Understanding Percentages: Calculation and Usage

• A percentage expresses a number as a fraction of 100.
• The percent sign, %, denotes it.
• Percentages show the relative size of one quantity compared to another.
• "Percent" comes from the Latin "per centum," meaning "out of one hundred."

Basic Percentage Calculation

• To find what percentage a number 'A' is of another number 'B', use: (A / B) * 100%.
• For example, if A = 20 and B = 50, then (20 / 50) * 100% = 40%, meaning 20 is 40% of 50.

Calculating Percentage of a Number

• 'P' percent of a number 'X' is found using: (P / 100) * X.
• For example, 25% of 80 is (25 / 100) * 80 = 20.

Percentage Increase

• Determine the relative increase in a quantity using the formula: [(New Value - Old Value) / Old Value] * 100%.
• If a price increases from $20 to $25, the percentage increase is [($25 - $20) / $20] * 100% = 25%.

Percentage Decrease

• Determine the relative decrease in a quantity using the formula: [(Old Value - New Value) / Old Value] * 100%.
• If a price decreases from $25 to $20, the percentage decrease is [($25 - $20) / $25] * 100% = 20%.

Converting Percentages to Decimals

• To convert a percentage to a decimal, divide by 100.
• For example: 75% = 75 / 100 = 0.75.

Converting Decimals to Percentages

• To convert a decimal to a percentage, multiply by 100.
• For example: 0.42 = 0.42 * 100 = 42%.

Converting Percentages to Fractions

• To convert a percentage to a fraction, write the percentage as a fraction with a denominator of 100 and simplify.
• For example: 60% = 60 / 100 = 3 / 5.

Converting Fractions to Percentages

• Convert a fraction to a percentage by first converting to a decimal, then multiplying by 100.
• As an example: 1 / 4 = 0.25 = 0.25 * 100 = 25%.

Percentage Change

• Express how much a quantity has changed relative to its initial value.
• The formula is: [(Final Value - Initial Value) / Initial Value] * 100%.
• A positive result indicates an increase; a negative result indicates a decrease.
• A company's revenue change from $100,000 to $120,000 is a percentage change of [($120,000 - $100,000) / $100,000] * 100% = 20%.
• A company's revenue change from $100,000 to $80,000 is a percentage change of [($80,000 - $100,000) / $100,000] * 100% = -20%.

Percentage Error

• Quantify the difference between an experimental and a true value.
• The formula is: [(|Experimental Value - True Value|) / True Value] * 100%.
• The absolute value ensures the error is always positive.
• If an experimental value is 52 and the true value is 50, the percentage error is [(|52 - 50|) / 50] * 100% = 4%.

Working with Multiple Percentages

• Apply multiple percentage increases or decreases sequentially.
• A price increase of 10% followed by 20%, isn't a simple 30% increase.
• With an original price of $100, a 10% increase makes it $110.
• A 20% increase on $110 is $22, bringing the final price to $132.
• The total percentage increase is [($132 - $100) / $100] * 100% = 32%.

Reverse Percentage

• Find the original value when a percentage of it is known.
• If X is P% of the original value: Original Value = X / (P / 100).
• If 20 is 25% of a number, the original number is 20 / (25 / 100) = 80.

Percentage Points

• Represents the simple difference between two percentages.
• A change from 40% to 50% is a 10 percentage point increase.
• This differs from a 10% increase, which would be 10% of 40% (an increase of 4).

Applications of Percentages

• Utilized in finance for calculating interest rates, discounts, and investment returns.
• Crucial in statistics for expressing data distributions and probabilities.
• Used in retail to determine discounts, markups, and sales tax.
• Applied in science to express concentrations, errors, and experimental result changes.