Profit, loss, and discount problems are a staple of every aptitude exam. The concepts are simple, but the questions can get tricky — especially dishonest dealer problems and successive discount questions. Let’s break it all down.
The Core Terms
- Cost Price (CP): What we paid to buy/produce an item
- Selling Price (SP): What we sold it for
- Marked Price (MP): The price tag on the item (also called “list price”)
- Profit: SP - CP (when SP > CP)
- Loss: CP - SP (when CP > SP)
In simple language, think of it like this: we buy at CP, we label it at MP, and we sell at SP (after possibly giving a discount on MP).
The CP → MP → SP Flow
Successive Discounts
Two discounts of 20% and 10% are NOT the same as a single 30% discount.
Method: Apply discounts one after another.
Example: MP = 1000, discounts of 20% and 10%.
- After 20% discount: 1000 × 0.8 = 800
- After 10% discount on 800: 800 × 0.9 = 720
- SP = 720
A single equivalent discount: 1000 - 720 = 280. So 28% discount, not 30%.
Formula (from our percentages chapter): Equivalent discount for a% and b% = a + b - ab/100 = 20 + 10 - (20×10)/100 = 30 - 2 = 28%
Buy X Get Y Free
This is just a discount in disguise.
Example: “Buy 4 get 1 free” means we pay for 4 but get 5 items.
- Effective discount = 1/5 = 20%
- We can also say: CP of 5 items = SP of 4 items
General rule: “Buy X get Y free” → Discount = Y/(X+Y) × 100%
Dishonest Dealer Problems
These are the trickiest and most fun problems. A dealer cheats on weight while claiming to sell at CP or with a small markup.
Type 1: Uses false weight, sells at CP
If a dealer claims to sell at cost price but uses a weight of w grams instead of 1000 grams:
Profit % = [(True weight - False weight) / False weight] × 100
Example: A shopkeeper claims to sell rice at cost price but uses an 800g weight instead of 1kg.
Profit % = (1000 - 800)/800 × 100 = 200/800 × 100 = 25%
In simple language, he’s paying for 800g and selling (claiming) it as 1000g. So for every 800g he buys, he makes a profit on 200g.
Type 2: False weight + markup or discount
Use the multiplier approach: Total SP factor = (True weight/False weight) × (100 ± markup or discount)/100
Example: A dealer uses 900g instead of 1kg and marks up by 10%. What’s his real profit%?
- Weight factor: 1000/900 = 10/9
- Price factor: 110/100 = 11/10
- Total: (10/9) × (11/10) = 11/9
- Profit = 11/9 - 1 = 2/9 = 22.22%
Worked Examples
Example 1: A shopkeeper bought 100 pens at Rs 10 each and sold them at Rs 12 each. Find profit%.
CP = 10, SP = 12. Profit = 2. Profit % = 2/10 × 100 = 20%
Example 2: An article is marked at Rs 800. After giving two successive discounts of 10% and 15%, what is the selling price?
SP = 800 × (90/100) × (85/100) = 800 × 0.9 × 0.85 = 800 × 0.765 = Rs 612
Example 3: A man buys an article for Rs 450 and sells it at a profit of 20%. What is the selling price?
SP = 450 × (120/100) = 450 × 1.2 = Rs 540
Or using fraction: 20% profit = 1/5 on CP. So profit = 450/5 = 90. SP = 450 + 90 = 540.
Example 4: By selling a watch for Rs 1140, a man loses 5%. At what price should he sell it to gain 5%?
At 5% loss: SP = 95% of CP → 1140 = 0.95 × CP → CP = 1140/0.95 = 1200. For 5% gain: SP = 105% of CP = 1.05 × 1200 = Rs 1260
Shortcut: New SP = Old SP × (100 + desired gain%) / (100 - loss%) = 1140 × 105/95 = Rs 1260
Example 5: A trader marks his goods 40% above CP and gives a discount of 25%. Find his profit or loss%.
Let CP = 100. MP = 100 × 1.4 = 140. SP = 140 × 0.75 = 105. Profit = 105 - 100 = 5. Profit % = 5%
Shortcut formula: When markup is m% and discount is d%: Profit/Loss % = m - d - (m×d)/100 = 40 - 25 - (40×25)/100 = 40 - 25 - 10 = 5% profit
The “Same SP” Trick
If two articles are sold at the same selling price, one at a% profit and the other at a% loss, there is ALWAYS a net loss.
Net Loss % = a²/100 %
Example: Two TVs are sold at Rs 10,000 each. One at 20% profit and the other at 20% loss. Net profit/loss?
Net loss % = (20)²/100 = 400/100 = 4% loss
Let’s verify:
- TV1 CP = 10000/1.2 = 8333.33
- TV2 CP = 10000/0.8 = 12500
- Total CP = 20833.33, Total SP = 20000
- Loss = 833.33, Loss% = 833.33/20833.33 × 100 = 4% ✓
Common Exam Patterns
- “Marked at X% above CP, discount of Y%” → Profit = X - Y - XY/100
- “Sold at same SP, one at profit, one at loss” → Always a net loss = a²/100%
- “Buy X get Y free” → Discount = Y/(X+Y) × 100
- “Dishonest dealer, false weight” → Profit = (True - False)/False × 100
- “Find SP to get desired profit given a loss/previous SP” → Find CP first, then calculate new SP
- “CP of X items = SP of Y items” → If X > Y: profit of (X-Y)/Y × 100%. If X < Y: loss.
Practice Problems
Q1: A shopkeeper marks an article at Rs 1200 and gives successive discounts of 20% and 10%. If he still makes a 20% profit, what was the cost price?
Q2: The cost price of 25 articles equals the selling price of 20 articles. Find the profit percentage.
Q3: A dishonest dealer sells goods at cost price but uses a weight of 850 grams instead of 1 kilogram. He also mixes 15% impurities. What is his actual profit percentage?
Answers:
A1: SP = 1200 × 0.8 × 0.9 = Rs 864. If this is 120% of CP: CP = 864/1.2 = Rs 720
A2: CP of 25 = SP of 20. Let CP of each = 1. Total CP of 25 = 25. SP of 20 = 25, so SP of each = 25/20 = 1.25. Profit per item = 0.25. Profit % = 0.25/1 × 100 = 25%
A3: For every 850g he buys, he adds 15% impurities: actual goods he sells = 850 × 1.15 = 977.5g. But he charges for 1000g at cost price. Profit% = (1000 - 850)/850 × 100 is the weight cheat. Adding impurities: he gets 977.5g of mixture but paid for only 850g. He sells 977.5g as 977.5g… wait, let’s think differently. He buys 850g for the cost of 850g. Adds impurities to make it 977.5g. Sells it claiming 1000g at CP per gram. So he charges for 1000g but his cost was for 850g. Profit = (1000-850)/850 × 100 ≈ 17.65%. But actually, the full calculation: he effectively sells 1000g worth at cost, having spent on 850g. So profit% = (1000-850)/850 × 100 = 150/850 × 100 = 17.65%