Percentage & Profit-Loss

Percentage & Profit-Loss

🟢 Lite

Key Formula/Rule

Percentage: Part ÷ Whole × 100. Profit-Loss: always relative to Cost Price (CP). Gain% = (Gain ÷ CP) × 100. Loss% = (Loss ÷ CP) × 100.

Memory Trick

“Per-Cent = Per 100” — just scale the part to a 100-point base. For profit/loss: always use CP as the denominator — this is the most common mistake. If SP > CP you made a gain; if SP < CP you made a loss.

1-Sentence Summary

Percentage is a way to express any fraction as if it were out of 100. Profit and loss measure how much you gained or lost relative to what something cost you (the CP), not what you sold it for.

30-Second Example

Q: An article costs ₹600 and is sold for ₹750. What is the gain%? A: Gain = 750 − 600 = ₹150. Gain% = (150 ÷ 600) × 100 = 25%

Q: What is 12% of 250? A: (12 ÷ 100) × 250 = 30

Must Remember

• Always calculate gain/loss relative to CP, not SP — this is the most common error in profit-loss questions.
• Marked Price (MP): The listed price before discount. Selling Price (SP): What the customer actually pays.
• Discount% = (MP − SP) ÷ MP × 100 (discount is always % of MP, not CP).
• SP = CP × (1 + Gain%)/100 for a gain, SP = CP × (100 − Loss)/100 for a loss.
• False weights: A shopkeeper uses 900g as 1kg → his gain% = (1000−900)/900 × 100 = 11.11%.

Exam Tips for CUET

• CUET mixes percentage with profit-loss in the same question frequently. Identify which base to use before calculating.
• If an article has two successive discounts (e.g., 20% and 10% off), the final price = MP × 0.80 × 0.90 — multiply the multipliers, don’t subtract.
• When CP is not given but SP and gain% are: CP = SP ÷ (1 + Gain/100).
• A question with “loss% first, then gain% on same cost” — the final result is always a LOSS because the bases are different.

Common Pitfalls

• Computing gain% using SP as the base: Wrong. Gain% = (SP − CP)/CP × 100, not ÷ SP.
• Adding percentages directly: 20% markup then 20% discount on MP gives a net LOSS: 1.20 × 0.80 = 0.96 of original CP.
• Confusing MP with SP: MP is the tag price; SP is the final price after discount.

🟡 Standard

Concept

Percentages are just fractions dressed up to look scary. “45% of 200” just means 45 out of every 100 parts of 200 — so you’re scaling down first, then multiplying. Once that clicks, profit and loss are just additions or subtractions on top of the Cost Price (CP).

The key thing to remember: gain or loss is always calculated relative to the Cost Price, not the Selling Price (SP). This trips up a lot of students. If something costs ₹100 (CP) and you sell it for ₹120 (SP), your gain is ₹20 — which is 20% of ₹100, not of ₹120.

Now here’s where it gets more interesting: shops don’t always sell at cost. They mark up the price first (called the Marked Price or MP), then offer a discount on that. So the price a customer actually pays is MP minus the discount. The shopkeeper’s profit or loss is still measured against the original CP.

Key Formulas

Formula	Use
Percentage = (Part ÷ Whole) × 100	Convert any fraction to a %
Gain% = ((SP − CP) ÷ CP) × 100	Find profit percentage
Loss% = ((CP − SP) ÷ CP) × 100	Find loss percentage
Discount % = ((MP − SP) ÷ MP) × 100	Find discount percentage
SP with Gain = CP × (1 + Gain%)/100	Find selling price from CP and gain%
Break-even: SP = CP	No profit, no loss

Worked Example

Q: An article costs ₹800. The shopkeeper marks it up by 40% and then offers a 10% discount. What is the final selling price?

Step 1: Find Marked Price (MP = CP + 40% of CP) MP = 800 + (40 ÷ 100) × 800 = 800 + 320 = ₹1,120

Step 2: Apply 10% discount on MP Discount = (10 ÷ 100) × 1120 = ₹112 SP = MP − Discount = 1120 − 112 = ₹1,008

Answer: ₹1,008

Common Errors

• Finding gain% using SP as the base → Always use CP as the denominator: Gain% = (Gain ÷ CP) × 100
• Confusing MP with SP → MP is the listed price before discount; SP is what the customer actually pays
• Adding percentages directly when a discount follows a markup → Can’t just do 40% − 10% = 30%. You must apply them sequentially: MP = CP × 1.40, then SP = MP × 0.90

🔴 Extended

Full Concept

Why Multiplying Factors Works

Here’s the “aha!” moment most textbooks skip. When you say “increase by 20%”, you’re really saying “the new value is 100% + 20% = 120% of the original.” As a multiplier, that’s 1.20 (or 120 ÷ 100). When you decrease by 10%, you’re keeping 90%, so the multiplier is 0.90.

So if something goes up 20% then down 10%, the net multiplier is 1.20 × 0.90 = 1.08 — an 8% net increase, not 10%. This is the most common mistake in CUET: students subtract the percentages (20% − 10% = 10%) and get it wrong every time. You can only add or subtract percentage changes directly when they’re on the same base. When the base keeps changing (as it does with successive changes), you must multiply the multipliers.

Percentage Change When Base Changes

This is subtle but important. If X increases by 25%, it becomes 1.25X. If you then decrease that result by 25%, you get 1.25X × 0.75 = 0.9375X. That’s a net decrease of 6.25%, not zero! The 25% decrease was applied to the new, larger base (1.25X), so it took away more than the original 25% increase gave.

This is why the phrase “increased by x%, then decreased by x%” does NOT cancel out. The decrease is always on the larger number.

Population Growth and Depreciation

These are just percentage applications in disguise. Population grows at, say, 10% per year — that means each year the population is multiplied by 1.10. Depreciation works the same way but in reverse: a machine worth ₹1,00,000 depreciating at 20% per year is worth 1,00,000 × 0.80 = ₹80,000 after one year. After two years: 80,000 × 0.80 = ₹64,000. You can chain these just like successive percentage changes.

Price Halving and Doubling

If a price first increases by 25% and then decreases by 20%, what’s the net change? Multiplier = 1.25 × 0.80 = 1.00. Net change = 0% — the price returns to its original value. If it increased 100% (doubled), you need a 50% decrease to get back to original (1.00 ÷ 2.00 = 0.50). Notice it’s not 100% decrease — that would make it zero! These “undo” problems are common in CUET.

Multiple Approaches

Standard Method: Successive increase of r₁% then r₂%: Final = Original × (1 + r₁/100) × (1 + r₂/100)

Shortcut — Net Change Formula: For two successive changes: Net% ≈ r₁ + r₂ + (r₁ × r₂)/100

• Positive for both: add the cross term (+)
• Negative for both: add the cross term (+)
• One positive, one negative: subtract the cross term (−) Example: +20%, −10% → Net ≈ 20 − 10 + (20 × −10)/100 = 10 − 2 = 8% increase ✓

CUET-Level Problems

Q1: The population of a town is 2,00,000. It increases at 5% per annum. What will it be after 3 years? Working: Year 1: 2,00,000 × 1.05 = 2,10,000 Year 2: 2,10,000 × 1.05 = 2,20,500 Year 3: 2,20,500 × 1.05 = 2,31,525 Answer: 2,31,525

Q2: A trader mixes two qualities of rice costing ₹40/kg and ₹60/kg in the ratio 3:2. He sells the mixture at ₹55/kg. Find his gain%. Working: Cost of 5 kg mixture = (3×40) + (2×60) = 120 + 120 = ₹240 Cost per kg = 240 ÷ 5 = ₹48 Selling price per kg = ₹55 Gain% = ((55 − 48) ÷ 48) × 100 = (7 ÷ 48) × 100 ≈ 14.58% Answer: 14.58%

Tricky Cases

• “x% of y equals y% of x” — These are always equal! Because x% of y = (x/100)×y and y% of x = (y/100)×x, both equal xy/100. Don’t be fooled by answer choices that make one seem bigger.
• Percentage vs Percentage Point: If something goes from 20% to 25%, that’s a 5 percentage point increase, but a (5/20)×100 = 25% increase in percentage terms. CUET often tests the difference.
• 100% increase = 2× original, not 1× original. A 100% increase means you add another whole copy, so 100 + 100 = 200% of the original.

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