The vocabulary
Cost price (CP): what the seller paid.
Selling price (SP): what the buyer paid.
Marked price (MP): the price on the tag, before any discount.
Profit = SP − CP, and profit % is always on CP.
Discount = MP − SP, and discount % is always on MP.
Use multipliers
Profit of p%: SP = CP × (1 + p/100). A 25% profit means SP = 1.25 × CP.
Loss of l%: SP = CP × (1 − l/100).
Mark-up of m% followed by a discount of d%: SP = CP × (1 + m/100)(1 − d/100).
Example. Marked 40% above cost with a 25% discount: 1.4 × 0.75 = 1.05, a 5% profit.
Successive discounts
Two discounts of a% and b% are equal to one discount of a + b − ab/100.
Example. 20% then 10%: 20 + 10 − 2 = 28%, not 30%.
Classic traps
Same selling price, equal profit and loss. If two items are sold at the same price, one at x% profit and the other at x% loss, there is always an overall loss of x²/100 %.
Example: x = 20 gives a 4% loss.
Cost of n items equals the selling price of m items. Profit % = (n − m) / m × 100.
Example: CP of 12 = SP of 10 gives (12 − 10)/10 = 20% profit.
False weights. A dealer who claims to sell at cost price but gives w grams instead of 1000 gains (1000 − w) / w × 100%.
Example: 900 g for a kilogram gives 100/900 = 11.11%.
Change in SP. If an extra ₹x on the SP would change the profit from a% to b%, then (b − a)% of the CP equals x.
Worked example
An article sold at a 10% loss fetches ₹900. Its CP is 900 ÷ 0.9 = ₹1,000.
If it had been sold for ₹1,150 instead, the profit would have been 150/1000 = 15%.
Shortcuts and tricks
The standard way is to set the cost price at 100 and follow each step. These shortcuts compress the steps.
Multipliers instead of steps
Example: goods are marked 25% above cost and sold at a 20% discount.
Standard way: CP = 100, so MP = 125. A 20% discount removes 25, so SP = 100: no profit, no loss.
Shortcut: 1.25 × 0.8 = 1, so SP equals CP.
Same selling price, equal gain and loss
Example: two items sell for ₹990 each, one at a 10% gain and one at a 10% loss.
Standard way: the cost prices are 990 ÷ 1.1 = 900 and 990 ÷ 0.9 = 1,100, a total of 2,000. The total SP is 1,980, a loss of 20 on 2,000 = 1%.
Shortcut: equal % gain and loss on the same SP is always a loss of x²/100 = 1%.
False weights
Example: a trader sells at cost price but gives 800 g for every kilogram.
Standard way: if 1 kg costs ₹100, then 800 g costs the trader ₹80 and is sold for ₹100, a gain of 20 on 80 = 25%.
Shortcut: gain% = shortfall ÷ amount given × 100 = 200/800 × 100 = 25%.
Cost of x equals selling price of y
Example: the CP of 15 items equals the SP of 12 items.
Standard way: let each item cost ₹1. Then 12 items cost ₹12 and sell for ₹15, a profit of 3/12 = 25%.
Shortcut: profit% = (x − y)/y × 100 = 3/12 × 100 = 25%.
Exam tips
Never take profit % on the selling price unless the question says so.
Convert "gain of a quarter" to 25% and "loss of one-sixth" to 16⅔%.
For two items with different CPs, add the CPs and SPs before computing the overall percentage. Don't average the percentages.