Profit, loss and discount questions are percentage questions wearing shopkeeper clothes. The arithmetic is the same as every other percentage problem; what changes is that three different prices are in play at once, and each percentage is measured against a different one. Choose the wrong base and every later step is wrong, while the answer still looks tidy.
Get the bases right and the entire family collapses into one or two multiplications.
Three prices move through every transaction.
The model omits overheads such as transport, tax and packaging unless the question names them. When it does, add them to the cost price first, because profit is still measured against total cost.
The core definitions. Profit is what is left when the selling price exceeds the cost price; loss is the shortfall when it does not.
Working backwards. The formulas invert cleanly, and most questions test the inverse.
| Given | Find SP | Find CP |
|---|---|---|
| Profit of p percent | SP = CP × (1 + p ÷ 100) | CP = SP ÷ (1 + p ÷ 100) |
| Loss of l percent | SP = CP × (1 − l ÷ 100) | CP = SP ÷ (1 − l ÷ 100) |
| Discount of d percent | SP = MP × (1 − d ÷ 100) | MP = SP ÷ (1 − d ÷ 100) |
Successive discounts. Two discounts are never added. Convert each to a multiplier and multiply: discounts of 10 percent and 20 percent give 0.90 × 0.80 = 0.72, which is a single discount of 28 percent, not 30.
Mark-up then discount. A shop marks up by 40 percent and then discounts by 25 percent: the combined multiplier is 1.40 × 0.75 = 1.05, a profit of 5 percent. The order does not change the product; it only fixes which base each percentage uses.
A shop marks a jacket at 2000. It offers a 20 percent discount, and still makes a profit of 25 percent on the cost. Find the cost price.
Start at the marked price, because that is where the discount is measured. The discount multiplier is 0.80, so the selling price is 2000 × 0.80 = 1600.
Now move from the selling price to the cost. A profit of 25 percent means the selling price is 1.25 times the cost, so the cost is the selling price divided by 1.25.
Check forward: 25 percent of 1280 is 320, and 1280 + 320 = 1600, which matches the discounted selling price. The cost price is 1280.
An article bought for 800 is sold for 720. What is the loss percentage?
A marked price of 1500 is reduced by a 20 percent discount. What is the selling price?
Two successive discounts of 10 percent and 20 percent are equivalent to a single discount of what percentage?
An item is sold for 600 at a profit of 20 percent. What is the cost price?
Without looking back: which price is the base for profit percent, and which is the base for a discount?
The answers: profit and loss percent are measured on the cost price, and discount is measured on the marked price.
Profit percent is measured against the price, while a discount is measured against the price. A 20 percent discount on a marked price of 1500 gives a selling price of . Successive discounts of 10 percent and 20 percent give a net multiplier of .
A trader marks an article 40 percent above its cost and then allows a discount of 25 percent on the marked price. What is his profit percentage?
Let the cost price be 100, which turns every percentage into an amount. A mark-up of 40 percent makes the marked price 140. A discount of 25 percent on the marked price gives a selling price of 140 × 0.75 = 105.
The trader makes 5 percent. The combined multiplier is 1.40 × 0.75 = 1.05, which confirms the same result in one step.
The trap is 15 percent, obtained by subtracting the discount from the mark-up. That subtraction is invalid because the two percentages have different bases: the 40 percent is taken on cost, and the 25 percent is taken on the marked price.
Profit and loss are single-step percentage changes. The next lesson handles a change that repeats itself over time, where the percentage is applied again and again to a growing base. That is simple and compound interest, and the same multiplier logic drives both.