Profit and loss percentages and reverse calculations

Learning outcome

Calculate profit or loss percentages on cost and work backwards from a selling price to the cost or original purchase price.

Percentage base and reasoning

In this lesson, a stated profit or loss percentage always uses cost price (CP) as its base. Include any stated overheads before calculating that percentage. Selling price (SP) is the actual receipt, and all costs and selling prices here are positive.

Profit percentage = (profit ÷ CP) × 100.

Loss percentage = (loss ÷ CP) × 100.

The denominator matters: these percentages compare the gain or shortfall with the money invested, not with the money received. Report a loss rate as a positive percentage labelled “loss.”

If the profit rate is p%, SP = CP × (1 + p/100).

If the loss rate is l%, SP = CP × (1 - l/100).

Why? Think of cost as 100 equal parts. Profit adds p parts to that base; loss removes l parts. The multiplier therefore includes the original cost as well as the change.

Reverse calculations undo multiplication by division. Thus CP = SP ÷ (1 + p/100) for profit, or CP = SP ÷ (1 - l/100) for loss. Do not subtract the stated profit percentage from SP: that would apply the percentage to the wrong base.

For a positive selling price under loss, use 0 < l < 100. At a 100% loss the multiplier is zero, so it cannot be used for reverse division. A percentage on zero cost is also undefined.

Worked examples

Example 1 — Find the rate. An item costs ₹960 and sells for ₹1,104. Profit = 1104 - 960 = ₹144. Profit percentage = (144 ÷ 960) × 100 = 15%. The base is ₹960.

Example 2 — Reverse a loss. An item sells for ₹1,539 at a 19% loss. The seller recovers 81% of cost. CP = 1539 ÷ 0.81 = ₹1,900. Check: loss = 1900 × 0.19 = ₹361, and 1900 - 361 = ₹1,539.

Example 3 — Separate purchase price from cost. A machine sells for ₹2,484 at an 8% profit after ₹180 of stated overheads. Effective CP = 2484 ÷ 1.08 = ₹2,300. Purchase price = 2300 - 180 = ₹2,120. Check: profit = 2300 × 0.08 = ₹184, giving receipts of ₹2,484.

Common mistakes

Do not divide profit by SP when asked for profit percentage on cost. Do not reverse a percentage change using subtraction. If overheads are included in the profit base, recover the total cost before subtracting overheads to find the purchase price.

Practice questions

1. An item costs ₹1,440 and sells for ₹1,260. Find the loss percentage.
2. The effective cost of an item is ₹1,750. Find the selling price needed for a 16% profit.
3. An item sells for ₹2,142 at a 26% profit. Find its cost.
4. A repaired item sells for ₹1,944 at a 10% loss on total cost, including ₹160 repairs. Find its purchase price.

Worked answers

1. Loss = 1440 - 1260 = ₹180. Loss percentage = (180 ÷ 1440) × 100 = 12.5%.
2. Profit = 1750 × 0.16 = ₹280. Required SP = 1750 + 280 = ₹2,030.
3. SP represents 126% of cost, so CP = 2142 ÷ 1.26 = ₹1,700. Check: 1700 × 1.26 = ₹2,142.
4. Effective CP = 1944 ÷ 0.90 = ₹2,160. Since this includes repairs, purchase price = 2160 - 160 = ₹2,000.