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Percentage

Chapter Outline & Learning Goals

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

1. Convert between percentages, fractions, and decimals instantly.
2. Solve successive percentage change problems without confusion.
3. Master percentage applications: profit-loss, population change, exam marks, and data interpretation.
4. Spot and avoid the classic traps examiners set in SSC CGL Tier-II, IBPS PO Mains, CAT, and CSAT.
5. Use fraction shortcuts to calculate in seconds instead of minutes.

The Hook: The Discount That Wasn't a Discount

A shopkeeper raises the price of a jacket by 25%, then announces a "25% discount" during a sale. Most people think the jacket is back to its original price. It isn't. If the original price was ₹800, after a 25% hike it becomes ₹1,000. A 25% discount on ₹1,000 is ₹250 off — final price ₹750. You just "saved" money and still paid ₹50 less than the original... wait, actually the shopkeeper *lost* ₹50? No — check again: the shopkeeper bought it cheaper. The point is this: equal percentage up and down never cancels out, and examiners build entire questions on this one illusion. Master this chapter, and these traps become free marks.

1. What Percentage Really Means

"Per cent" comes from the Latin *per centum* — "per hundred." So 45% simply means 45 out of 100, or the fraction 45/100, or the decimal 0.45. That's it. Every percentage problem in every exam is, at its core, a fraction problem wearing a costume.

The three forms of the same value:

The single most important skill in this chapter is fraction conversion. If you see 12.5% and your brain instantly says "1/8," you will solve problems three times faster than the candidate next to you.

🧠 Active Recall: Without looking above — what is 40% as a fraction? What fraction is 0.75? Write both down before reading on.

2. The Fraction Table You Must Memorise

Advanced exams (CAT, PO Mains) reward speed, and speed comes from this table:

Example: What is 14.28% of 700? Don't multiply 0.1428 × 700. Just compute 700/7 = 100. Done in two seconds.

🧠 Active Recall: Close the table. What is 37.5% as a fraction? (Hint: it's 3/8 — half of 75%.) What is 62.5%? (5/8.)

3. Percentage of a Number & Percentage Change

Basic formula:

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

Percentage change is where exams live:

$$\%\text{ change} = \frac{\text{New value} - \text{Old value}}{\text{Old value}} \times 100$$

The denominator is always the original (old) value — this is the #1 trap. If a price rises from ₹80 to ₹100, the increase is 20/80 = 25%, not 20/100 = 20%.

Reverse question: If a value increases by 25% to become 500, what was it originally? New value = 125% of old = 5/4 × old. So old = 500 × 4/5 = 400. Learn to move backwards fluently — Tier-II and PO Mains love reverse problems.

🧠 Active Recall: A salary rises from ₹4,000 to ₹4,600. What is the percentage increase? (Answer: 600/4000 = 15%.)

4. Successive Percentage Change — The Heart of This Chapter

When two changes a% and b% happen one after another, the net change is:

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

(Use negative signs for decreases.)

Example: +20% then −10%: Net = 20 − 10 + (20 × −10)/100 = 10 − 2 = +8%.

Notice: +20% then −20% gives 20 − 20 − 4 = −4%, not zero. This is the jacket illusion from the hook — equal up-and-down changes always end in a net *decrease* of (a²/100)%.

Fraction method (often faster): +25% means multiply by 5/4; −20% means multiply by 4/5. Net multiplier = 5/4 × 4/5 = 1 → no change. Beautiful when fractions cancel.

🧠 Active Recall: Population increases 10% in year 1 and 20% in year 2. Net change? (Answer: 10 + 20 + 2 = 32%.)

5. Key Application Areas

A. Population / Depreciation: If a population grows r% per year, after n years: $P_n = P_0(1 + r/100)^n$. Depreciation uses $(1 - r/100)^n$. For 2 years, expand: 2 years at 10% growth → multiplier 1.1² = 1.21 → 21% total growth.

B. Exam marks: If passing requires 40% and a student scores 180 but fails by 20 marks, the pass mark is 200, so maximum marks = 200 × 100/40 = 500.

C. Percentage points vs. percent: If interest rates rise from 5% to 7%, that is a rise of 2 percentage points but a 40% increase (2/5). CSAT and CAT deliberately confuse these two phrases. Read the wording twice.

D. Data Interpretation: "A is what percent more than B?" → (A − B)/B × 100. "B is what percent less than A?" → (A − B)/A × 100. Same numbers, different base, different answer. The base is whatever comes after "than."

🧠 Active Recall: A = 120, B = 100. A is what % more than B? B is what % less than A? (Answers: 20% and 16.66% — different bases!)

6. Mnemonic: "BASE Follows THAN"

> "The BASE always follows THAN."

Whenever a question says "X is what percent more/less than Y," the value after *than* (Y) is your denominator. Tattoo this on your brain — it kills the most common error in percentage DI questions.

Second mnemonic for successive change: "SUM plus PRODUCT over hundred" — a + b + ab/100. Say it as: *"Sum it, then tip it."*

7. Classic Examiner Traps

1. Up-down illusion: +x% then −x% is never zero; it's −x²/100 %.
2. Wrong base: Using the new value as denominator in percentage change.
3. Percentage points vs. percent: 5% → 6% is +1 point, +20%.
4. "More than" vs. "less than": different denominators (see mnemonic).
5. Mixing ratios: If A:B = 3:4, then B is 33.33% more than A — not 25%. (Difference 1 over base 3.)

Summary Table

📌 Review this chapter after 2 days. Redo the fraction table from memory, solve one successive-change problem, and re-test yourself on the "more than / less than" base rule. Spaced repetition is what converts this chapter from "understood" to "unshakeable."