Name the price before you use a percentage
Profit and loss questions go wrong when three different prices are treated as one number. Keep them separate:
- Cost price (CP) is what the seller paid.
- Marked price (MP) is the tag price before any discount.
- Selling price (SP) is what the buyer actually pays.
Profit and loss percentages are taken on cost price. Discount percentages are taken on marked price. A discount is not a loss unless you also know the cost.
Profit on cost, discount on the tag
Profit % = (SP − CP) / CP × 100
Discount % = (MP − SP) / MP × 100
An item costs 400. It is marked at 500 and sold at a 10% discount. The discount is 10% of 500, so the selling price is 450. Profit is 50 on a cost of 400, which is 12.5%. It is not a 10% profit, and it is not a 10% loss.
Two discounts in a row
Successive discounts follow the same net-change rule as successive percentages. Discounts of a% and b% give a single equivalent discount of:
a + b − (a × b) / 100
Discounts of 20% and 10% are not 30% off. Equivalent discount = 20 + 10 − (20 × 10) / 100 = 28%. On a marked price of 1,000, the selling price is 1,000 × 0.80 × 0.90 = 720, which is 28% below 1,000.
Apply this only to the marked price. You still need the cost price before you can call the result a profit or a loss.
False weight
A seller who promises 1 kg but hands over 900 g, while charging the honest cost of 1 kg, is making a profit on the missing 100 g.
Cost of what was actually given = 0.9 × C. Money received = C. Profit = (C − 0.9C) / 0.9C = 0.1 / 0.9 = 11.11%.
If that same seller also adds a stated 20% profit on the full kilogram, the buyer pays 1.2C for goods that cost the seller 0.9C. Actual profit = (1.2C − 0.9C) / 0.9C = 0.3 / 0.9 = 33.33%. The stated 20% is not the real gain.
A short checklist
- Write CP, MP, and SP as three labels before any percentage.
- Take profit or loss on CP. Take discount on MP.
- Chain two discounts with a + b − ab/100, or multiply the factors (0.80 × 0.90).
- For a short weight, divide the gain by the quantity actually handed over, not by the quantity printed on the pack.
Once the base of each percentage is named, these questions are a few multiplications. The marks are lost in the setup, not in the arithmetic.