Percentages

Percentages

Percentages represent parts per hundred and are essential for many real-world applications. Here's what you need to know:

Basic Concepts

A percentage is a ratio expressed as a fraction of 100:

$$n\% = \frac{n}{100}$$

Converting Between Forms

• Percentage to Decimal: Divide by 100 (move decimal point 2 places left)
• Example: 75% = 75 ÷ 100 = 0.75
• Decimal to Percentage: Multiply by 100 (move decimal point 2 places right)
• Example: 0.35 = 0.35 × 100 = 35%
• Fraction to Percentage: Divide numerator by denominator, then multiply by 100
• Example: $$\frac{3}{4} = 0.75 = 75\%$$
• Percentage to Fraction: Write as a fraction with denominator 100, then simplify
• Example: 25% = $$\frac{25}{100} = \frac{1}{4}$$

Key Percentage Calculations

Finding a Percentage of a Number

To find n% of x: $$\frac{n}{100} \times x$$ or $$n\% \times x$$

Example: 15% of 80 = 0.15 × 80 = 12

Finding What Percentage One Number is of Another

To find what percentage x is of y: $$\frac{x}{y} \times 100\%$$

Example: What percentage is 18 of 72? $$\frac{18}{72} \times 100\% = 25\%$$

Finding the Original Number Given a Percentage

If n% of x is y, then $$x = \frac{y}{n\%} = \frac{y \times 100}{n}$$

Example: 30% of what number is 21? $$x = \frac{21}{0.3} = 70$$

Percentage Change

Increase

If a value increases from x to y, the percentage increase is:

$$\frac{y - x}{x} \times 100\%$$

Example: A price increases from $80 to $100. The percentage increase is $$\frac{100 - 80}{80} \times 100\% = 25\%$$

Decrease

If a value decreases from x to y, the percentage decrease is:

$$\frac{x - y}{x} \times 100\%$$

Example: A price decreases from $50 to $40. The percentage decrease is $$\frac{50 - 40}{50} \times 100\% = 20\%$$

Successive Percentage Changes

When applying multiple percentage changes, multiply the factors rather than adding percentages:

Example: A 10% increase followed by a 10% decrease is NOT a 0% change:

$100 × 1.10 = $110, then $110 × 0.90 = $99, which is a 1% overall decrease

Common Applications

Discounts and Sales Tax

For a discount of n%: Final price = Original price × (1 - n%)

Example: 30% off $80: $80 × (1 - 0.3) = $80 × 0.7 = $56

For sales tax of n%: Final price = Original price × (1 + n%)

Example: 8% tax on $50: $50 × (1 + 0.08) = $50 × 1.08 = $54

Interest

Simple interest: Interest = Principal × Rate × Time

Example: $1000 at 5% for 3 years: $1000 × 0.05 × 3 = $150

Compound interest: Final amount = Principal × (1 + Rate)^Time

Example: $1000 at 5% compounded annually for 3 years: $1000 × (1 + 0.05)^3 = $1000 × 1.157625 = $1157.63

Common Pitfalls and Tips

• Remember that percentage points and percentages are different (e.g., an increase from 5% to 7% is a 2 percentage point increase but a 40% increase)
• When calculating percentage change, always divide by the original amount
• Be careful with successive percentage changes—they don't add
• For mental math, know common percentages: 50% = 1/2, 25% = 1/4, 20% = 1/5, 10% = 1/10