Percentage
1. Meaning of Percentage
Percent means 'per hundred' or 'out of 100.' The symbol is %.
If you score 85% in a test, you got 85 marks out of every 100 marks.
Percent is a fraction with denominator 100: 85% = 85/100.
Worked Example: In a school, 65 students out of 250 have bicycles. What percentage is this?
Fraction = 65/250 = 26/100 = 26%.
Common Mistake: Writing the fraction as 65/250 = 65%. Wrong! You must convert to denominator 100 first.
2. Converting Fractions to Percentages
Method: Multiply the fraction by 100%.
| Fraction | Calculation | Percentage |
|---|---|---|
| 3/4 | (3/4) x 100% = 75% | 75% |
| 2/5 | (2/5) x 100% = 40% | 40% |
| 7/20 | (7/20) x 100% = 35% | 35% |
| 1/3 | (1/3) x 100% = 33 1/3% | 33 1/3% |
Worked Example: Convert 5/8 to a percentage.
(5/8) x 100% = 500/8 % = 125/2 % = 62.5%.
3. Converting Percentages to Fractions
Method: Write the percentage as a fraction with denominator 100, then simplify.
| Percentage | Fraction | Simplified |
|---|---|---|
| 25% | 25/100 | 1/4 |
| 60% | 60/100 | 3/5 |
| 12.5% | 125/1000 | 1/8 |
| 33 1/3% | (100/3)/100 = 100/300 | 1/3 |
Exam Focus (2 marks): 'Express 37 1/2% as a fraction in simplest form.'
37 1/2% = (75/2)% = (75/2) / 100 = 75/200 = 3/8.
4. Converting Decimals to Percentages
Method: Multiply the decimal by 100%. Equivalent to shifting the decimal point two places to the right.
| Decimal | Percentage |
|---|---|
| 0.35 | 35% |
| 0.07 | 7% |
| 1.25 | 125% |
| 0.005 | 0.5% |
Worked Example: Convert 0.625 to a percentage.
0.625 x 100% = 62.5%.
5. Converting Percentages to Decimals
Method: Divide the percentage by 100. Equivalent to shifting the decimal point two places to the left.
| Percentage | Decimal |
|---|---|
| 45% | 0.45 |
| 8% | 0.08 |
| 150% | 1.5 |
| 2.5% | 0.025 |
Common Mistake: Writing 5% as 0.5. Correct: 5% = 5/100 = 0.05.
6. Simple Applications of Percentage
Finding a Percentage of a Quantity
Worked Example: Find 15% of 200.
15% of 200 = (15/100) x 200 = 30.
Finding Quantity when Percentage is Given
Worked Example: 20% of a number is 50. Find the number.
20% of x = 50 => (20/100) x X = 50 => x = (50 x 100)/20 = 250.
Percentage of One Quantity of Another
Worked Example: What percentage is 30 of 120?
(30/120) x 100% = 25%.
Real-Life Application
Exam Focus (4 marks): 'Ravi scored 68 marks out of 80 in Maths and 72 out of 100 in Science. In which subject did he perform better?'
Maths: (68/80) x 100% = 85%. Science: (72/100) x 100% = 72%.
He performed better in Maths.
Common Mistake: Comparing raw marks (68 vs 72) and saying Science is better. Always compare percentages when the totals are different.
7. Comparison Table: Conversions
| From | To | Operation |
|---|---|---|
| Fraction | Percentage | Multiply by 100% |
| Percentage | Fraction | Divide by 100, simplify |
| Decimal | Percentage | Multiply by 100% |
| Percentage | Decimal | Divide by 100 |
8. Self-Test
- Convert to percentage: (a) 7/10 (b) 3/8 (c) 0.45.
- Convert to fraction in simplest form: (a) 75% (b) 62.5% (c) 16 2/3%.
- Convert to decimal: (a) 32% (b) 4.5%.
- Find: (a) 30% of 450 (b) 12 1/2% of 80.
- 25% of a number is 60. Find the number.
- What percentage is 56 of 70?
- In an exam, Amit scored 84 out of 120. What is his percentage score?
- A shirt costs 800. It is sold at a 15% discount. Find the discounted price.
9. Answers to Self-Test
- (a) 70% (b) 37.5% (c) 45%.
- (a) 3/4 (b) 5/8 (c) 1/6.
- (a) 0.32 (b) 0.045.
- (a) (30/100) x 450 = 135 (b) (12.5/100) x 80 = 10.
- (25/100) x X = 60 => X = (60 x 100)/25 = 240.
- (56/70) x 100% = 80%.
- (84/120) x 100% = 70%.
- Discount = 15% of 800 = 120. Discounted price = 800 - 120 = 680.
