Profit and Loss Formulas: CP, SP and Profit Percentage

Every profit and loss question is about two prices: the cost price (CP), what the seller paid, and the selling price (SP), what the buyer paid. If SP is higher there is a profit; if it is lower there is a loss. Percentages are always worked out on CP unless the question says otherwise.

To find	Formula
Profit or loss	SP − CP, or CP − SP
Profit % or loss %	(Profit or loss ÷ CP) × 100
SP from CP	CP × (100 + profit%) ÷ 100, or CP × (100 − loss%) ÷ 100
CP from SP	SP × 100 ÷ (100 + profit%), or SP × 100 ÷ (100 − loss%)

Worked example 1: profit percentage

An item bought for ₹640 is sold for ₹800. The profit is ₹160, and 160 ÷ 640 × 100 = 25%.

The fastest way to use the formulas is to turn each percentage into a multiplier. A 10% profit means SP = 1.10 × CP; a 15% loss means SP = 0.85 × CP. Going from CP to SP you multiply; going back from SP to CP you divide by the same multiplier.

Worked example 2: CP from SP after a profit

A bag is sold for ₹1,188 at a 10% profit. CP = 1,188 ÷ 1.10 = ₹1,080.

Worked example 3: CP from SP after a loss

A lamp sold for ₹1,530 makes a 15% loss. CP = 1,530 ÷ 0.85 = ₹1,800.

Worked example 4: changing the target percentage

Selling a toy for ₹450 causes a 10% loss. What price gives a 15% profit? First recover CP: 450 ÷ 0.90 = ₹500. Then SP = 500 × 1.15 = ₹575. Never add 25% to ₹450, because both percentages are measured on CP, not on the old SP.

Some questions compare counts instead of prices: "the CP of 12 articles equals the SP of 10 articles". Take the price of one article as the unit. The seller recovers the cost of 12 articles by selling only 10, so the profit is the cost of 2 articles on a cost of 10.

If the CP of x articles = the SP of y articles, profit % = (x − y) ÷ y × 100. A negative result is a loss.

Worked example 5: 12 and 10

Profit % = (12 − 10) ÷ 10 × 100 = 20%.

When the question gives rates such as "buys 6 for ₹5 and sells 5 for ₹6", find the price of one article on each side. Using fractions keeps the arithmetic exact.

Worked example 6: buying and selling rates

CP per article = ₹5/6 and SP per article = ₹6/5. Profit % = (6/5 − 5/6) ÷ (5/6) × 100 = (36 − 25) ÷ 25 × 100 = 44%.

Same selling price, equal gain and loss. Two items sell for the same price, one at x% profit and one at x% loss. The CP of the losing item is higher, so the loss outweighs the gain. The net result is always a loss of x² ÷ 100 percent.

Worked example 7: 20% each way

Two cycles sell for the same price, one at 20% profit and one at 20% loss. The net loss is 20 × 20 ÷ 100 = 4%, whatever the selling price.

Equal profit and loss at two prices. If selling at one price gives the same rupee profit as the rupee loss from selling at another price, CP is the average of the two prices.

Worked example 8: halfway

The profit at ₹920 equals the loss at ₹680, so CP = (920 + 680) ÷ 2 = ₹800. Check: 920 − 800 = 120 and 800 − 680 = 120.

✘ Calculating the percentage on SP instead of CP.
✘ Finding CP by subtracting the profit percentage from SP. Divide by the multiplier instead.
✘ Applying the x² ÷ 100 loss when the cost prices, not the selling prices, are equal.
✘ In article questions, dividing by x instead of y.

When a seller also gives discounts, move on to marked price and discount problems and the successive discount formula. Cheating with weights is covered in dishonest dealer problems, and the formula list is in the Quant cheat code.

Q1. An article bought for ₹1,250 is sold for ₹1,500. What is the profit percentage? easy

(a) 16.67% (b) 25% (c) 20% (d) 18%

✅ Answer: (c) 20%

💡 Explanation: Profit = 1,500 − 1,250 = ₹250. Profit percentage is always on cost price: 250 ÷ 1,250 × 100 = 20%.

Q2. A shopkeeper sells a fan for ₹2,240 and gains 12%. What did the fan cost him? easy

(a) ₹2,000 (b) ₹1,971 (c) ₹2,100 (d) ₹1,960

✅ Answer: (a) ₹2,000

💡 Explanation: A 12% profit means SP = 1.12 × CP. So CP = 2,240 ÷ 1.12 = ₹2,000. Taking 12% off the SP instead gives the wrong ₹1,971.

Q3. The cost price of 15 pens equals the selling price of 12 pens. What is the profit percentage? medium

(a) 20% (b) 3% (c) 15% (d) 25%

✅ Answer: (d) 25%

💡 Explanation: The seller gains the cost of 15 − 12 = 3 pens on every 12 pens sold. Profit percentage = 3 ÷ 12 × 100 = 25%.

Q4. The profit made by selling a watch for ₹1,560 equals the loss made by selling it for ₹1,240. What is its cost price? medium

(a) ₹1,320 (b) ₹1,400 (c) ₹1,450 (d) ₹1,380

✅ Answer: (b) ₹1,400

💡 Explanation: If the profit and the loss are equal, the cost price lies exactly halfway between the two selling prices: (1,560 + 1,240) ÷ 2 = ₹1,400.

Q5. Two phones are sold for ₹9,900 each, one at a 10% gain and the other at a 10% loss. What is the overall result? medium

(a) No profit, no loss (b) 1% gain (c) 1% loss (d) 2% loss

✅ Answer: (c) 1% loss

💡 Explanation: When two items have the same selling price and equal percentage gain and loss, there is always a loss of x² ÷ 100 percent. Here 10 × 10 ÷ 100 = 1% loss.

Q6. By selling a chair for ₹1,140 a trader loses 5%. At what price should he sell it to gain 5%? hard

(a) ₹1,197 (b) ₹1,200 (c) ₹1,254 (d) ₹1,260

✅ Answer: (d) ₹1,260

💡 Explanation: CP = 1,140 ÷ 0.95 = ₹1,200. For a 5% gain, SP = 1,200 × 1.05 = ₹1,260. Adding 10% to ₹1,140 is wrong because both percentages are on CP.

Is profit percentage calculated on cost price or selling price?

On cost price, unless the question explicitly says the profit is on selling price. That is why you divide the selling price by (100 + profit%) to get back to the cost, and never simply subtract the percentage from the selling price.

Is there an overall loss whenever gain and loss percentages are equal?

Only when the two selling prices are equal. If instead the two cost prices are equal, a 20% gain on one and a 20% loss on the other cancel out, giving no profit and no loss overall.

All problems and solutions on this page are original. No outside facts are relied on; every numeric answer is recomputed by software before publishing.

How this page was prepared. Drafted with AI assistance (Claude (Anthropic AI model)) from the sources listed above. 14 calculations on this page were recomputed by software and matched. Automated checks also covered structure, internal links and duplicate text. Software checks cannot confirm facts. Reviewed by praveenkumar on 2026-10-20 against the sources listed above.

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