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Percentage — The One Chapter That Runs Half of Quant

Chapter Outline & Learning Goals

By the end of this chapter, you will be able to:

1. Convert between fractions, decimals, and percentages instantly.
2. Solve percentage increase/decrease and successive change problems.
3. Handle exam favourites: marks-based questions, population change, price-consumption, and election problems.
4. Use fraction shortcuts (like 12.5% = 1/8) to solve questions in seconds.
5. Spot the traps examiners set with "percentage of" vs "percentage more than".

The Hook: A Shopkeeper Who Lost Money by Giving a Discount

A shopkeeper bought a shirt for ₹400. He marked it at ₹600 and then, feeling generous during a sale, gave a 20% discount. "I still made a profit," he smiled.

Did he? 20% of ₹600 is ₹120. Selling price = ₹480. Profit = ₹80. Yes, he did.

But here's the twist: his friend marked the same shirt at ₹500 and gave only a 10% discount. Selling price = ₹450. Profit = ₹50.

The first shopkeeper gave a bigger discount and still earned more. Why? Because percentages don't live alone — they live on a base. Change the base, and the same percentage tells a completely different story.

That single idea — *percentage of what?* — is the heart of this chapter, and the heart of almost every trap SSC and Banking exams set for you.

1. What Is a Percentage, Really?

Percentage means "per hundred" (from Latin *per centum*). When we say 40%, we simply mean 40 out of every 100.

$$x\% = \frac{x}{100}$$

So:

• 25% = 25/100 = 1/4 = 0.25
• 60% = 60/100 = 3/5 = 0.60
• 7.5% = 7.5/100 = 3/40 = 0.075

A percentage is not a new kind of number. It is just a fraction wearing a uniform — the uniform of 100. That's why percentages are so useful: they let us compare things fairly. Scoring 18 out of 25 in one test and 30 out of 40 in another — which is better? Convert both to percentages: 72% vs 75%. Now the comparison is instant.

🧠 Active Recall: Without looking above — what does 12.5% equal as a fraction? And what is 3/8 as a percentage? (Answers: 1/8 and 37.5%. If you hesitated, memorise the table coming in Section 3.)

2. The Learner's Struggle (You're Not Alone)

Most students don't fail percentage questions because they can't calculate. They fail because of three silent mistakes:

1. Wrong base: "A is 25% more than B" does NOT mean "B is 25% less than A." (We'll prove this soon.)
2. Adding percentages blindly: Two successive increases of 10% and 20% are NOT a 30% increase.
3. Percentage vs percentage points: If interest rates rise from 4% to 6%, that's a rise of 2 *percentage points*, but a 50% *percentage increase*.

Every formula in this chapter exists to kill one of these three mistakes. Keep them in mind as we go.

3. The Fraction–Percentage Bridge (Your Biggest Weapon)

Toppers don't calculate 37.5% of 640 by multiplying 0.375 × 640. They see 37.5% and think 3/8, then do 640 ÷ 8 × 3 = 240 in their heads.

Memorise this bridge table — it pays off in every single Quant paper:

Mnemonic (the weird one): Imagine a pizza shop where the owner is a miser named "Aath-ka-Pizza" (Mr. Eighth). He cuts his pizza into 8 slices and prices them: 1 slice = ₹12.5, 3 slices = ₹37.5, 5 slices = ₹62.5, 7 slices = ₹87.5. The odd slices (1, 3, 5, 7) give the odd prices (12.5, 37.5, 62.5, 87.5). Remember the miser, and the whole 1/8 family is yours forever.

🧠 Active Recall: Cover the table. What fraction is 62.5%? What percentage is 1/12? (Answers: 5/8 and 8.33%.)

4. The Core Formula: Finding Percentages and Values

Everything in this chapter grows from one line:

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

From this, three question types are born:

Type 1 — Find the percentage: What % of 80 is 56?
→ (56/80) × 100 = 70%

Type 2 — Find the part: What is 35% of 240?
→ 240 × 35/100 = 84 (or smarter: 35% = 25% + 10% = 60 + 24 = 84)

Type 3 — Find the whole: 45 is 15% of what number?
→ Whole = 45 × 100/15 = 300

🧠 Active Recall: 60 is 12% of which number? Try it before reading on. (Answer: 60 × 100/12 = 500.)

5. Percentage Increase and Decrease — and the Base Trap

$$\text{\% Increase} = \frac{\text{Increase}}{\text{Original Value}} \times 100$$

$$\text{\% Decrease} = \frac{\text{Decrease}}{\text{Original Value}} \times 100$$

The denominator is always the original (starting) value — this is where the trap lives.

The classic trap: If A's salary is 25% more than B's, by what % is B's salary less than A's?

Let B = 100. Then A = 125.
B is less than A by 25, but the base is now A = 125:
→ (25/125) × 100 = 20%, not 25%!

The golden rule: If A is x% more than B, then B is $\frac{x}{100+x} \times 100$% less than A.
If A is x% less than B, then B is $\frac{x}{100-x} \times 100$% more than A.

🧠 Active Recall: A number is 20% less than 80. By what % is 80 more than that number? (Answer: number = 64; (16/64) × 100 = 25%.)

6. Successive Percentage Change

Prices rise 10% this year and 20% next year. Total rise = 30%? No.

Start with 100. After 10% rise → 110. After 20% rise on 110 → 132. Total rise = 32%.

Formula: For successive changes of a% and b%:

$$\text{Net change} = a + b + \frac{ab}{100}\ \%$$

(Use negative signs for decreases.)

• 10% rise then 20% rise: 10 + 20 + (10×20)/100 = +32%
• 20% rise then 20% fall: 20 − 20 + (20×−20)/100 = −4% (a net loss!)

That last result shocks everyone: rise 20%, then fall 20%, and you're still 4% poorer. Why? The fall acts on a bigger base. This is exactly the kind of question examiners love.

🧠 Active Recall: A price rises 25% and then falls 25%. Net change? (Answer: 25 − 25 − 625/100 = −6.25%.)

7. Exam Favourite Models

(a) Price–Consumption: If the price of sugar rises by 25%, by how much % should consumption be cut so that expenditure stays the same?
→ Reduction = $\frac{x}{100+x} \times 100 = \frac{25}{125} \times 100$ = 20%

(b) Population: A town's population is 80,000. It grows 5% every year. Population after 2 years?
→ 80,000 × (1.05)² = 80,000 × 1.1025 = 88,200

(c) Marks: A student needs 40% to pass. He scores 180 marks and fails by 20 marks. Maximum marks?
→ Pass marks = 180 + 20 = 200 = 40% of total → Total = 200 × 100/40 = 500

(d) Elections: In an election between two candidates, the winner got 62% of valid votes and won by 4,800 votes. Total valid votes?
→ Margin = 62% − 38% = 24% → 24% of votes = 4,800 → Total = 20,000

🧠 Active Recall: In the election model, why do we subtract 62 − 38 instead of using 62 directly? (Because the *margin of victory* is the difference between the two candidates' shares.)

8. Summary Table

Master conversions: 1/8 = 12.5%, 1/6 = 16.66%, 1/7 = 14.28%, 1/9 = 11.11%, 1/11 = 9.09%, 1/12 = 8.33%.

🔁 Review Reminder

You have just built the foundation for Profit & Loss, Simple & Compound Interest, and Data Interpretation — all of them are percentage chapters in disguise.

Review this chapter after 2 days. When you do, first try to write the fraction–percentage table and the successive-change formula from memory. If both come out clean, you're exam-ready. If not, re-read Sections 3 and 6 — they carry the most marks per minute of study.