Percent and Percentage
1. Meaning of Percent
'Per cent' means 'out of 100' or 'for every hundred.' The symbol is %.
x% = x/100 (x divided by 100).
'Percentage is a way of expressing a number as a FRACTION of 100. It makes comparison EASY.'
Conversions:
| Convert | Method | Example |
|---|---|---|
| Fraction to % | Multiply by 100 | 3/5 = 3/5 × 100 = 60% |
| Decimal to % | Multiply by 100 | 0.45 = 0.45 × 100 = 45% |
| % to fraction | Divide by 100 | 75% = 75/100 = 3/4 |
| % to decimal | Divide by 100 | 12.5% = 12.5/100 = 0.125 |
| Ratio to % | Convert to fraction, then multiply by 100 | 3 : 4 = 3/4 × 100 = 75% |
| % to ratio | Write as fraction, simplify | 40% = 40/100 = 2 : 5 |
2. Finding Percentage of a Given Number
Worked Example: Find 15% of 250.
15% of 250 = (15/100) × 250 = 3750/100 = 37.5
Worked Example: A student scored 42 out of 50 in a test. Find the percentage.
Percentage = (42/50) × 100 = 84%
3. Expressing One Quantity as a Percentage of Another
Formula: (First quantity / Second quantity) × 100%
Worked Example: What percent of 80 is 20?
(20/80) × 100 = 25%
Worked Example: In a class of 45 students, 27 are girls. What percentage are boys?
Boys = 45 — 27 = 18 Percentage of boys = (18/45) × 100 = 40%
4. Percentage Increase and Decrease
Formula: Percentage increase = (Increase in value / Original value) × 100% Percentage decrease = (Decrease in value / Original value) × 100%
Worked Example: The price of a book increased from Rs 120 to Rs 150. Find the percentage increase.
Increase = 150 — 120 = 30 Percentage increase = (30/120) × 100 = 25%
Worked Example: A shirt costing Rs 800 is now available for Rs 680. Find the percentage decrease.
Decrease = 800 — 680 = 120 Percentage decrease = (120/800) × 100 = 15%
5. Finding Original Value After Increase/Decrease
Worked Example: After a 12% increase, the population of a town is 56,000. What was the original population?
Let original population = P. P + 12% of P = P(1 + 12/100) = P × 1.12 = 56000 P = 56000/1.12 = 50000
Worked Example: A laptop is sold at a 15% discount for Rs 34,000. Find its marked price.
Let marked price = M. M — 15% of M = M × 0.85 = 34000 M = 34000/0.85 = 40000
6. Applications of Percentage
Passing Percentage
Worked Example: A student scored 78 marks and failed by 12 marks. If the passing percentage is 45%, find the maximum marks.
Passing marks = 78 + 12 = 90 45% of Max marks = 90 Max marks = 90 × 100/45 = 200
Percentage of Pure Substance
Worked Example: 15 g of salt is dissolved in 285 g of water. Find the percentage of salt in the solution.
Total solution = 15 + 285 = 300 g Percentage of salt = (15/300) × 100 = 5%
Income and Expenditure
Worked Example: A person spends 70% of his income. If his income increases by 20% and his expenditure increases by 10%, find the percentage change in savings.
Let original income = Rs 100. Original expenditure = Rs 70. Original savings = Rs 30. New income = 100 × 1.20 = Rs 120. New expenditure = 70 × 1.10 = Rs 77. New savings = 120 — 77 = Rs 43. Increase in savings = 43 — 30 = Rs 13. Percentage increase in savings = (13/30) × 100 = 43⅓%
Common Mistakes and Fixes
| Mistake | Fix |
|---|---|
| 'x% of y = y% of x is FALSE' | It IS true: x% of y = (x/100) × y = xy/100 = (y/100) × x = y% of x |
| 'Percentage increase is calculated on the NEW value' | Always calculate on the ORIGINAL value |
| 'Discount of 20% + 30% = 50%' | SUCCESSIVE discounts are NOT additive. Final = Original × 0.8 × 0.7 |
| '100% means nothing' | 100% means the WHOLE. 100% of a number = the number itself |
ICSE Exam Focus (4–6 marks)
- 2-mark questions: Convert fractions/decimals to percentages and vice versa
- 3-mark questions: Simple applications (marks, expenditure)
- 4-mark questions: Percentage increase/decrease word problems
- 6-mark questions: Multi-step applications (population growth, successive discounts)
Self-Test
Q1. Convert 7/8 to percentage. A1. 7/8 × 100 = 700/8 = 87.5%.
Q2. Find 12½% of 360. A2. 12½% = 12.5% = 12.5/100 = 1/8. 1/8 × 360 = 45.
Q3. What percent of 125 is 30? A3. (30/125) × 100 = 3000/125 = 24%.
Q4. A man spends 75% of his income. If his income increases by 30%, by what percentage should his expenditure increase so that his savings remain the same? A4. Let income = 100. Savings = 25, Expenditure = 75. New income = 130. Need new savings = 25. New expenditure = 105. Increase in expenditure = 30. Percentage increase = (30/75) × 100 = 40%.
Q5. The value of a car depreciates 12% every year. If its present value is Rs 5,50,000, what was its value one year ago? A5. Let value one year ago = V. V(1 — 12/100) = 550000. V × 0.88 = 550000. V = 550000/0.88 = Rs 6,25,000.
Q6. In an election, a candidate gets 65% of the valid votes. If 15% of the total votes are invalid and the total votes are 20,000, how many votes did the winning candidate get? A6. Valid votes = 85% of 20000 = 17000. Winning candidate's votes = 65% of 17000 = 11050.
