Ratios & Percentages Cheat Sheet

This cheat sheet covers the fundamentals of ratios and percentages, including how to simplify ratios, solve various types of percentage problems, and calculate percent increases and decreases.

Core Principles

A ratio compares two or more numbers.
Ratios can be simplified like fractions by dividing all parts by a common factor.
Quantities in a ratio must have the same units for comparison.
Percentages can be solved using the formula: Part/Base = Percent/100.
There are three types of percentage problems: finding the part, the base, or the percent.
Percent increases are calculated by adding the percentage to 100%.
Percent decreases are calculated by subtracting the percentage from 100%.

Action Steps

Identify the numbers or quantities to be compared for a ratio.
Find a common divisor for all parts of the ratio to simplify.
Ensure units are the same before comparing quantities in a ratio.
Determine if the problem asks for the 'part', 'base', or 'percent'.
Set up the proportion using the formula: Part/Base = Percent/100.
Substitute known values and solve for the unknown variable.
For increases, add the percentage to 100%; for decreases, subtract from 100%.
Apply the adjusted percentage in the proportion to solve.

Formulas

Ratio Simplification: $a:b:c \div d = (a/d):(b/d):(c/d)$
Unit Conversion for Ratios: Ensure consistent units before comparing.
Percentage Formula: $\frac{Part}{Base} = \frac{Percent}{100}$
Finding the Part: $x = \frac{Base \times Percent}{100}$
Finding the Base: $x = \frac{Part \times 100}{Percent}$
Finding the Percent: $x = \frac{Part \times 100}{Base}$
Percent Increase: New Value = Original Value $\times (1 + \text{Increase Rate})$
Percent Decrease: New Value = Original Value $\times (1 - \text{Decrease Rate})$

Key Terms

Ratio: A comparison of two or more numbers, often expressed in the form a:b:c.
Simplify Ratio: To reduce a ratio to its lowest terms by dividing all parts by their greatest common divisor.
Percentage: A rate or proportion per hundred.
Part: A portion of the whole or base amount in a percentage problem.
Base: The total or original amount in a percentage problem.
Markup: An increase in price or value, often expressed as a percentage.
Markdown: A decrease in price or value, often expressed as a percentage.

Real World Examples

A student scores 96 out of 120 on a test.: Calculate the percentage score: (96/120) * 100 = 80%.
3% of 15 defective propellers are found.: Find the total number of propellers: 15 is 3% of what number? (15/x) = (3/100) -> x = 500.
A refrigerator costs $550 with a 6% sales tax.: Calculate the sales tax amount: (x/550) = (6/100) -> x = $33.
A chair is marked down by 15% and sells for $127.46.: Find the original price: $127.46 is 85% of what original price? (127.46/x) = (85/100) -> x = $149.95.