Percentages

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Definition: Percent means “per hundred.” $x%$ = $\frac{x}{100}$

Quick Conversions:

Fraction	Percentage	Decimal
1/2	50%	0.5
1/3	33.33%	0.333
1/4	25%	0.25
1/5	20%	0.2
1/8	12.5%	0.125
1/10	10%	0.1
1/20	5%	0.05

Essential Formulas:

1. Basic Percentage: $x%$ of $N = \frac{x}{100} \times N$

2. Percentage Change: $\text{Change} % = \frac{\text{New} - \text{Old}}{\text{Old}} \times 100$

3. Successive Percentage Change:

   • Two increases of $a%$ and $b%$: Net = $(1 + \frac{a}{100})(1 + \frac{b}{100}) - 1$
   • Increase then decrease: Net depends on order!

4. Reverse Percentage: If $x%$ of $N = P$, find $N = \frac{P \times 100}{x}$

⚡ CAT Tip: For successive percentage changes, use multiplication, not addition. Two 20% increases ≠ 40% increase (it’s actually 44%). Remember: percentage change compounds.

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding.

Working with Percentages

Finding the Original Value: If price increases by 25% to ₹500, original = $\frac{500}{1.25} = ₹400$

If price decreases by 20% to ₹400, original = $\frac{400}{0.80} = ₹500$

Percentage vs Absolute Change:

Initial	Final	% Change	Absolute Change
50	60	+20%	+10
50	45	-10%	-5
40	50	+25%	+10

Comparison Using Percentages: Don’t compare percentage changes on different bases blindly. ₹10 off ₹100 is 10%, but ₹10 off ₹50 is 20% — the same absolute change means different percentage impacts depending on base.

Applications in CAT:

1. Profit and Loss:

   • Profit % = $\frac{\text{SP} - \text{CP}}{\text{CP}} \times 100$
   • Loss % = $\frac{\text{CP} - \text{SP}}{\text{CP}} \times 100$
   • If profit% = x%, then SP/CP = $(1 + x/100)$

2. Discount Problems:

   • Two successive discounts of $a%$ and $b%$: Final price = $100 \times (1 - \frac{a}{100})(1 - \frac{b}{100})$
   • A discount of 50% followed by 20% off = $100 \times 0.5 \times 0.8 = ₹40$ (not ₹30!)

3. Population/Growth Problems:

   • If population grows at $r%$ per year, after $n$ years: $P_n = P_0 \times (1 + \frac{r}{100})^n$

Percentage Point vs Percent: If interest rates go from 8% to 10%, the increase is 2 percentage points, but the percent increase is $\frac{2}{8} \times 100 = 25%$.

⚡ Common Mistake: Students confuse “percentage point change” with “percentage change.” Always check what’s being compared.

Base Selection: For questions like “X is what percent of Y?”, base is always Y (the “of” quantity). For “X is P% more than Y”, X = $(1 + \frac{P}{100}) \times Y$.

Average Percentage: If different quantities have different percentages, weighted average depends on the base of each quantity, not the percentages themselves.

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🔴 Extended — Deep Study (3mo+)

Comprehensive theory for serious exam preparation.

Advanced Percentage Applications

Data Interpretation Percentage: In DI, percentages appear constantly:

• Year-over-year growth: $\frac{\text{Year } N - \text{Year } (N-1)}{\text{Year } (N-1)} \times 100$
• Market share changes
• Contribution to total (percent of pie chart)
• Break-even analysis

Compound vs Simple Interest:

• Simple Interest: $SI = P \times R \times T / 100$
• Compound Interest (annual): $A = P(1 + R/100)^T$
• For semi-annual compounding: $A = P(1 + R/200)^{2T}$
• Difference grows with time and rate

Depreciation: If value decreases by $r%$ per year: $V_n = V_0 \times (1 - \frac{r}{100})^n$

Markup vs Margin:

• Markup % = $\frac{\text{Profit}}{\text{Cost Price}} \times 100$
• Margin % = $\frac{\text{Profit}}{\text{Selling Price}} \times 100$
• Same profit amount gives different percentages depending on base!

Percentage in Ratios: If A:B = x:y, then A is $\frac{x}{x+y} \times 100%$ of total, and B is $\frac{y}{x+y} \times 100%$.

Error Analysis:

• Measured value vs True value
• Absolute error = |Measured - True|
• Relative/Percentage error = $\frac{\text{Absolute error}}{\text{True value}} \times 100$

Break-Even Analysis:

• Break-even point: Revenue = Cost
• Contribution = Selling Price - Variable Cost
• Break-even quantity = $\frac{\text{Fixed Cost}}{\text{Contribution per unit}}$

Salary-Increase Chain: A gets 20% hike, B gets 10% hike. If A’s new salary is 10% more than B’s new salary, find original ratio:

• Let A = $a$, B = $b$
• $1.2a = 1.1 \times 1.1b$ (since B is 10% more than A originally… depends on question)
• Set up equations carefully based on exact wording

Mixture Problems: If you mix solutions of two concentrations $x%$ and $y%$ in ratio $r:s$, the concentration of mixture is $\frac{rx + sy}{r + s}%$.

⚡ Advanced Tip: When multiple percentage changes occur, working with base-100 method often simplifies. Assume base of 100 for original value, apply successive changes, read final value. For two changes: 100 → $100(1 + a/100)(1 + b/100)$. For successive discounts: 100 → $100(1-d_1/100)(1-d_2/100)$.

Percentage Distribution: If A’s share is $x%$, B’s share is $y%$, and C gets the rest:

• C’s share = $(100 - x - y)%$
• Useful in profit distribution, inheritance, market share problems

Taxation Problems:

• Adding tax: $SP = CP \times (1 + \frac{\text{tax}%}{100})$
• Tax-inclusive pricing: $CP = \frac{SP}{1 + \frac{\text{tax}%}{100}}$

CAT 2023 Pattern: Percentages appear in:

• DI sets involving growth rates, market share, comparisons
• QA direct percentage calculations
• Profit/Loss/Discount problems
• Data comparison questions

Quick Reference — Common Conversions to Memorize:

• $12.5% = \frac{1}{8}$
• $6.25% = \frac{1}{16}$
• $33.33% \approx \frac{1}{3}$
• $16.67% \approx \frac{1}{6}$
• $66.67% \approx \frac{2}{3}$
• $14.29% \approx \frac{1}{7}$
• $11.11% \approx \frac{1}{9}$

⚡ Exam Strategy: For percentage problems in CAT, identify whether you’re looking for old value, new value, or change. Use the formula: $\text{New} = \text{Old}(1 + \frac{%}{100})$ for increase or $\text{New} = \text{Old}(1 - \frac{%}{100})$ for decrease. For successive changes, chain the multiplications.

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📊 CAT Exam Essentials

Detail	Value
Sections	VARC (24 Qs), DILR (20 Qs), QA (22 Qs)
Time	2 hours (40 min per section)
Total	66 questions, 198 marks
Marking	+3 correct, −1 wrong (MCQ); no penalty for TITA
Mode	Computer-based, multiple sessions
Percentile	Normalized — 99+ needed for top IIMs

🎯 High-Yield Topics for CAT

• Reading Comprehension — 16-20 marks in VARC
• Para Summary + Odd Sentence — 8-12 marks
• DI Sets (Tables + Caselets) — 10-15 marks in DILR
• Arithmetic (Percentages + Profit/Loss) — 8-12 marks in QA
• Geometry + Mensuration — 6-10 marks
• Logarithm + Sequences — 6-10 marks

📝 Previous Year Question Patterns

• Q: “The passage is primarily concerned with…” [2024 VARC — RC passage]
• Q: “If f(x) = x² - 5x + 6, the value of f(3) is…” [2024 QA — Arithmetic]
• Q: “How many ways can 5 people be arranged around a round table…” [2024 DILR — Circular]

💡 Pro Tips

• VARC is the top priority — strong RC skills can push you to 99+ percentile quickly
• DILR: attempt 2 full sets out of 4-5 sets — accuracy matters more than coverage
• QA: arithmetic (time-speed-work) + geometry carry ~40% of QA marks
• Take 3-4 full mocks before the exam to find your section-wise pacing

🔗 Official Resources

• IIM CAT Official
• [CAT Syllabus](https://iimcat.ac.in/exam pattern)

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