Percentage

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Percentage is one of the most frequently tested topics in SSC CGL Quantitative Aptitude. The word “percent” means “per hundred” — a fraction with denominator 100. Mastery of percentage calculations is essential because this concept underlies many other quantitative topics including profit-loss, simple/compound interest, and data interpretation.

Basic Formulas:

$$\text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100$$

$$\text{Part} = \frac{\text{Percentage}}{100} \times \text{Whole}$$

$$\text{Whole} = \frac{\text{Part}}{\text{Percentage}} \times 100$$

Key Conversions:

• 1% = $\frac{1}{100} = 0.01$
• 50% = $\frac{1}{2} = 0.5$
• 25% = $\frac{1}{4} = 0.25$
• 20% = $\frac{1}{5} = 0.2$
• 10% = $\frac{1}{10} = 0.1$
• 5% = $\frac{1}{20} = 0.05$

Percentage Increase/Decrease:

$$\text{Percentage Increase} = \frac{\text{New} - \text{Old}}{\text{Old}} \times 100$$

$$\text{Percentage Decrease} = \frac{\text{Old} - \text{New}}{\text{Old}} \times 100$$

⚡ Exam Tip: When a value changes from 50 to 75, the percentage increase is $\frac{75-50}{50} \times 100 = 50%$. But if it changes back from 75 to 50, the percentage decrease is $\frac{75-50}{75} \times 100 = 33.33%$. The base for percentage change is always the ORIGINAL value before the change.

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding and problem-solving practice.

Successive Percentage Change:

When a value undergoes multiple percentage changes, the net change is NOT simply the sum:

If price increases by 20% then decreases by 10%: $$Final = \text{Original} \times 1.20 \times 0.90 = 1.08 \times \text{Original}$$ Net increase = 8%

For two successive changes $r_1$% and $r_2$%: $$Net change = r_1 + r_2 + \frac{r_1 \times r_2}{100}$$

Note: If both changes are increases, the formula works directly. If one is decrease, use negative sign.

Reversal of Percentage:

If a value increases by P%, to get back to original value, it must decrease by: $$\frac{100 \times P}{100 + P}%$$

Example: 50% increase requires 33.33% decrease to return to original.

If value decreases by P%, to get back to original, it must increase by: $$\frac{100 \times P}{100 - P}%$$

Example: 20% decrease requires 25% increase to return to original.

Population公式:

$$P_t = P_0 \times \left(1 + \frac{r}{100}\right)^t$$ Where $r$ = annual growth rate, $t$ = number of years

If population grows at 5% per year from 2,00,000:

• After 3 years: $2,00,000 \times (1.05)^3 = 2,00,000 \times 1.157625 = 2,31,525$

Fraction-Percentage Equivalents:

Fraction	Percentage
1/1	100%
1/2	50%
1/3	33.33%
2/3	66.67%
1/4	25%
3/4	75%
1/5	20%
2/5	40%
3/5	60%
4/5	80%
1/6	16.67%
1/7	14.29%
1/8	12.5%
1/9	11.11%
1/10	10%
1/15	6.67%

Percentage in Profit-Loss:

• Markup percentage: $\frac{MP - CP}{CP} \times 100$
• Discount percentage: $\frac{MP - SP}{MP} \times 100$
• Profit percentage: $\frac{SP - CP}{CP} \times 100$ (always on Cost Price)
• Loss percentage: $\frac{CP - SP}{CP} \times 100$ (always on Cost Price)

⚡ SSC CGL-Specific Tip: In SSC CGL, questions often combine percentage with ratio. If A is 30% more than B, then A/B = 130/100 = 13/10. If A is 40% less than B, then A/B = 60/100 = 3/5.

Common Student Mistakes:

• Confusing “percentage points” with “percentage”
• Mixing up increase vs decrease calculations (using wrong base)
• Forgetting that successive changes multiply, not add

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Compound Percentage Formula:

For n years of growth at r% per year: $$P_n = P_0 \times \left(1 + \frac{r}{100}\right)^n$$

For decline: use $\left(1 - \frac{r}{100}\right)^n$

Average Percentage Rate:

If values grow at different rates over different periods: Example: A value 100 grows by 20% in year 1, then 30% in year 2:

• Year 1 end: $100 \times 1.20 = 120$
• Year 2 end: $120 \times 1.30 = 156$ Net increase: 56%

Not the same as average of 20% and 30% (which would be 25%).

Percentage Distribution:

If A:B:C = 3:4:5 and total is 240:

• Sum of ratios = 3+4+5 = 12
• A’s share = $\frac{3}{12} \times 240 = 60$
• B’s share = $\frac{4}{12} \times 240 = 80$
• C’s share = $\frac{5}{12} \times 240 = 100$

Venn Diagram Percentage:

In problems with overlapping sets:

• n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
• If percentages are given: Convert percentage to actual values first

Election Problems:

In voting/election percentage problems: $$\text{Votes for winner} = \frac{\text{Vote share}}{100} \times \text{Total votes cast}$$ $$\text{Margin of victory} = (\text{Share of winner} - \text{Share of runner-up}) \times \text{Total votes cast}$$

SSC CGL Tier II Advanced:

More complex percentage problems in Tier II:

• Percentage of percentage (e.g., 20% of 30% of X)
• Comparison of percentages in different groups with different bases
• Finding original number when percentage and resulting value are given

Sample SSC CGL Questions:

Q1. If A’s salary is 40% more than B’s and B’s salary is 30% less than C’s, what is A’s salary compared to C’s? Solution: Let C = 100 B = 100 - 30% of 100 = 70 A = 70 + 40% of 70 = 70 + 28 = 98 A’s salary = 98% of C’s = 2% less than C

Q2. The population of a town increased by 10% in first year, 20% in second year, and decreased by 5% in third year. If initial population is 10,000, find final population. Solution: Year 1 end: $10,000 \times 1.10 = 11,000$ Year 2 end: $11,000 \times 1.20 = 13,200$ Year 3 end: $13,200 \times 0.95 = 12,540$

Q3. A number is first increased by 25%, then decreased by 20%. What is the net percentage change? Solution: $1.25 \times 0.80 = 1.00$ Net change = 0% (back to original — this is because 25% increase requires 20% decrease to cancel)

General: A% increase followed by B% decrease gives net $(1 + A/100)(1 - B/100) - 1$ times original.

⚡ Advanced Tip: When comparing percentages across different bases, convert to actual values first. If Group A has 20% of 50 students passing, and Group B has 25% of 80 students passing, actual numbers are 10 and 20. Group B has more students passing in absolute terms despite lower percentage.

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