Fractions in Disguise · Lesson 7 of 12
Profit and Loss
“Calculate profit and loss on cost, recover prices and combine transactions accurately.”
• Distinguish cost price, marked price and selling price. • Calculate profit or loss and express it as a percentage of cost price. • Find a selling price from a given profit or loss percentage. • Recover cost price and combine transactions using their actual amounts.
A sweater passes from a manufacturer to a wholesaler, then to a shopkeeper and finally to a customer. At each stage, someone pays money to obtain it and may later receive money by selling it. A price can therefore play different roles for different people. The wholesaler’s selling price is the shopkeeper’s cost price. Before calculating profit, you need to decide whose transaction you are describing and what that person paid and received.
Suppose a shopkeeper pays ₹300 for a sweater, displays a price of ₹480 and finally sells it for ₹430 after bargaining. Three amounts are involved, but profit depends on the actual ₹430 received and the ₹300 paid. The displayed ₹480 helps describe the discount; it does not replace the actual selling price.
The mathematics is percentage change applied to buying and selling. For the standard profit and loss calculations here, the buying cost is the reference quantity.
The amount paid to buy an item, viewed from the buyer’s position in the transaction. It is commonly abbreviated CP.
The displayed or quoted price before any discount. It is commonly abbreviated MP.
The actual amount received for selling an item, before considering any separately identified tax collected for the government. It is commonly abbreviated SP.
If the selling price exceeds the cost price, the difference is profit. If it is smaller, the difference is loss. If the two prices are equal, there is neither profit nor loss in this buying-and-selling comparison.
For the sweater, the profit is ₹430 − ₹300 = ₹130. To describe how large that profit is relative to the money used to buy the sweater, compare ₹130 with ₹300. This is why the denominator is cost price when the question asks for profit percentage on cost.
The excess of selling price over cost price when SP > CP. In these transaction problems, profit means gross profit unless further expenses are explicitly included.
The excess of cost price over selling price when CP > SP.
Problem
A sweater costs ₹300 and sells for ₹430. Find the profit and percentage profit on cost.
- 1.Profit = SP − CP = ₹430 − ₹300 = ₹130.
- 2.Percentage profit on cost = (130/300) × 100 = 43⅓%.
- 3.This is approximately 43.33% when rounded to two decimal places.
- 4.Check: one third of ₹300 is ₹100 and another 10% is ₹30, giving a total profit of ₹130.
- 5.The marked price is not needed for this calculation because the actual selling price is already known.
Problem
Rice costs ₹35 per kg. Ten kilograms sell for ₹300. Find the loss percentage.
- 1.Use equal quantities on both sides: the cost of 10 kg is 10 × ₹35 = ₹350.
- 2.The loss is ₹350 − ₹300 = ₹50.
- 3.Loss percentage = (50/350) × 100 = 100/7% = 14 2/7%, approximately 14.29%.
- 4.Alternatively, selling price per kg is ₹300 ÷ 10 = ₹30. The loss per kg is ₹5, so 5/35 × 100 gives the same percentage.
- 5.Comparing ₹35 for one kilogram directly with ₹300 for ten kilograms would mix quantities and give a meaningless result.
Once a desired profit percentage is known, think in terms of how much of cost must be recovered. A 20% profit means the seller receives the original 100% of cost plus another 20%. Thus the selling price is 120% of cost.
At an 18% loss, the seller recovers only 82% of cost. The multiplier is 0.82. Calculating 18% of cost finds the loss amount, while calculating 82% finds the selling price. Label the quantity you are seeking before you multiply.
Problem
A notebook costs ₹36. What selling price gives 20% profit on cost?
- 1.Profit amount = 20% of ₹36 = 0.20 × 36 = ₹7.20.
- 2.Add profit to cost: SP = ₹36 + ₹7.20 = ₹43.20.
- 3.Using the multiplier directly gives 1.20 × ₹36 = ₹43.20.
- 4.Check: ₹7.20 ÷ ₹36 = 0.20, so the profit is exactly 20% of cost.
Problem
A vase costs ₹2650 and is sold at an 18% loss. Find its selling price.
- 1.An 18% loss leaves 100% − 18% = 82% of the cost to be recovered.
- 2.SP = 0.82 × ₹2650 = ₹2173.
- 3.The loss amount is 0.18 × ₹2650 = ₹477.
- 4.Check: ₹2173 + ₹477 = ₹2650. The selling price is below cost, as a loss requires.
Recovering cost price is a reverse-percentage problem. If a crayon box sells for ₹50 at 25% profit on cost, the ₹50 represents 125% of cost. Divide by 1.25 to recover the original 100%.
Do not subtract 25% of the selling price. The stated 25% was calculated on the unknown cost price, not on the selling price. The two bases differ precisely because a profit has been added.
Problem
A crayon box sells for ₹50 at 25% profit on cost. Find the cost price.
- 1.Write the relation: SP = 1.25 × CP.
- 2.Substitute the selling price: 50 = 1.25 × CP.
- 3.Divide: CP = 50 ÷ 1.25 = ₹40.
- 4.Check: the profit is ₹10, and ₹10 is 25% of ₹40. Subtracting 25% of ₹50 would have given ₹37.50, the wrong base.
Problem
Strawberries sell for ₹80 per kg at a loss of 12% on cost. Find the cost per kg.
- 1.A 12% loss means the selling price is 88% of the cost.
- 2.0.88 × CP = ₹80, so CP = 80 ÷ 0.88 = 1000/11 rupees per kg.
- 3.The exact cost is ₹90.909090… per kg, approximately ₹90.91 per kg.
- 4.Using the exact value, the loss is 120/11 rupees per kg. Its ratio to 1000/11 is 0.12, confirming 12%.
For several transactions, add the actual costs and actual selling amounts first. Then compare the totals. You cannot usually average the individual profit and loss percentages, because they may be based on different costs.
Even equal selling prices do not imply equal cost prices. An item sold at a gain and another sold at a loss for the same amount were bought at different prices. Recover both costs before deciding the overall result.
Problem
Two animals sell for ₹80,000 each. One sale gives a 5% profit on cost and the other a 10% loss. Find the overall result.
- 1.For the profitable sale, CP = 80,000 ÷ 1.05 = 1,600,000/21 rupees, approximately ₹76,190.48.
- 2.For the loss-making sale, CP = 80,000 ÷ 0.90 = 800,000/9 rupees, approximately ₹88,888.89.
- 3.Keep exact values for the total: total CP = 10,400,000/63 rupees, approximately ₹165,079.37.
- 4.Total SP = ₹160,000. Total loss = total CP − total SP = 320,000/63 rupees, approximately ₹5079.37.
- 5.Overall loss percentage = (320,000/10,400,000) × 100 = 40/13%, approximately 3.08%.
- 6.The two cost bases were unequal, so averaging +5% and −10% would not give the correct overall percentage.
Choose a product and invent a manufacturer’s selling price, a wholesaler’s selling price and a retailer’s selling price. For each seller after the manufacturer, identify the previous selling price as the new cost price. Calculate each profit on that seller’s own cost. A price is not permanently a cost or a selling price; its role depends on whose transaction you are studying.
Quiz
A seller buys for ₹75 and sells for ₹90. What is the profit percentage on cost?
Which amount is the base for standard profit percentage on cost?
A 15% loss means the selling price equals what percentage of cost?
A product sells for ₹240 at 20% profit on cost. What is its cost?
Why can averaging individual profit percentages give a wrong overall result?
Practice Problems
- A geometry box costs ₹75 and sells for ₹110. Find profit and profit percentage on cost. Solution: Profit = ₹35. Profit percentage = 35/75 × 100 = 46⅔%, approximately 46.67%.
- A chair costs ₹475 to make. At what price should it be sold for 50% profit on that cost? Solution: Profit = 0.5 × 475 = ₹237.50. Selling price = ₹475 + ₹237.50 = ₹712.50.
- A product costs ₹1250 and is sold at a 16% loss. Find selling price. Solution: The selling price is 84% of cost: 0.84 × 1250 = ₹1050. Loss is ₹200, and 200/1250 = 0.16.
- An article sells for ₹720 at a 10% loss. Find its cost and the selling price needed for 15% profit on that cost. Solution: ₹720 is 90% of cost, so CP = 720 ÷ 0.9 = ₹800. A 15% profit requires SP = 1.15 × 800 = ₹920.
- One item costs ₹400 and sells for ₹500. Another costs ₹600 and sells for ₹540. Find the overall profit or loss percentage. Solution: Total cost = ₹1000; total selling amount = ₹1040. Overall profit = ₹40, so profit percentage = 40/1000 × 100 = 4%. The separate +25% and −10% percentages should not be simply averaged.
Key Takeaways
• Cost, marked and selling prices describe different roles in a transaction. • Profit is selling price minus cost; loss is cost minus selling price. • Standard profit and loss percentages here use cost as their base. • Use 1 + p/100 for a selling price at p% profit and 1 − l/100 at l% loss. • Divide by the appropriate multiplier to recover cost. • For an overall result, combine actual money amounts before calculating a percentage.