Ratios & Percentages Cheat Sheet

Understand and apply the concepts of ratios and percentages to simplify comparisons and solve real-world problems involving proportions, increases, and decreases.

Core Principles

A ratio compares two or more numbers, often simplified to lowest terms.
Ratios can involve fractions or decimals, requiring simplification.
Quantities in a ratio must have the same units for comparison.
Percentages represent parts of a whole (100) and are used for calculations involving proportions.
Three types of percentage problems exist: finding the part, the base, or the percent.
Percent increases involve adding to 100%; decreases involve subtracting from 100%.

Action Steps

For ratios, ensure units are consistent before simplifying.
Divide all parts of a ratio by their greatest common divisor.
To simplify decimals in ratios, multiply by powers of 10 to make them whole numbers.
Identify 'Part', 'Base', and 'Percent' in word problems.
Substitute known values into the percentage formula and solve for the unknown.
For increases, add the percentage to 100% before using the formula.
For decreases, subtract the percentage from 100% before using the formula.

Formulas

Ratio: a:b:c ...
Simplified Ratio: Divide all parts by a common factor.
Percentage Formula: $ \frac{\text{Part}}{\text{Base}} = \frac{\text{Percent}}{100} $
Increase: Percent = 100% + Increase%
Decrease: Percent = 100% - Decrease%

Key Terms

Ratio: A comparison of two or more numbers.
Simplify Ratio: To reduce a ratio to its lowest terms by dividing by a common factor.
Percent: A rate or proportion per hundred.
Part: A portion of the whole or base amount.
Base: The total or original amount, equivalent to 100%.
Markup: An increase in price or value.
Markdown: A decrease in price or value.

Real World Examples

Student test scores: Calculating the percentage of correct answers (e.g., 96 out of 120 is 80%).
Product pricing: Determining sales tax, discounts, or markup rates (e.g., $33 sales tax on a $550 item at 6%).
Population changes: Calculating percentage increase or decrease in a group over time (e.g., a 40% population increase).
Inventory management: Finding original prices after markdowns or calculating markup rates (e.g., finding the original $149.95 price of a chair marked down by 15%).