Complete Guide to Percentages
1. What is a Percentage?
A percentage is a number expressed as a fraction of 100. The symbol "%" represents percentage. For example, 25% means 25 out of 100, or 25/100 = 0.25.
2. Converting Between Percentages, Decimals, and Fractions
Decimal to Percentage
Multiply the decimal by 100 and add the % symbol.
Example: Convert 0.75 to a percentage.
Solution: 0.75 × 100 = 75%
Percentage to Decimal
Divide the percentage by 100 (or move the decimal point two places to the left).
Example: Convert 32% to a decimal.
Solution: 32 ÷ 100 = 0.32
Fraction to Percentage
Divide the numerator by the denominator and multiply by 100.
Example: Convert 3/8 to a percentage.
Solution: (3 ÷ 8) × 100 = 0.375 × 100 = 37.5%
Percentage to Fraction
Write the percentage over 100 and simplify.
Example: Convert 45% to a fraction.
Solution: 45% = 45/100 = 9/20 (simplified)
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/4 | 0.25 | 25% |
| 1/3 | 0.333... | 33.33...% |
| 1/2 | 0.5 | 50% |
| 2/3 | 0.666... | 66.66...% |
| 3/4 | 0.75 | 75% |
3. Common Percentage Calculations
Finding the Percentage of a Number
Example: Find 15% of 80.
Solution: 15% of 80 = (15/100) × 80 = 0.15 × 80 = 12
Finding a Number When a Percentage is Known
Example: 24 is 30% of what number?
Solution: If 24 is 30% of X, then X = (24 / 30) × 100 = 80
Finding What Percentage One Number is of Another
Example: What percentage of 75 is 15?
Solution: Percentage = (15 / 75) × 100% = 0.2 × 100% = 20%
4. Percentage Increase and Decrease
Percentage Increase
Example: A shirt's price increases from $40 to $50. What is the percentage increase?
Solution: % Increase = ((50 - 40) / 40) × 100% = (10 / 40) × 100% = 0.25 × 100% = 25%
Percentage Decrease
Example: A car's value depreciates from $20,000 to $17,000. What is the percentage decrease?
Solution: % Decrease = ((20,000 - 17,000) / 20,000) × 100% = (3,000 / 20,000) × 100% = 0.15 × 100% = 15%
Finding the Final Value After a Percentage Change
For decrease: New Value = Original Value × (1 - Percentage/100)
Example 1: Find the new price of a $80 item after a 25% increase.
Solution: New Price = $80 × (1 + 25/100) = $80 × 1.25 = $100
Example 2: Find the sale price of a $60 shirt after a 30% discount.
Solution: Sale Price = $60 × (1 - 30/100) = $60 × 0.7 = $42
Finding the Original Value Before a Percentage Change
For decrease: Original Value = New Value / (1 - Percentage/100)
Example 1: After a 20% increase, the price of a laptop is $960. What was the original price?
Solution: Original Price = $960 / (1 + 20/100) = $960 / 1.2 = $800
Example 2: After a 15% decrease, the weight of a package is 17 kg. What was the original weight?
Solution: Original Weight = 17 kg / (1 - 15/100) = 17 kg / 0.85 = 20 kg
5. Successive Percentage Changes
When multiple percentage changes occur successively, they don't simply add up.
Example: A stock increases by 20% and then decreases by 10%. What is the overall percentage change?
Solution:
- Find the multiplier for each change: 1.2 for 20% increase, 0.9 for 10% decrease
- Multiply these factors: 1.2 × 0.9 = 1.08
- Subtract 1 and multiply by 100%: (1.08 - 1) × 100% = 8%
The stock experienced an overall increase of 8%, not 10% (20% - 10%).
Alternative approach: If we start with a value of 100:
- After 20% increase: 100 × 1.2 = 120
- After 10% decrease: 120 × 0.9 = 108
- Overall change: (108 - 100) / 100 × 100% = 8%
6. Percentage Points vs. Percentages
A percentage point (pp) is the arithmetic difference between two percentages.
Example: If interest rates increase from 5% to 8%:
This is an increase of 3 percentage points (8% - 5% = 3pp)
But it's a 60% increase in the rate itself ((8% - 5%) / 5% × 100% = 60%)
7. Compound Interest (Percentage in Finance)
Where:
A = Final amount
P = Principal (initial investment)
r = Annual interest rate (in decimal)
n = Number of times interest is compounded per year
t = Time in years
Example: Calculate the final amount when $1,000 is invested for 3 years at 5% annual interest, compounded quarterly.
Solution:
- P = $1,000, r = 0.05, n = 4, t = 3
- A = $1,000(1 + 0.05/4)^(4×3)
- A = $1,000(1 + 0.0125)^12
- A = $1,000(1.0125)^12
- A = $1,000 × 1.1605 = $1,160.50
8. Markup and Markdown (Retail Percentages)
Markup (based on cost)
Example: A retailer buys a product for $40 and sells it for $60. What is the markup percentage?
Solution: Markup % = ((60 - 40) / 40) × 100% = (20 / 40) × 100% = 50%
Profit Margin (based on selling price)
Example: Using the same values: $40 cost, $60 selling price.
Solution: Profit Margin % = ((60 - 40) / 60) × 100% = (20 / 60) × 100% = 33.33%
Margin % = Markup % / (100% + Markup %)
Markup % = Margin % / (100% - Margin %)
9. Percentage Error and Accuracy
Example: A measurement gives 52 cm when the actual length is 50 cm. Calculate the percentage error.
Solution: % Error = (|52 - 50| / |50|) × 100% = (2 / 50) × 100% = 4%
10. Mixture Problems (Weighted Averages)
Where V represents volume and C represents concentration
Example: If 200 ml of 25% alcohol solution is mixed with 300 ml of 40% alcohol solution, what is the concentration of the final mixture?
Solution:
- V₁ = 200 ml, C₁ = 25%, V₂ = 300 ml, C₂ = 40%
- Final % = (200 × 25% + 300 × 40%) / (200 + 300)
- Final % = (5,000 + 12,000) / 500
- Final % = 17,000 / 500 = 34%
Test Your Understanding: Percentage Quiz
Question 1: Convert 3/5 to a percentage.
Question 2: What is 35% of 240?
Question 3: A shirt originally priced at $50 is on sale for $40. What is the percentage discount?
Question 4: If the price of an item increases by 20% and then decreases by 20%, the final price is:
Question 5: 42 is what percentage of 168?