Profit and Loss Problems Explained — Formulas, Tricks and Interview Questions
Every business on the planet — from a street-side chai stall to Amazon — lives and breathes profit and loss. When a shopkeeper marks up a price, offers a discount, or sells goods in bulk, they're doing profit and loss calculations in their head. These aren't abstract math problems; they're the engine of commerce. That's exactly why aptitude tests and technical interviews use them: they reveal whether you can think clearly about numbers under pressure.
The Four Foundation Terms You Must Know Cold
Before you touch a single formula, you need to own these four terms. Confusing any two of them is the #1 reason candidates lose marks.
Cost Price (CP): The price at which you BUY something. This is your investment. If you bought a phone for ₹10,000, your CP is ₹10,000.
Selling Price (SP): The price at which you SELL something. If you sold that phone for ₹12,000, your SP is ₹12,000.
Profit: When SP > CP. You sold for MORE than you paid. Profit = SP − CP.
Loss: When CP > SP. You sold for LESS than you paid. Loss = CP − SP.
Here's the anchor image to burn into memory: think of CP as the floor. If your SP is above the floor, you're in profit territory. If it's below the floor, you're in a hole — a loss. Everything in profit and loss revolves around the relationship between these two numbers. Get this instinct right and every formula that follows will feel obvious, not memorised.
=== PROFIT AND LOSS — FOUNDATION FORMULAS === Given: Cost Price (CP) = ₹10,000 ← what you paid to buy the phone Selling Price (SP) = ₹12,000 ← what you got when you sold it Step 1 — Identify whether it's Profit or Loss: SP (12,000) > CP (10,000) → It's a PROFIT situation Step 2 — Calculate Profit: Profit = SP − CP = 12,000 − 10,000 = ₹2,000 Step 3 — Calculate Profit %: Profit % = (Profit / CP) × 100 = (2,000 / 10,000) × 100 = 20% ────────────────────────────────────────── Now flip it — Loss scenario: CP = ₹10,000 SP = ₹8,500 CP (10,000) > SP (8,500) → It's a LOSS situation Loss = CP − SP = 10,000 − 8,500 = ₹1,500 Loss % = (Loss / CP) × 100 = (1,500 / 10,000) × 100 = 15% ══════════════════════════════════════════ KEY RULE: Profit % and Loss % are ALWAYS calculated on the COST PRICE — not the Selling Price. Forgetting this = wrong answer. ══════════════════════════════════════════
Loss scenario → Loss = ₹1,500 | Loss % = 15%
The Master Formula Sheet — Every Variation in One Place
Once you know CP and SP, everything else is derived. Here's every formula you'll ever need, explained with the reasoning behind each one.
Finding SP when CP and Profit% are given: SP = CP × (1 + Profit%/100) Think of it as: SP = CP + (Profit% of CP). You're adding the gain on top.
Finding SP when CP and Loss% are given: SP = CP × (1 − Loss%/100) You're subtracting the loss from what you paid.
Finding CP when SP and Profit% are given: CP = SP × 100 / (100 + Profit%) You're reverse-engineering the original price.
Finding CP when SP and Loss% are given: CP = SP × 100 / (100 − Loss%)
These four formulas handle 90% of all profit and loss questions. The trick is recognising WHICH two pieces of information the question gives you, then picking the right formula. Always label what you KNOW and what you NEED before you start calculating.
=== WORKED PROBLEMS — ALL 4 FORMULA TYPES === ───────────────────────────────────────────── PROBLEM 1: Find SP "A shopkeeper buys a bag for ₹800 and wants to make a 25% profit. What should he sell it for?" Known: CP = ₹800, Profit% = 25 Need: SP = ? Formula: SP = CP × (1 + Profit%/100) SP = 800 × (1 + 25/100) SP = 800 × (1 + 0.25) SP = 800 × 1.25 SP = ₹1,000 ✓ Check: Profit = 1000 − 800 = 200 Profit% = (200/800) × 100 = 25% ✓ ───────────────────────────────────────────── PROBLEM 2: Find SP (Loss) "A TV bought for ₹15,000 is sold at a 10% loss. Find the selling price." Known: CP = ₹15,000, Loss% = 10 Need: SP = ? Formula: SP = CP × (1 − Loss%/100) SP = 15,000 × (1 − 10/100) SP = 15,000 × 0.90 SP = ₹13,500 ───────────────────────────────────────────── PROBLEM 3: Find CP (given SP and Profit%) "A watch is sold for ₹1,320 at a profit of 10%. What was the cost price?" Known: SP = ₹1,320, Profit% = 10 Need: CP = ? Formula: CP = SP × 100 / (100 + Profit%) CP = 1,320 × 100 / (100 + 10) CP = 132,000 / 110 CP = ₹1,200 ✓ Check: Profit = 1320 − 1200 = 120 Profit% = (120/1200) × 100 = 10% ✓ ───────────────────────────────────────────── PROBLEM 4: Find CP (given SP and Loss%) "A laptop is sold for ₹34,000 at a loss of 15%. Find the original cost price." Known: SP = ₹34,000, Loss% = 15 Need: CP = ? Formula: CP = SP × 100 / (100 − Loss%) CP = 34,000 × 100 / (100 − 15) CP = 3,400,000 / 85 CP = ₹40,000 ✓ Check: Loss = 40,000 − 34,000 = 6,000 Loss% = (6,000/40,000) × 100 = 15% ✓
Problem 2 → SP = ₹13,500 (10% loss on ₹15,000)
Problem 3 → CP = ₹1,200 (SP ₹1,320 at 10% profit)
Problem 4 → CP = ₹40,000 (SP ₹34,000 at 15% loss)
Marked Price, Discounts and the Real-World Shop Problem
Real interviews love adding one more layer: the concept of Marked Price (MP) and Discount. Here's where it gets interesting.
Marked Price (MP): The price tag on the shelf — what the seller ADVERTISES. Also called List Price.
Discount: A reduction given on the Marked Price. Discount is always calculated on the MP — not on CP.
SP = MP × (1 − Discount%/100)
The juicy interview question combines all three: a shopkeeper marks up the price above CP, then gives a discount — and you need to figure out the final profit or loss.
Here's the mental model: Imagine a shirt costs ₹500 (CP). The shopkeeper writes ₹800 on the tag (MP — a 60% markup). Then there's a sale: 25% discount on ₹800. The customer pays ₹600 (SP). The shopkeeper still made ₹100 profit on a ₹500 shirt. He gave a discount AND still profited. That's the trick these questions test.
=== MARKED PRICE + DISCOUNT — COMBINED PROBLEM === PROBLEM: "A shopkeeper marks a bicycle 40% above its cost price of ₹2,500 and then gives a discount of 20%. Find: (a) Marked Price (b) Selling Price (c) Profit or Loss (d) Profit/Loss %" ───────────────────────────────────────────── Given: Cost Price (CP) = ₹2,500 Markup = 40% above CP Discount = 20% on Marked Price Step 1 — Find Marked Price (MP): MP = CP × (1 + Markup%/100) = 2,500 × (1 + 40/100) = 2,500 × 1.40 = ₹3,500 Step 2 — Find Selling Price (SP): SP = MP × (1 − Discount%/100) = 3,500 × (1 − 20/100) = 3,500 × 0.80 = ₹2,800 Step 3 — Profit or Loss? SP (2,800) > CP (2,500) → PROFIT Profit = SP − CP = 2,800 − 2,500 = ₹300 Step 4 — Profit %: Profit % = (Profit / CP) × 100 = (300 / 2,500) × 100 = 12% ───────────────────────────────────────────── SHORTCUT FORMULA (for combined markup + discount): Net % change = M − D − (M × D)/100 where M = markup%, D = discount% = 40 − 20 − (40 × 20)/100 = 40 − 20 − 800/100 = 40 − 20 − 8 = +12% ← positive means PROFIT Same answer! Use the shortcut in exams when time is tight. ─────────────────────────────────────────────
Selling Price = ₹2,800
Profit = ₹300
Profit % = 12%
Shortcut result = +12% (Profit) ✓
Tricky Problem Types That Appear in Every Aptitude Test
Knowing the formulas is only half the battle. Aptitude tests throw a few classic curveballs that trip up even prepared candidates. Here are the three most common trap-style questions, fully worked through.
Type 1 — Two items, same SP, one profit one loss: Whenever a problem says 'two articles each sold at the same price, one at X% profit and one at X% loss' — there is ALWAYS a net loss. Always. This is a mathematical certainty.
Net Loss % = (Common %/10)² = X²/100
Type 2 — Dishonest shopkeeper using false weights: If a shopkeeper uses a 900g weight and claims it's 1000g, he's effectively gaining 100g per kg. Profit% = (True weight − False weight) / False weight × 100.
Type 3 — Successive profit/loss (buying then reselling): If you buy at CP, sell at a profit to Person B, and Person B resells at another profit, the final price chains multiplicatively — not additively.
=== CLASSIC TRAP PROBLEMS — FULLY WORKED === ───────────────────────────────────────────── TRAPPROBLEM 1: Two items, same SP, X% profit & X% loss "Two mobiles are sold at ₹9,900 each. One at 10% profit, another at 10% loss. Is there an overall profit or loss? By how much?" Many people say ZERO. It's NOT zero. There is always a LOSS. For the item sold at 10% PROFIT: CP₁ = SP × 100/(100 + Profit%) = 9,900 × 100/110 = ₹9,000 For the item sold at 10% LOSS: CP₂ = SP × 100/(100 − Loss%) = 9,900 × 100/90 = ₹11,000 Total CP = 9,000 + 11,000 = ₹20,000 Total SP = 9,900 + 9,900 = ₹19,800 Overall → Loss = 20,000 − 19,800 = ₹200 Shortcut: Loss % = X²/100 = 10²/100 = 1% Loss amount = 1% of Total CP = 1% of 20,000 = ₹200 ✓ ───────────────────────────────────────────── TRAP PROBLEM 2: False weight shopkeeper "A shopkeeper uses a weight of 800g instead of 1kg (1000g) but sells goods at the listed CP. What is his profit %?" He gives 800g but charges for 1000g. He gains 200g on every 800g he actually gives out. Profit % = (Gain / False weight) × 100 = (200 / 800) × 100 = 25% Intuition check: For every 800g he spends, he gets paid for 1000g — that's a 200g gain on an 800g investment = 25%. ───────────────────────────────────────────── TRAP PROBLEM 3: Successive transactions "A buys a guitar for ₹5,000 and sells to B at 20% profit. B sells to C at 10% profit. What does C pay?" A's SP = B's CP = 5,000 × 1.20 = ₹6,000 B's SP = C's CP = 6,000 × 1.10 = ₹6,600 C pays ₹6,600. WRONG approach (don't do this!): Adding 20% + 10% = 30% → 5,000 × 1.30 = 6,500 ✗ Percentages in successive steps multiply, not add.
Trap 2: Shopkeeper earns 25% profit using false weights
Trap 3: C pays ₹6,600 (not ₹6,500 — percentages multiply)
| Concept | Calculated On | Formula | When CP > SP or SP > CP? |
|---|---|---|---|
| Profit % | Cost Price (CP) | (Profit / CP) × 100 | SP > CP |
| Loss % | Cost Price (CP) | (Loss / CP) × 100 | CP > SP |
| Discount % | Marked Price (MP) | (Discount / MP) × 100 | Always reduces MP |
| Markup % | Cost Price (CP) | ((MP − CP) / CP) × 100 | MP > CP always |
| SP from CP + Profit% | Cost Price | CP × (100 + P%) / 100 | Gives SP above CP |
| CP from SP + Profit% | Selling Price given | SP × 100 / (100 + P%) | Reverse-engineers CP |
| Net % (Markup + Discount) | Cost Price | M − D − (M×D)/100 | Can be + or − or 0 |
🎯 Key Takeaways
- Profit% and Loss% are ALWAYS on the Cost Price — never on Selling Price or Marked Price. This single rule eliminates the most common aptitude exam mistake.
- When the same percentage applies to both profit and loss on identical selling prices, there is always a net loss of X²/100 — this is a mathematical certainty, not a coincidence.
- The shortcut Net % = M − D − (M×D)/100 handles any combined markup and discount in one line — learn it for timed exams where working through longhand costs you marks.
- Successive percentage changes multiply, not add. Convert each percentage to a multiplier first (P% profit → ×(1 + P/100)), then chain-multiply. This works for every successive transaction problem.
⚠ Common Mistakes to Avoid
- ✕Mistake 1: Calculating Profit% on Selling Price instead of Cost Price — Symptom: your answer doesn't match any option in the MCQ — Fix: Always anchor percentage calculations to CP. Ask yourself 'what did I originally PAY?' and use THAT number as the denominator.
- ✕Mistake 2: Adding successive percentages instead of multiplying them — Symptom: you get an answer like 30% for a 20% then 10% chain, but the correct answer is 32% — Fix: Convert each percentage to a multiplier (20% profit → ×1.20, 10% profit → ×1.10) and multiply the multipliers. Never add percentages across different bases.
- ✕Mistake 3: Assuming equal SP with equal profit% and loss% means break-even — Symptom: you write 'no profit no loss' and lose the mark — Fix: Memorise the law: same percentage profit and loss on the SAME selling price always results in a net loss of X²/100 percent. Prove it once with numbers, then never forget it.
Interview Questions on This Topic
- QA shopkeeper marks his goods 30% above cost price and offers a 10% discount. Does he make a profit or loss — and by exactly what percentage? Walk me through every step.
- QTwo articles are sold at ₹660 each — one at a profit of 10% and one at a loss of 10%. Find the total profit or loss on the entire transaction.
- QA dishonest dealer claims to sell goods at cost price but uses a weight of 960g instead of 1000g. What is his actual profit percentage — and how would your answer change if he also marked up his price by 5%?
Frequently Asked Questions
What is the difference between profit percentage and markup percentage?
Both are calculated on Cost Price — they're actually the same thing. Markup % and Profit % both equal (SP − CP) / CP × 100. The word 'markup' is just the retail industry's term for profit percentage. Be careful though: Discount % is calculated on Marked Price, not CP — that's the one that differs.
How do I find the cost price when only the selling price and profit percentage are given?
Use the formula: CP = SP × 100 / (100 + Profit%). For example, if SP is ₹1,100 and profit is 10%, then CP = 1100 × 100 / 110 = ₹1,000. The logic is that SP represents (100 + Profit%) parts, so you scale back down to 100 parts to find CP.
Why is there always a loss when two items are sold at the same price — one at X% profit and one at X% loss?
Because the cost prices are different. The item sold at a loss had a higher cost price (you paid more for it), while the item sold at a profit had a lower cost price. When you add the two cost prices together, the total always exceeds the total selling price. The net loss is exactly X²/100 percent — for 10%, that's 1%. You can only avoid this by selling at the same COST price, not the same selling price.
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