Unit 2 · Topic 03 · Comparing Quantities

Profit and Loss

Hook

Kabir bought a bicycle for ₹4,000 and sold it for ₹4,600. He said his profit percent was 15%.

Zara checked his division and found he'd divided by the selling price, not the cost price.

The real answer was different, and it mattered for the bigger sale he was planning next.

Watch the Lessons

The Story

Kabir had bought an old bicycle for ₹4,000 (his cost price, CP) and sold it for ₹4,600 (his selling price, SP), making a profit of ₹600. To find his profit percent, he divided 600 by 4,600, the selling price, getting about 13%. Zara pointed out the mistake immediately: profit percent must always be calculated on the COST price, since that's the amount actually invested, not the amount received.

She redid it: profit% = (profit/CP) × 100 = (600/4000) × 100 = 15%. "That's the real number," she said. "Dividing by the selling price always understates the true profit percentage, because the selling price is bigger."

Kabir tested this on a loss scenario too: a shopkeeper buys a chair for ₹1,200 and sells it for ₹1,000, a loss of ₹200. Loss% = (loss/CP) × 100 = (200/1200) × 100 ≈ 16.67%. "Same rule," he said, "loss is always measured against the cost price too."

Zara then showed him the reverse problem: if you know the CP and the profit percent, you can find the SP directly. A trader buys goods for ₹800 and wants a 20% profit. Profit = 20% of 800 = 160, so SP = 800 + 160 = 960. She wrote a shortcut formula: SP = CP × (1 + profit%/100). For a loss instead: SP = CP × (1 − loss%/100).

Kabir used the shortcut on a phone case bought for ₹250 with a target loss of 8% (a clearance sale): SP = 250 × (1 − 0.08) = 250 × 0.92 = 230. He checked it the long way and got the same ₹230.

"So every profit-and-loss question boils down to two things," Kabir said. "Whether you're finding profit%/loss% from CP and SP, or finding SP from CP and a target percent — and either way, the percent is ALWAYS relative to the cost price, never the selling price."

600/4000 gives the true 15% profit; 600/4600 understates it.

So What Just Happened?

Profit = SP − CP (when SP > CP). Loss = CP − SP (when CP > SP). Profit% = (Profit/CP) × 100. Loss% = (Loss/CP) × 100 — always divide by the cost price.

Dividing profit or loss by the selling price instead of the cost price is a common error and gives a systematically wrong percentage.

To find SP given CP and a profit percent: SP = CP × (1 + profit%/100). To find SP given CP and a loss percent: SP = CP × (1 − loss%/100).

To find CP given SP and a profit or loss percent, rearrange the same formula: CP = SP / (1 + profit%/100), or CP = SP / (1 − loss%/100).

Both percentages are always measured against the cost price.

Remember This

Profit% and Loss% are ALWAYS calculated on the cost price (CP), never the selling price (SP).
Profit = SP − CP, and Profit% = (Profit/CP) × 100.
Loss = CP − SP, and Loss% = (Loss/CP) × 100.
SP = CP × (1 + profit%/100) for a profit, or CP × (1 − loss%/100) for a loss.
To go backward from SP to CP, divide by (1 + profit%/100) or (1 − loss%/100) — do not just subtract or add the percent to SP directly.

SP = CP times (1 plus or minus the percent as a decimal).

Try It Yourself

A shopkeeper buys a fan for ₹1,500 and sells it for ₹1,800. Find the profit percent, making sure you divide by the cost price.

A trader wants a 12% profit on goods bought for ₹950. Compute the selling price needed using the shortcut formula.

Word Bank

Cost price (CP): The price at which an item was bought — the base for all profit and loss percent calculations.
Selling price (SP): The price at which an item is sold.
Profit: The positive difference SP − CP, when SP is greater than CP.
Loss: The positive difference CP − SP, when CP is greater than SP.
Profit percent: Profit expressed as a percentage of the cost price.

Questions

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