Percentages, Profit and Loss
Percentages are everywhere in GATE’s General Aptitude section. They’re fast, scoring, and the formulas are straightforward — once you know how to apply them without getting tripped up by successive percentage changes or discount chains.
🟢 Lite — Quick Review (1h–1d)
Core formulas to memorize:
- Percentage increase = (Actual Increase / Original) × 100
- SP = CP × (1 + profit%/100) or SP = CP × (1 − loss%/100)
- Successive percentage change: Net change ≈ sum of % ± (product of % / 100)
- Simple Interest (SI) = P × R × T / 100; Amount = P + SI
- Compound Interest (CI): A = P × (1 + R/100)^T; CI = A − P
- Discount: SP = MP × (1 − D/100)
⚡ GATE exam tip: When two successive percentage changes happen (e.g., 20% up then 10% down on the new value), the net change is NOT simply 10%. Use the multiplicative approach: 1.20 × 0.90 = 1.08 → net 8% increase.
⚡ Quick trick: To find 15% of a number, find 10% then add 5% (which is half of 10%). These mental shortcuts save time.
⚡ Common trap: Don’t confuse ” Markup on cost” with “Markup on selling price.” Always identify the base.
🟡 Standard — Regular Study (2d–2mo)
Percentage Fundamentals
Percent means “per hundred.” Converting between fractions and percentages:
- 1/8 = 12.5% = 0.125
- 1/6 ≈ 16.67%
- 3/8 = 37.5%
- 2/3 ≈ 66.67%
Percentage vs. Percentage Points: If something goes from 10% to 12%, that’s a 20% increase (relative), but a 2 percentage point increase (absolute). GATE loves testing this distinction.
Base identification is critical:
- “Profit of 20% on cost price” → SP = 1.20 × CP
- “Profit of 20% on selling price” → SP − CP = 0.20 × SP → CP = 0.80 × SP → SP = CP / 0.80 = 1.25 × CP
GATE Example (2019, 1 mark): A shopkeeper offers a 15% discount on the marked price of a product, and still makes a 10% profit. If the cost price is ₹500, find the marked price.
Solution: SP = 500 × 1.10 = ₹550. Also SP = MP × (1 − 15/100) = MP × 0.85. So MP = 550 / 0.85 = ₹647.06 (approx ₹647).
Successive Percentage Changes
When a quantity changes by x% then y% (applied to the new value):
- Net multiplier = (1 + x/100) × (1 + y/100)
- For two increases: net % = x + y + xy/100
- For increase then decrease by same %: net is always a loss of (x²/100)%
Example: Price increases by 20%, then decreases by 20%. Net change?
Net = 1.20 × 0.80 = 0.96 → 4% loss (not 0%!)
Profit and Loss Deep Dive
Key formulas:
| Concept | Formula |
|---|---|
| Profit % | (SP − CP)/CP × 100 |
| Loss % | (CP − SP)/CP × 100 |
| SP (given CP & profit%) | CP × (100 + profit%)/100 |
| CP (given SP & profit%) | SP × 100/(100 + profit%) |
Break-even: Occurs when SP = CP (0% profit, 0% loss).
False weights: If a shopkeeper uses a false weight (claims to sell 1 kg but actually sells only x kg for the price of 1 kg), their gain % = ((1/x) − 1) × 100%.
GATE Example: A merchant uses a scale that weighs 900g for 1kg. What is his gain %?
Solution: Sells 900g as 1000g. Gain = 100g on cost of 900g. Gain % = (100/900) × 100 = 11.11%.
Simple vs. Compound Interest
| Simple Interest | Compound Interest | |
|---|---|---|
| Formula | SI = P × R × T / 100 | A = P(1 + R/100)^T |
| Interest each year | Same | Decreasing (effective rate changes) |
| Grows linearly | Yes | No (exponential) |
⚡ GATE trick — Equal SI and CI question: If SI = CI for 2 years at rate R%, then R = 200/2 = 100/R? Wait, the formula: SI for 2 years = 2PR/100. CI for 2 years = P[(1+R/100)² − 1] = P[R/100 + R²/10000]. Setting equal: 2PR/100 = PR/100 + PR²/10000 → PR/100 = PR²/10000 → R = 100/2 = 50. So R = 50%.
Discounts and Marked Price
- Single discount equivalent to successive discounts d₁% and d₂%: = d₁ + d₂ − (d₁×d₂)/100
- Three successive discounts: extend the formula iteratively
GATE Example: Two successive discounts of 20% and 10% are equivalent to a single discount of:
20 + 10 − (20×10)/100 = 30 − 2 = 28%.
🔴 Extended — Deep Study (3mo+)
Compound Interest with Different Compounding Periods
When interest is compounded quarterly, semi-annually, or monthly, adjust the rate and time:
- Quarterly: Rate per quarter = R/4, periods = 4T
- Monthly: Rate per month = R/12, periods = 12T
- Effective Annual Rate (EAR): EAR = (1 + R/100)^n − 1 (where n = compounding frequency per year)
Depreciation
Straight-line depreciation: Value after T years = P × (1 − dT/100), where d = annual depreciation rate.
Declining balance depreciation: Value after T years = P × (1 − d/100)^T.
Sales Tax / VAT Problems
If marked price = MP, tax rate = t%, and discount = d%:
- SP before tax = MP × (1 − d/100)
- SP after tax = SP_before_tax × (1 + t/100)
Population/Mixture in Percentage Terms
For population growth/decline:
- After T years at r%: P_T = P_0 × (1 ± r/100)^T
The “Change of Base” in Percentages
GATE Advanced Example (2017, 2 marks): The price of a commodity increases by 20% in January, decreases by 10% in February, and increases by 15% in March. What is the net percentage change from the beginning of January to end of March?
Solution: 1.20 × 0.90 × 1.15 = 1.242 → 24.2% increase.
Partnership and Share Distribution
When partners invest for different time periods, profits are divided in ratio of (capital × time). This is essentially a weighted average problem in disguise.
Present Worth / True Discount
For bills of exchange:
- True Discount (TD): Difference between nominal (face) value and present worth
- Present Worth (PW): FV / (1 + rt/100) where r = rate%, t = time in years
- Banker’s Discount: Discount calculated on the face value (not on PW)
Multi-Item Shopkeeper Problems
When a shopkeeper sells at a profit of x% on some items and loss of y% on others, overall profit/loss % = (Profit − Loss) / Total Cost × 100.
If equal cost is involved: Net % = (x − y) / 2 (if x% profit and y% loss on equal cost).
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