Understanding Percentages

• A percentage represents a number as a fraction of 100
• The percent sign "%" denotes percentage

Core Concept

• Percentage means "per hundred"
• Percentages represent a ratio with "100" as the denominator

Calculating Percentages

• Use the formula (A / B) × 100% to determine what percentage a number 'A' is of another number 'B'

Converting Percentages to Decimals

• A percentage becomes a decimal when divided by 100
• For example, 75% is 0.75

Converting Decimals to Percentages

• Convert a decimal to a percentage by multiplying by 100
• For example, 0.25 becomes 25%

Converting Percentages to Fractions

• Percentages are converted to fractions by expressing the percentage as a fraction with 100 as the denominator, simplifying if needed
• For example, 60% is 3 / 5

Converting Fractions to Percentages

• Fractions are converted to percentages when multiplied by 100%
• For example, 1 / 4 becomes 25%

Finding a Percentage of a Quantity

• Multiply the quantity by the decimal or fraction equivalent of the percentage
• Finding 20% of 50 involves calculating 0.20 × 50, resulting in 10

Percentage Increase

• Calculate using: [(New Value - Original Value) / Original Value] × 100%
• As an example, a price increase from $20 to $25 is a 25% increase

Percentage Decrease

• Calculate using: [(Original Value - New Value) / Original Value] × 100%
• As an example, a price decrease from $25 to $20 is a 20% decrease

Working with Percentage Change

• Calculate the new value directly when a value changes by a percentage
• The new value after increasing x% is Original Value × (1 + x/100)
• The new value after decreasing x% is Original Value × (1 - x/100)

Percentage Error

• Represents the difference between measured and actual values
• Determined by: [(|Approximate Value - Exact Value|) / |Exact Value|] × 100%

Simple Interest

• Calculated solely on the principal amount
• Calculated using: (Principal × Rate × Time) / 100

Compound Interest

• Calculated on the principal amount and accumulated interest
• Calculated using: Amount = Principal × (1 + Rate/100)^Time
• Subtract the principal from the final amount to find the compound interest

Successive Percentage Change

• The net percentage change after consecutive increases of x% and y% is x + y + (xy / 100)
• The net percentage change after an increase of x% and decrease of y% is x - y - (xy / 100)

Applications in Statistics

• Percentages represent proportions and statistical data distributions
• They facilitate comparisons with data sets of varied total sizes

Applications in Finance

• Used to describe interest rates, investment returns, and financial ratios
• Profit margin is calculated as (Profit / Revenue) × 100%

Applications in Everyday Life

• Applicable to sales discounts, taxes, and statistical data
• Helpful in calculating tips, splitting bills, and recognizing price changes

Tips and Tricks

• Simplify complicated percentage problems by breaking them into smaller steps
• Use cross-multiplication to determine unknown values in percentage equations

Common Mistakes to Avoid

• Avoid confusing the formulas for calculating percentage increase and decrease
• Always convert percentages into decimals or fractions for calculations
• Know the base value when calculating percentages