Percentage

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What: Percentage covers conversion between percentages, fractions and decimals, percentage change, successive percentages, and the standard CDS applications — population growth, depreciation, salary changes, exam-score percentages.
Why it matters: Percentage is the most universally applied head — it appears directly in 3–5 CDS questions per paper and underlies Profit and Loss, SI/CI, and data interpretation.
Key fact: For two successive percentage changes $x\%$ and $y\%$, the net effect is $x + y + \tfrac{xy}{100}\%$. Use this for back-to-back hikes, discounts, or population growth.

Percentage is the workhorse of CDS arithmetic. It is the chapter that quietly powers Profit and Loss, Simple and Compound Interest, mixture problems, and statistical interpretation. Master the conversion tables and the successive-percentage formula, and you will collect 3–5 marks per paper here plus a multiplier on every related chapter.

This page is built from CDS Previous Year Questions across 2000–2021 plus NCERT Class 8 Proportional Reasoning. Pair with Decimal Fractions (the bridge to percentages) and Ratio and Proportion.

What This Topic Covers

CDS scope: (1) conversion — percentage ↔ fraction ↔ decimal; (2) percentage of a number — basic computation; (3) percentage change — increase, decrease, "what percent of"; (4) successive percentages — back-to-back changes; (5) word applications — population, salary, election votes, exam scoring, depreciation; and (6) error analysis — percentage error in measurement.

Why This Topic Matters

Percentage is one of the four chapters that appear in every CDS paper without exception.
The successive-percentage formula $x + y + xy/100$ is the highest-ROI shortcut in CDS arithmetic.
Fluency here lifts your speed in Profit and Loss, SI/CI, mixtures, and statistics.

Exam Pattern & Weightage

Year / Paper	No.	Subtopics Tested
2007-II	3	Salary changes, election votes, exam marks
2009-II	2	Successive percentage, population
2010-I/II	4	Conversion, population growth, depreciation
2011-I/II	3	Salary hike, exam-mark percentages
2012-I/II	4	Successive change, electorate, expenditure
2013-I/II	3	Population, percentage error
2014-I/II	4	Salary, mixture composition, votes
2015-I/II	3	Successive discount, depreciation
2017-I/II	4	Population, expenditure, mixture
2018-I/II	3	Salary, exam, population
2019-II	3	Successive percentages, votes
2020-I/II	3	Mixture, salary, exam
2021-I/II	3	Population, depreciation

⚡ CDS Alert

The election-votes archetype: "candidate A got 40% of votes and lost by 4000 votes" type. Always set up the equation as $0.6V - 0.4V = 4000 \Rightarrow V = 20000$. The percentages relate to total votes, not to each candidate separately.

Core Concepts

Definition

Percent = Per Hundred $$x\% = \frac{x}{100}, \qquad x\% \text{ of } y = \frac{x \cdot y}{100}$$

Conversion Reflexes

Quick Conversions $\tfrac{1}{2} = 50\%,\; \tfrac{1}{3} = 33.\overline{3}\%,\; \tfrac{1}{4} = 25\%,\; \tfrac{1}{5} = 20\%,\; \tfrac{1}{6} = 16.\overline{6}\%,\; \tfrac{1}{8} = 12.5\%,\; \tfrac{1}{9} = 11.\overline{1}\%,\; \tfrac{1}{10} = 10\%,\; \tfrac{1}{11} = 9.\overline{09}\%,\; \tfrac{1}{12} = 8.\overline{3}\%,\; \tfrac{1}{16} = 6.25\%,\; \tfrac{1}{20} = 5\%,\; \tfrac{1}{25} = 4\%,\; \tfrac{1}{50} = 2\%$

Percentage Change

Increase / Decrease $$\%\text{ change} = \frac{\text{final} - \text{initial}}{\text{initial}} \times 100$$

Positive value indicates an increase; negative, a decrease. Always divide by the initial (original) value, not the final.

Successive Percentage Changes

Net Effect of Two Successive Changes $$\text{Net} = x + y + \frac{xy}{100} \quad (\%)$$

Signs matter: an increase is positive, a decrease is negative. So a 20% increase followed by a 10% decrease gives net $20 - 10 + \tfrac{20 \times (-10)}{100} = 10 - 2 = 8\%$ increase.

⚠ Common Trap

A 20% increase followed by a 20% decrease is not 0% net change. It is $20 - 20 - \tfrac{400}{100} = -4\%$, i.e. 4% net decrease. CDS plants this every other paper.

Population, Depreciation, Compounding

Compound Growth / Decay $$P_n = P_0 \left(1 + \frac{r}{100}\right)^n \quad (\text{growth}); \qquad P_n = P_0 \left(1 - \frac{r}{100}\right)^n \quad (\text{decay})$$

Salary, Exam, and Election Patterns

Salary hike: "salary increased by 20%, then decreased by 10%". Apply successive percentage formula. Election: "winner got 60%, loser 40%, margin = 4000". Set total = $V$; solve $0.6V - 0.4V = 4000$. Exam: "scored 35% but failed by 20 marks; pass mark is 40%". Pass marks − scored = $0.4M - 0.35M = 20$, giving total marks $M = 400$.

Worked Examples

Example 1 — Successive Percentages (2012-II)

Q: The price of a commodity rises by 25% and then falls by 20%. What is the net change?

Apply $x + y + xy/100$ with $x = 25,\, y = -20$.
Net $= 25 - 20 + \tfrac{25 \cdot (-20)}{100} = 5 - 5 = 0\%$.
Answer: no net change. (Verify: 100 → 125 → 100. ✓)

Example 2 — Election Votes (2010-II)

Q: In an election between two candidates, the winner got 60% of valid votes and won by 1200 votes. Find total valid votes.

Winner: $0.6V$. Loser: $0.4V$. Margin: $0.6V - 0.4V = 0.2V = 1200$.
$V = 6000$ valid votes.

Example 3 — Exam Pass Mark (2011-II)

Q: A student scored 30% and failed by 25 marks. Another scored 50% and passed with 35 marks more than the pass mark. Find the pass mark.

Let total marks be $M$ and pass mark be $P$.
From student 1: $P - 0.3M = 25 \implies P = 0.3M + 25$.
From student 2: $0.5M - P = 35 \implies P = 0.5M - 35$.
Equate: $0.3M + 25 = 0.5M - 35 \implies 0.2M = 60 \implies M = 300$.
So $P = 0.3 \times 300 + 25 = 90 + 25 = 115$ marks. ($115/300 \approx 38.3\%$ pass.)

Example 4 — Population Growth (2017-II)

Q: The population of a town increases by 10% per year. If the present population is 24200, what was it 2 years ago?

Apply $P_{\text{now}} = P_{\text{then}} (1.10)^2$.
$24200 = P_{\text{then}} \cdot 1.21 \implies P_{\text{then}} = 24200/1.21 = 20000$.

Example 5 — Salary Change (2014-I)

Q: A man's salary is decreased by 50%, then increased by 50%. What is the net change?

Apply $x + y + xy/100$ with $x = -50,\, y = +50$.
Net $= -50 + 50 + \tfrac{(-50)(50)}{100} = 0 - 25 = -25\%$.
Answer: 25% net decrease. (Verify: 100 → 50 → 75. ✓)

Example 6 — Mixture Percentage (2017-I)

Q: A mixture of 60 L contains milk and water in the ratio 2 : 1. How much water must be added to make milk 50% of the mixture?

Milk $= \tfrac{2}{3} \cdot 60 = 40$ L. Water $= 20$ L.
Let $x$ L water be added. New total $= 60 + x$. Milk stays 40.
Need $\tfrac{40}{60 + x} = 0.5 \implies 60 + x = 80 \implies x = 20$ L.

Example 7 — Depreciation (2019-II)

Q: A machine depreciates by 10% per year. If its value after 3 years is 14580, find its original value.

Apply $P_3 = P_0 (0.9)^3 = 0.729 P_0$.
$14580 = 0.729 P_0 \implies P_0 = 14580 / 0.729 = 20000$.

How CDS Tests This Topic

Six archetypes: (1) salary change with successive percentages, (2) election votes with given margin, (3) exam pass-mark from two students' results, (4) population growth/decay compounding, (5) depreciation, (6) mixture composition expressed as percentage. Spot the archetype — every CDS percentage question fits one.

Exam Shortcuts (Pro-Tips)

Shortcut 1 — Memorise Fraction Equivalents

Knowing $\tfrac{1}{n} = $ percent for $n = 2, 3, 4, \ldots, 20$ is half the battle. $\tfrac{1}{8} = 12.5\%$, $\tfrac{1}{16} = 6.25\%$, $\tfrac{1}{20} = 5\%$. When you see $12.5\%$ in a problem, mentally substitute $\tfrac{1}{8}$ and the arithmetic collapses.

Shortcut 2 — Successive Change Formula

Two-Step Net Change $$\text{Net}\% = x + y + \frac{xy}{100}$$

Always sign the changes correctly. Increase positive, decrease negative.

Shortcut 3 — Use Decimal Multipliers

A 20% increase = multiply by 1.20. A 30% decrease = multiply by 0.70. Chain multipliers for successive changes: $1.20 \times 0.70 = 0.84$, i.e. 16% net decrease. Faster than the formula for some.

Shortcut 4 — Reverse Percentages

"After a 25% increase, price is 250. Original?" Don't subtract 25%; divide by 1.25. Original $= 250/1.25 = 200$.

Shortcut 5 — Percent of Percent

$x\%$ of $y$ = $y\%$ of $x$. So 18% of 50 = 50% of 18 = 9. Useful when one factor is a "friendly" percentage like 50% or 25%.

Common Question Patterns

Pattern 1 — Salary / Price Successive Change

Two or more percentage changes applied in sequence. Apply $x + y + xy/100$ (or multiplier chain). Sign each change correctly.

Pattern 2 — Election with Given Margin

Two candidates, one percentage, one margin in votes. Set total $V$; solve linear equation.

Pattern 3 — Pass Mark from Two Students

One student fails by some marks at scored percentage. Another passes by some marks at higher percentage. Two equations, two unknowns (total $M$, pass mark $P$).

Pattern 4 — Population Growth / Depreciation

Compound formula $P_n = P_0 (1 \pm r/100)^n$. For growth, "+"; for depreciation, "−". CDS questions usually have integer $n$ and clean answers.

Pattern 5 — Mixture Composition

Given mixture A:B ratio or A% concentration, add or remove component to reach a target percentage. Equation: target fraction × new total = preserved quantity.

Preparation Strategy

Week 1. Memorise fraction-to-percent table for $\tfrac{1}{2}$ through $\tfrac{1}{20}$. Drill 20 problems on percentage of a number and percentage change. Build reflex on reverse percentages (dividing by 1.20 instead of subtracting).

Week 2. Successive percentages. Drill 20 problems on the $x + y + xy/100$ formula. Cover all six archetypes — salary, election, exam, population, depreciation, mixture. Layer in Profit and Loss and SI/CI since both lean heavily on percentage fluency.

Mock testing. Take timed papers. Track whether you slip on sign errors (decrease vs increase) or on the $xy/100$ term. Both are fixable with drill. Use CDS mock tests for paced practice.

Drill Percentage in Timed Conditions

Try CDS mocks loaded with successive percentage, election, exam, and depreciation problems. Master the six archetypes and you will never miss marks here.

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Frequently Asked Questions

What is the formula for the net effect of two successive percentage changes?

If $x\%$ is followed by $y\%$, the net change is $x + y + \tfrac{xy}{100}\%$. Sign each correctly — increase positive, decrease negative. Example: a 25% increase followed by a 20% decrease gives net $25 - 20 - \tfrac{500}{100} = 0\%$, i.e. no change.

Why is a 20% increase and a 20% decrease not zero net change?

Apply the formula: $20 - 20 + \tfrac{20 \cdot (-20)}{100} = -4\%$. A 20% increase followed by a 20% decrease leaves a 4% net decrease. The reason: the decrease is applied to a larger base, so it subtracts more than the original increase added.

How do I reverse a percentage change?

If a value increased by $x\%$ to reach final $F$, the original was $\tfrac{F}{1 + x/100}$. Do not simply subtract $x\%$ — that would give a different (wrong) answer. Example: after a 25% increase the price is 250. Original $= 250/1.25 = 200$, not $250 - 0.25 \cdot 250 = 187.5$.

How do I solve "election with given margin" problems?

Set total valid votes $= V$. Express each candidate's votes as a percentage of $V$. The difference equals the given margin. Example: winner 60%, loser 40%, margin 1200 ⇒ $0.6V - 0.4V = 1200 \Rightarrow V = 6000$.

How do I handle population growth or depreciation over multiple years?

Use the compound formula $P_n = P_0 (1 \pm r/100)^n$. "+ " for growth, "−" for depreciation. $n$ is the number of years. CDS questions usually have integer $n$ and clean answers, often involving $(1.10)^2 = 1.21,\; (1.10)^3 = 1.331,\; (0.9)^3 = 0.729$.

What does "$x\%$ of $y = y\%$ of $x$" mean and how is it useful?

It is the commutative property of multiplication in percent form: $\tfrac{x \cdot y}{100} = \tfrac{y \cdot x}{100}$. Useful when one number is a "friendly" percentage. Example: 24% of 75 = 75% of 24 = 18 — much faster than computing 24% directly.

Which CDS Maths topics depend on Percentage?

Three big ones: Profit and Loss (markup, discount), Simple and Compound Interest (rate as percentage), and mixture problems (composition as percentage). Sharpening percentage fluency lifts your speed across all three.