**Percentage Basics — 48-Hour Exam Crash Guide**

Percentage Basics — 48-Hour Exam Crash Guide

What Is This?

Percentage means per hundred. It’s a way to express a number as a fraction of 100.
- Exam role: Percentages appear in every quantitative exam—aptitude tests, finance certifications, job interviews, and school exams.
- Question types: - What is 15% of 240? - If a price rises by 20%, what is the new price? - A test score increased from 70% to 85%. What is the percentage increase?

Why It Matters

• Exams that test it: SAT, ACT, GRE, GMAT, banking exams (IBPS, SBI), job aptitude tests, school finals.
• Frequency: 80%+ of quantitative sections include at least one percentage question.
• Marks: Typically 1–3 marks per question, but errors compound in multi-step problems.
• Skill tested: Your ability to convert, compare, and manipulate proportions under time pressure.

Core Concepts

You must own these before attempting any question:

1. Percentage as a fraction: 15% = 15/100 = 0.15.

2. Examiner trap: Confusing percentage (15%) with decimal (0.15). Always convert first.

3. Base value: The number you’re taking a percentage of.

4. 20% of 50 → 50 is the base.

5. Key rule: The base is always the denominator in the fraction.

6. Percentage change: The difference between two values, expressed as a percentage of the original.

7. Formula: (New Value – Original Value) / Original Value × 100%.

8. Examiner trap: Using the wrong base (e.g., dividing by the new value instead of the original).

9. Reverse percentages: Finding the original value after a percentage change.

10. After a 25% increase, the price is $120. What was the original price?
11. Key rule: New Value = Original × (1 ± % Change).

The Rule-Book (How It Works)

1. Basic Percentage Calculation

Rule: Percentage of a number = (Percentage / 100) × Number.
- Example: 12% of 250 = (12/100) × 250 = 30.

Sub-rules: - Increase: New Value = Original × (1 + Percentage/100).
- Decrease: New Value = Original × (1 – Percentage/100).
- Edge case: 0% of anything = 0. 100% of a number = the number itself.

Mnemonic: "% means per hundred, so divide by 100 first."

2. Percentage Change

Rule: Percentage Change = (Change / Original) × 100%.
- Change = New Value – Original Value.
- Examiner trap: If the question says "increased by 20%", the change is +20%. If it says "increased to 120%", the new value is 120% of the original.

Sub-rules: - Successive changes: Multiply the factors, don’t add percentages.
- A price rises 10%, then falls 10% → Net change = (1.10 × 0.90) – 1 = –1% (a 1% loss).
- Negative change: A decrease of 15% = –15%.

3. Reverse Percentages

Rule: Original Value = New Value / (1 ± Percentage Change).
- Key signal: The question gives the new value and asks for the original.
- Examiner trap: Forgetting to divide by (1 + %) for increases or (1 – %) for decreases.

Example: - After a 20% increase, the price is $180. Original price? - Original = 180 / 1.20 = $150.

Exam / Job / Audit Weighting

Difficulty Level

Beginner (but examiners add twists to trap careless students).

Must-Know Rules, Formulas, Standards

1. Basic Percentage:

2. % of X = (Percentage / 100) × X.

3. Percentage Change:

4. (New – Original) / Original × 100%.

5. Reverse Percentage:

6. Original = New / (1 ± % Change).

Worked Examples (Step-by-Step)

Example 1 (Easy)

Question: What is 30% of 150?

Steps: 1. Convert 30% to decimal: 30% = 0.30.
2. Multiply by the base: 0.30 × 150 = 45.

Answer: 45.
Rule applied: % of X = (Percentage / 100) × X.

Example 2 (Medium)

Question: A shirt costs $40. After a 15% discount, what is the new price?

Steps: 1. Calculate discount amount: 15% of $40 = 0.15 × 40 = $6.
2. Subtract from original: $40 – $6 = $34.

Shortcut: New Price = Original × (1 – % Discount) = 40 × 0.85 = $34.

Answer: $34.
Rule applied: New Value = Original × (1 – % Decrease).

Example 3 (Hard)

Question: After a 25% increase, a salary is $3,750. What was the original salary?

Steps: 1. Recognize this is a reverse percentage problem.
2. New Salary = Original × (1 + 25%) = Original × 1.25.
3. Original = New Salary / 1.25 = 3,750 / 1.25 = $3,000.

Answer: $3,000.
Rule applied: Original = New / (1 + % Increase).

Common Exam Traps & Mistakes

Shortcut Strategies & Exam Hacks

1. Mental math for 10% and 1%:
2. 10% of X = X / 10.
3. 1% of X = X / 100.

4. Example: 15% of 240 = (10% × 240) + (5% × 240) = 24 + 12 = 36.

5. Reverse percentages trick:

6. If a value increases by x%, the original is New / (1 + x/100).

7. If it decreases by x%, the original is New / (1 – x/100).

8. Successive changes:

9. Multiply the factors: (1 + % Increase) × (1 – % Decrease).

10. Example: Price rises 20%, then falls 10% → 1.20 × 0.90 = 1.08 → 8% net increase.

11. Signal words:

12. "Of" → Multiply (20% of 50).
13. "Increased by" → Add to 100% (1 + 20% = 1.20).
14. "Decreased to" → New value is given (price decreased to 80% of original).

Question-Type Taxonomy

Practice Set (MCQs)

Question 1

What is 18% of 250? A) 45 B) 40 C) 36 D) 50

Correct Answer: A) 45 Explanation: 18% of 250 = (18/100) × 250 = 45.
Why distractors are tempting: - B) 40 → 16% of 250 (misread the percentage).
- C) 36 → 18% of 200 (wrong base).
- D) 50 → 20% of 250 (rounded up).

Question 2

A laptop’s price decreased by 15% to $850. What was the original price? A) $1,000 B) $977.50 C) $1,050 D) $950

Correct Answer: A) $1,000 Explanation: Original = New / (1 – % Decrease) = 850 / 0.85 = $1,000.
Why distractors are tempting: - B) $977.50 → Incorrectly divided by 0.875 (misapplied the percentage).
- C) $1,050 → Added 15% to $850 (reverse logic).
- D) $950 → Guessed a round number.

Question 3

If a number increases by 20% and then decreases by 20%, what is the net change? A) 0% B) 4% decrease C) 4% increase D) 2% decrease

Correct Answer: B) 4% decrease Explanation: Net change = (1.20 × 0.80) – 1 = –0.04 → 4% decrease.
Why distractors are tempting: - A) 0% → Assumed increases and decreases cancel out.
- C) 4% increase → Misapplied the sign.
- D) 2% decrease → Incorrect multiplication (1.20 × 0.80 = 0.96, not 0.98).

Question 4

A student scored 45 out of 60 in a test. What percentage did they score? A) 60% B) 75% C) 80% D) 90%

Correct Answer: B) 75% Explanation: (45 / 60) × 100% = 75%.
Why distractors are tempting: - A) 60% → Divided 45 by 75 (misread the denominator).
- C) 80% → Rounded 45/60 to 4/5 = 80% (wrong simplification).
- D) 90% → Divided 60 by 45 (reversed the fraction).

Question 5

After a 10% increase, a population is 55,000. What was the original population? A) 50,000 B) 52,500 C) 55,000 D) 60,500

Correct Answer: A) 50,000 Explanation: Original = New / (1 + % Increase) = 55,000 / 1.10 = 50,000.
Why distractors are tempting: - B) 52,500 → Divided by 1.05 (wrong percentage).
- C) 55,000 → Assumed no change.
- D) 60,500 → Added 10% to 55,000 (reverse logic).

30-Second Cheat Sheet

1. % = per hundred → Always divide by 100 first.
2. Base is the denominator → X% of Y = (X/100) × Y.
3. Percentage change = (New – Original) / Original × 100%.
4. Reverse percentages → Original = New / (1 ± % Change).
5. Successive changes → Multiply factors, don’t add percentages.
6. Signal words:
7. "Of" → Multiply.
8. "Increased by" → Add to 100%.
9. "Decreased to" → New value given.
10. 10% trick → Move decimal one place left (10% of 250 = 25).

Learning Path

1. Day 1 (0–12 hours):
2. Master the core concepts (percentage as a fraction, base value, percentage change).
3. Memorize the 3 must-know formulas.

4. Work through the easy and medium examples.

5. Day 1 (12–24 hours):

6. Drill reverse percentages and successive changes.
7. Practice mental math shortcuts (10%, 1%, 5%).

8. Attempt the MCQ practice set (timed: 1 min per question).

9. Day 2 (24–36 hours):

10. Review common traps and exam hacks.
11. Rework hard examples without notes.

12. Simulate timed conditions (e.g., 5 questions in 5 minutes).

13. Day 2 (36–48 hours):

14. Take a mock test with mixed percentage questions.
15. Focus on speed and accuracy (exams penalize careless errors).
16. Memorize the 30-second cheat sheet.

Related Topics

1. Fractions and Decimals – Percentages are fractions/decimals in disguise (15% = 0.15 = 3/20).
2. Ratio and Proportion – Percentages are ratios with a base of 100 (2:5 = 40%).
3. Profit and Loss – Uses percentage change to calculate gains/losses (20% profit on cost price).