100; SP = MP − discount |
| Simple interest | SI = PRT ÷ 100; amount = P + SI |
| Compound interest | A = P(1 + R/100)ⁿ; half-yearly: rate R/2, periods 2n |
| Present worth (discounting, SI) | PW = 100 × amount ÷ (100 + RT); true discount = amount − PW |
| Same SP, x% gain on one and x% loss on the other | Always a net loss of x²/100 per cent |
| Doubling under SI | T = 100 ÷ R years |
NET asks CP-SP-MP chains, successive discounts, SI vs CI gaps and the same-selling-price gain/loss trap. One clean formula usually settles each in under a minute.
Typical MCQ: “Two chairs are each sold for ₹990, one at 10% profit and the other at 10% loss. What is the overall result?” A loss of 1% (10²/100). The cost prices are ₹900 and ₹1,100, so ₹1,980 is received for ₹2,000 spent.
Typical MCQ: “At what rate of simple interest does a sum double in 8 years?” 12.5% per annum. Doubling needs SI = P, so R = 100 ÷ 8.
Profit on CP, discount on MP. Equal-SP gain and loss of x% = a loss of x²/100 per cent. Over two years, CI − SI = P(R/100)².