Profit, Loss and Discount
Profit, Loss and Discount
Commercial arithmetic gives 2 questions in the Part-III maths paper. At this level they go well beyond one-step formulas, into successive discounts, false weights and the fixed-percentage traps. Almost every one of them turns on identifying which price the percentage is measured against.
The Three Bases
Three different prices appear in these questions and each percentage attaches to a specific one.
| Quantity | Percentage measured on |
|---|---|
| Profit or loss | Cost price |
| Discount | Marked price |
| Interest | Principal |
Profit % = (SP − CP) ÷ CP × 100
SP = CP × (100 + Profit %) ÷ 100
SP = MP × (100 − Discount %) ÷ 100
✗ An article bought at 400 and sold at 500 gives a profit of 100/500 = 20% | ✓ Profit is measured on cost price: 100/400 = 25%
Successive Discounts
Two discounts one after the other are not additive. Use the successive formula from the percentage topic with both values negative.
Discounts of 20 per cent and 10 per cent give −20 − 10 + 2 = −28, so the single equivalent discount is 28 per cent, not 30.
The order of the two discounts does not change the final price, which is itself a question sometimes set to test understanding rather than calculation.
The Standard Traps
- Equal profit and loss on two articles. If two items are sold at the same price, one at x per cent profit and the other at x per cent loss, the seller always makes a net loss of x²/100 per cent, never breaks even.
- Profit on selling price. If a question says profit is 25 per cent of the selling price, convert it to cost price first: it equals 33.33 per cent of cost.
- False weight. If a shopkeeper uses a 900 gram weight for a kilogram, the gain per cent is error divided by true value, so 100/900, about 11.11 per cent.
The first of these is the most frequently set, and the wrong option is always no profit no loss.
Marked Price Problems
A common shape: an article marked at Rs 500 is sold at a 10 per cent discount and still yields a 12.5 per cent profit. Find the cost price.
- Selling price = 500 × 90/100 = Rs 450.
- SP is 112.5 per cent of CP, so CP = 450 × 100/112.5 = Rs 400.
Work in the fixed order — marked price to selling price to cost price. Jumping straight from marked price to cost price is what produces wrong answers.
लाभ क्रय मूल्य पर, छूट अंकित मूल्य पर — आधार बदलना ही गलती है।
TRE pointer: The equal-profit-and-loss question is close to guaranteed in a Part-III paper because it looks like it should give zero and does not. Fix the result: always a loss of x²/100 per cent. The false-weight variant is the second most common. Both are single-formula questions once recognised, which is exactly why they are worth memorising when a wrong answer costs −1/3 and a right one is a full mark.
60-Second Recap
- Profit and loss are on cost price; discount is on marked price.
- SP = CP(100 + P%)/100 and SP = MP(100 − D%)/100.
- Successive discounts of 20% and 10% equal a single 28%, and order does not matter.
- Selling two items at the same price with equal profit and loss percentages gives a loss of x²/100 per cent.
- Profit stated on selling price must be converted to cost price before use.
- False weight gain = error ÷ true value × 100.
Frequently Asked Questions
If two articles are sold at the same price, one at 10 per cent profit and one at 10 per cent loss, what is the result?
A net loss of one per cent, from the formula x squared over a hundred. The transaction never breaks even, however equal the two percentages look, because the profit and the loss are calculated on different cost prices. No profit no loss is the planted wrong option.
Are two successive discounts of 20 and 10 per cent the same as 30 per cent?
No, they come to twenty-eight per cent, because the second discount applies to the already reduced price. Use a plus b plus ab over a hundred with both values negative. The order of the two discounts makes no difference to the final price.
How do I find the gain when a shopkeeper uses a false weight?
Divide the error by the true value and multiply by a hundred. Selling nine hundred grams as a kilogram means an error of a hundred over a true value of nine hundred, giving about eleven point one one per cent gain, not ten per cent.
What if profit is given as a percentage of the selling price?
Convert it to cost price before using any formula. A profit of twenty-five per cent on selling price means the cost is seventy-five per cent of the selling price, so the profit is thirty-three point three three per cent of cost. Every standard formula assumes cost price.