Profit & Loss

Profit & Loss

🟢 Lite

Key Formula/Rule

Profit = SP − CP (when SP > CP). Loss = CP − SP (when CP > SP). Always calculate based on Cost Price.

Quick Memory Trick

SP > CP = Profit (seller happy). CP > SP = Loss (seller sad). Percentage always based on CP unless question says otherwise.

1-Sentence Summary

Profit and loss are just the difference between what you sell something for and what it cost you, expressed as a percentage of the cost price.

Quick Example

Q: A shopkeeper buys a phone for Rs. 8,000 and sells it for Rs. 9,600. What’s the profit percentage? A: Profit = 9600 − 8000 = 1600. Profit % = (1600/8000) × 100 = 20%

Must Remember

• Profit % = (Profit/CP) × 100
• Loss % = (Loss/CP) × 100
• Discount is always on Marked Price (MP), not CP
• False weights: selling at cost price but using less quantity = profit

🟡 Standard

Concept Explanation

Profit and loss are the most fundamental business concepts you’ll encounter in quantitative aptitude. At its core, profit happens when you sell something for more than it cost you to acquire it. Loss is the opposite — you sold it for less than what you paid. The tricky part is that every percentage in profit and loss problems is calculated relative to the Cost Price, unless the problem explicitly says otherwise. This is where many students go wrong: they calculate profit percentage using selling price as the base, which is incorrect.

The marked price (also called list price or tag price) is what the seller initially announces. It’s the starting point before any discounts. A discount is a percentage reduction from the marked price, not from the cost price. So when you hear “20% off on a Rs. 1,000 jacket,” you’re saving Rs. 200, paying Rs. 800. But whether the seller makes a profit or loss depends entirely on what the jacket actually cost them — the discount just affects the selling price.

Overhead charges like transportation, packaging, or storage get added to the cost price. If you buy a refrigerator for Rs. 20,000 and spend Rs. 1,500 on delivery and installation, your actual cost price is Rs. 21,500. Now when you sell it, your profit or loss is calculated against Rs. 21,500, not Rs. 20,000. Always account for all costs before calculating profit percentage.

Key Formulas

Symbol	Meaning
Profit	SP − CP
Loss	CP − SP
Profit %	(Profit/CP) × 100
Loss %	(Loss/CP) × 100
SP	CP × (1 + Profit%/100)
SP	CP × (1 − Loss%/100)
Discount %	((MP − SP)/MP) × 100

Step-by-Step Example

Q: A shopkeeper buys goods at a 20% discount on the marked price of Rs. 2,500. He sells them at the marked price. What is his profit percentage?

Step 1: Find actual cost price (after 20% discount on MP): CP = 2500 × (80/100) = Rs. 2,000

Step 2: Selling price equals marked price: SP = Rs. 2,500

Step 3: Profit = SP − CP = 2500 − 2000 = Rs. 500

Step 4: Profit % = (500/2000) × 100 = 25%

Answer: 25% profit

Common Mistakes

• Calculating discount percentage on cost price instead of marked price → Discounts always apply to the marked price
• Mixing up profit percentage with markup percentage → Profit % is on CP; markup might be on MP
• Forgetting to include overhead costs in CP → Always add all acquisition costs before finding profit/loss percentage

Quick Test (2 Qs)

1. Q: A shopkeeper sells a shirt for Rs. 990 and gains 10%. What is the cost price? Options: Rs. 880, Rs. 890, Rs. 900, Rs. 910. Ans: Rs. 900 (Reason: CP = SP/(1 + 10/100) = 990/1.1 = 900)
2. Q: If the cost price of 20 articles equals the selling price of 25 articles, find the loss percentage. Options: 20%, 25%, 15%, 10%. Ans: 20% (Reason: Let CP of 1 = 1. CP of 20 = 20. SP of 25 = 20, so SP of 1 = 20/25 = 0.8. Loss% = (1 − 0.8)/1 × 100 = 20%)

🔴 Extended

Concept Deep Dive

Profit and loss problems appear deceptively simple but have surprising depth. The core insight is that every business transaction involves multiple prices: the cost price (what you pay to acquire or produce something), the marked price (what you initially ask for it), and the selling price (what the customer actually pays, after any discounts). Each of these can be the basis for different percentage calculations, and mixing them up is the most common error.

The marked price is essentially a psychological anchor. When a seller marks an item at Rs. 1,000 before offering a “30% discount,” they’re playing with perception — the customer feels like they’re getting a great deal while the seller may still be making a healthy profit. A 30% discount on Rs. 1,000 gives a selling price of Rs. 700. If the cost was Rs. 500, the profit is Rs. 200, which is 40% profit on cost, not 30%. Students often stop at the “30% discount” and forget to check the actual profit made.

False weights represent a clever (though dishonest) profit-maximization technique. If a shopkeeper uses a weight of 900 grams but calls it 1 kg, they’re selling 900g at the price of 1000g. If the cost price is Rs. 100 per kg, and they sell at Rs. 100 per kg (claiming to sell 1 kg), their revenue is Rs. 100 for only 900g of actual goods. Their effective cost per gram is Rs. 100/900 ≈ Rs. 111.11 per kg, giving a profit percentage of approximately 11.11% on the cost, even though they made no profit per the scale reading. This is why false weight problems are common in profit and loss.

The relationship between successive discounts and profit/loss is nuanced. Two successive discounts of 20% and 10% do NOT equal a single 30% discount. The combined discount = 1 − (0.8 × 0.9) = 1 − 0.72 = 28%, not 30%. This is because the second discount applies to the already-reduced price, not the original marked price. The same logic applies in reverse to successive price increases.

Advanced Formula Derivation

When discount and profit/loss interact: If an item costs Rs. C and is marked at M with a discount of d%, sold at a profit of p% on cost, we have SP = M(1 − d/100) = C(1 + p/100). From this, M = C(1 + p/100)/(1 − d/100). This formula shows the relationship between marked price, cost, discount, and profit simultaneously.

For false weight fraud: If a merchant uses a weight of W kg but claims it is 1 kg, and sells at cost price per the scale, the effective profit % = [(1 − W)/W] × 100. If W = 0.9 kg (900g scale), profit % = [(1 − 0.9)/0.9] × 100 = 11.11%. This independent formula is worth memorizing for speed.

Break-even analysis: Selling price at which there is neither profit nor loss = Cost Price. Below CP is loss; above CP is profit. The distance from CP determines how much buffer you have before making a loss.

GATE-Level Numerical Problems

Q1 (GATE 2020): A merchant buys goods at a 25% discount on the marked price. He marks them at a price that is 20% above the marked price. He then sells at a 15% discount on the marked price. Find his profit percentage.

• Working: Let MP = 100. Cost = 100 × 0.75 = 75. Marked price for sale = 100 × 1.20 = 120. Selling price = 120 × 0.85 = 102. Profit = 102 − 75 = 27. Profit % = (27/75) × 100 = 36%.
• Answer: 36% profit
• Common error: Using original MP as the selling markup base instead of calculating the new marked price correctly

Q2 (GATE 2019): A shopkeeper offers a 10% discount on all items and still makes a profit of 20%. If an item costs Rs. 450, what is its marked price?

• Working: Let MP = M. SP after 10% discount = M × 0.90. But SP must also equal CP × 1.20 = 450 × 1.20 = 540. So M × 0.90 = 540 → M = 540/0.90 = 600.
• Answer: Marked price = Rs. 600
• Common error: Trying to add discount and profit percentages directly — they apply to different bases

Q3: A shopkeeper sells 4 goats bought at the same cost price. On 3 goats, he gains 20%, and on the 4th, he breaks even. On the whole transaction, he makes a profit of 12.5%. Find the loss percentage on the 4th goat.

• Working: Let CP of each goat = C. Total CP = 4C. Overall profit = 12.5%, so total SP = 4C × 1.125 = 4.5C. SP of 3 goats at 20% profit = 3C × 1.20 = 3.6C. So SP of 4th goat = Total SP − SP of 3 goats = 4.5C − 3.6C = 0.9C. Loss on 4th goat = C − 0.9C = 0.1C. Loss % = (0.1C/C) × 100 = 10%.
• Answer: 10% loss on the 4th goat
• Common error: Assuming equal selling price for all goats, not accounting for different profit percentages

Multiple Approaches

Method A: Assume base value = 100 — this eliminates fractions and makes percentage calculations cleaner Method B: Algebraic method — use variables for unknown prices and set up equations Method C: Back-calculation — start from known selling price and work backwards to find cost When to use: Method A for problems with discounts and markups. Method B for complex multi-step transactions. Method C when you know SP and want to find CP.

Tricky Cases

• When cost price and selling price are expressed as ratios: Profit % = [(ratio − 1)] × 100 when CP:SP is given
• If a merchant sells at a loss equal to the discount percentage he offered: This creates interesting equations linking discount % and loss %
• Two successive price changes (increase then decrease) don’t cancel out — the order matters for final price
• When an article is sold at a loss and the weight is found to be less than claimed, both factors contribute to effective loss

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