Percentage — Study Notes for HSSC CET
Overview
Percentage is one of the most fundamental topics in Quantitative Ability and forms the backbone for several other topics like Profit & Loss, Simple & Compound Interest, and Data Interpretation. In HSSC CET, you can expect 2–4 direct or indirect questions on percentages, making it a high-weightage area.
The word "percent" comes from Latin "per centum" meaning "per hundred." A percentage is simply a way of expressing a number as a fraction of 100. Mastering percentage calculations helps you solve problems faster, especially when combined with ratio, proportion, and commercial mathematics.
For HSSC CET success, you must be fluent in converting between fractions, decimals, and percentages, and apply percentage concepts to real-world problems like population change, price variations, and examination scores.
Key Concepts
- Basic Definition: x% means x out of 100, written as x/100 or 0.01x in decimal form.
- Conversion Rule: To convert a fraction to percentage, multiply by 100. To convert percentage to fraction, divide by 100.
- Percentage of a Number: x% of N = (x × N)/100. This is the most frequently used calculation.
- Percentage Change: When a quantity changes from old value to new value, percentage change = [(New − Old)/Old] × 100.
- Successive Percentage Change: When two successive changes of a% and b% occur, the net effect = a + b + (ab/100)%.
- Base Value Matters: "A is what percent of B" means (A/B) × 100, while "A is what percent more than B" means [(A−B)/B] × 100. The denominator (base) decides the answer.
- Reverse Percentage: If after x% increase the value is V, then original = V × 100/(100+x).
Formulas / Key Facts
| Formula | Context |
|---|---|
| x% of N = xN/100 | Finding percentage of any number |
| Fraction to % = Fraction × 100 | Converting 3/4 to 75% |
| % to Decimal = %/100 | Converting 25% to 0.25 |
| % Increase = [(New−Old)/Old] × 100 | Price rose from ₹50 to ₹60 → 20% increase |
| % Decrease = [(Old−New)/Old] × 100 | Price fell from ₹60 to ₹48 → 20% decrease |
| Net % change = a + b + ab/100 | Two successive changes of a% and b% |
| If A is x% more than B, then B is less than A by [x/(100+x)] × 100% | Reverse comparison |
| If A is x% less than B, then B is more than A by [x/(100−x)] × 100% | Reverse comparison |
Must-Remember Fraction-Percentage Equivalents:
- 1/2 = 50%, 1/3 = 33.33%, 1/4 = 25%, 1/5 = 20%
- 1/6 = 16.67%, 1/8 = 12.5%, 1/10 = 10%, 1/12 = 8.33%
- 2/3 = 66.67%, 3/4 = 75%, 4/5 = 80%, 5/6 = 83.33%
Worked Examples
Example 1: Basic Percentage Calculation Find 35% of 240.
Solution: 35% of 240 = (35 × 240)/100 = 8400/100 = 84
Example 2: Percentage Change The price of rice increased from ₹40/kg to ₹52/kg. Find the percentage increase.
Solution: Increase = 52 − 40 = ₹12 % Increase = (12/40) × 100 = (12 × 100)/40 = 1200/40 = 30%
Example 3: Successive Percentage Change A shopkeeper increases prices by 20% and then gives a 10% discount. What is the net percentage change in price?
Solution: Using formula: Net change = a + b + ab/100 Here a = +20, b = −10 Net change = 20 + (−10) + (20 × −10)/100 = 20 − 10 − 2 = +8% (Net increase of 8%)
Example 4: Reverse Percentage After a 25% increase, the salary of an employee became ₹50,000. What was the original salary?
Solution: Let original salary = X X + 25% of X = 50,000 X × (125/100) = 50,000 X = 50,000 × 100/125 X = 5,000,000/125 X = ₹40,000
Example 5: Comparison Between Two Quantities If A's income is 20% more than B's income, then B's income is what percent less than A's income?
Solution: Using formula: If A is x% more than B, then B is less than A by [x/(100+x)] × 100% = [20/(100+20)] × 100 = (20/120) × 100 = 2000/120 = 16.67% (or 50/3%)
Common Mistakes
- Wrong base selection → Students calculate "A is what % of B" but use A in denominator instead of B. Always identify what the "base" or "reference" quantity is — it goes in the denominator.
- Sign errors in successive changes → When one change is increase (+) and another is decrease (−), students forget to use negative sign in the formula. Always assign + for increase and − for decrease.
- Confusing "more than" with "of" → "A is 25% more than B" is different from "A is 25% of B." The first means A = 1.25B; the second means A = 0.25B.
- Applying percentage change on new value → For reverse problems, students mistakenly apply percentage on the final value instead of working backwards. Use the formula: Original = Final × 100/(100 ± x).
- Not memorizing fraction equivalents → Calculating 1/8 as percentage during exam wastes time. Memorize common fractions to speed up calculations.
Quick Reference
- x% of N = xN/100; always remember the base is 100.
- % Change = (Difference/Original) × 100 — Original is always the base.
- Successive % change: a + b + ab/100 (use signs carefully).
- 1/4 = 25%, 1/5 = 20%, 1/8 = 12.5%, 1/3 = 33.33% — memorize these.
- "More than" problems: denominator is the smaller quantity.
- For reverse calculation: Original = Final × 100/(100 ± change%).