Profit and loss
Profit and loss are always per cent of cost price; discount is per cent of marked price. Use multiplying factors for quick answers.
Terms and formulas
| Term | Meaning or formula |
|---|---|
| Cost price (CP) | Price paid for the item, including overheads such as transport |
| Selling price (SP) | Price at which the item is sold |
| Marked price (MP) | Price printed on the tag; discount is given on it |
| Profit % / loss % | (SP − CP) ÷ CP × 100 / (CP − SP) ÷ CP × 100 |
| SP from CP | CP × (100 + gain%) ÷ 100, or CP × (100 − loss%) ÷ 100 |
| CP from SP | SP × 100 ÷ (100 + gain%), or SP × 100 ÷ (100 − loss%) |
| SP from MP | MP × (100 − discount%) ÷ 100 |
| Mark-up a%, then discount b% | Net profit % = a − b − ab/100 |
| Same SP; x% gain on one, x% loss on the other | Always a net loss of x²/100 % |
| False weight (sells at CP) | Gain % = error ÷ (true value − error) × 100 |
Worked example 1: profit with overheads
A trader buys 40 kg of onions at ₹25 a kg and spends ₹100 on transport. He sells all of them at ₹30 a kg. Find his profit per cent.
- Standard method: CP = 40 × 25 + 100 = ₹1,100 (transport is part of cost). SP = 40 × 30 = ₹1,200. Profit = ₹100.
- Profit % = 100 ÷ 1,100 × 100 = 9 1/11 % (about 9.09%, rounded to two decimal places).
- Shortcut (per kg): cost per kg = 25 + 100 ÷ 40 = ₹27.50; profit per kg = 30 − 27.50 = ₹2.50; 2.50 ÷ 27.50 = 1/11 = 9 1/11 %.
- Check: 1,100 + 1/11 of 1,100 = 1,100 + 100 = ₹1,200.
9 1/11 % profit (about 9.09%).
Worked example 2: from a loss to a gain
By selling a mixer-grinder for ₹2,340, a shopkeeper loses 10%. At what price should he sell it to gain 15%?
- Standard method: SP = 90% of CP, so CP = 2,340 × 100 ÷ 90 = ₹2,600. New SP = 2,600 × 115 ÷ 100 = ₹2,990.
- Shortcut (one line): new SP = old SP × (100 + new gain%) ÷ (100 − old loss%) = 2,340 × 115 ÷ 90 = 26 × 115 = ₹2,990.
- Check: 2,600 × 0.90 = ₹2,340 and 2,600 × 1.15 = ₹2,990.
₹2,990.
Worked example 3: mark-up and discount
A shopkeeper marks a saree 40% above its cost price of ₹2,000 and allows a discount of 15%. Find the selling price and the profit per cent.
- Standard method: MP = 2,000 × 1.40 = ₹2,800. Discount = 15% of 2,800 = ₹420. SP = 2,800 − 420 = ₹2,380.
- Profit = 2,380 − 2,000 = ₹380, so profit % = 380 ÷ 2,000 × 100 = 19%.
- Shortcut: net % = a − b − ab/100 = 40 − 15 − (40 × 15)/100 = 40 − 15 − 6 = 19% profit, so SP = 2,000 × 1.19 = ₹2,380.
- Check: 1.40 × 0.85 = 1.19.
SP ₹2,380; profit 19%.
Worked example 4: false weight
A dishonest shopkeeper sells rice at cost price but uses a 900 g weight in place of 1 kg. Find his gain per cent.
- Standard method: suppose 1 kg costs ₹100. He gives 900 g, which costs him ₹90, but charges ₹100. Gain = ₹10 on ₹90 = 11 1/9 %.
- Shortcut: gain % = error ÷ (true value − error) × 100 = 100 ÷ 900 × 100 = 11 1/9 %.
- Check: ₹90 increased by 1/9 is ₹90 + ₹10 = ₹100.
11 1/9 % (about 11.11%).
Bank, SSC and railway papers ask CP-SP-MP chains, successive discounts, false weights and the same-selling-price trap; placement tests combine two of these in one question. Most fall in one line with the multiplying-factor method.
2 marks: “A pressure cooker bought for ₹1,600 is sold for ₹1,840. Find the profit per cent.” Profit = ₹240; 240 ÷ 1,600 × 100 = 15%.
5 marks: “By selling a mixer-grinder for ₹2,340, a shopkeeper loses 10%. At what price should he sell it to gain 15%?” CP = 2,340 × 100 ÷ 90 = ₹2,600; SP = 2,600 × 1.15 = ₹2,990. Shortcut: 2,340 × 115 ÷ 90.
10 marks: “(a) Onions bought at ₹25 a kg (40 kg, ₹100 transport) and sold at ₹30 a kg: find the profit %. (b) Goods marked 40% above a cost of ₹2,000 are sold at 15% discount: find the SP and profit %. (c) Two items are sold at ₹990 each, one at 10% gain and one at 10% loss: find the overall result.” (a) 9 1/11 %. (b) ₹2,380 and 19%. (c) CPs ₹900 and ₹1,100; total ₹2,000 against ₹1,980 received: a 1% loss (x²/100).
Profit on CP, discount on MP. Mark-up a%, then discount b% → a − b − ab/100. Same SP with x% gain and x% loss → a loss of x²/100 %.