Ratios, Rates & Proportions Cheat Sheet

Key Facts

• A ratio compares two quantities by division, such as $a:b$, $a\text{ to }b$, or $\frac{a}{b}$.
• Equivalent ratios are made by multiplying or dividing both terms by the same nonzero number: $\frac{a}{b}=\frac{ka}{kb}$ for $k\neq 0$.
• A rate compares quantities with different units, such as $\frac{120\text{ miles}}{2\text{ hours}}$.
• A unit rate has a denominator of $1$, so $\frac{120\text{ miles}}{2\text{ hours}}=\frac{60\text{ miles}}{1\text{ hour}}$.
• A proportion is an equation showing two ratios are equal: $\frac{a}{b}=\frac{c}{d}$.
• In a true proportion, the cross products are equal: if $\frac{a}{b}=\frac{c}{d}$, then $ad=bc$.
• To find a percent, use $\text{percent}=\frac{\text{part}}{\text{whole}}\times 100\%$.
• A scale factor compares matching lengths: $\text{scale factor}=\frac{\text{new length}}{\text{original length}}$.

Vocabulary

Ratio

A comparison of two quantities by division, often written as $a:b$, $a\text{ to }b$, or $\frac{a}{b}$.

Rate

A ratio that compares quantities with different units, such as miles per hour or dollars per pound.

Unit Rate

A rate with a second quantity of $1$, such as $\frac{5\text{ dollars}}{1\text{ pound}}$.

Proportion

An equation that states two ratios are equal, such as $\frac{a}{b}=\frac{c}{d}$.

Cross Product

The product found by multiplying diagonally in a proportion, where $ad$ and $bc$ are equal when $\frac{a}{b}=\frac{c}{d}$.

Scale Factor

The multiplier that changes a figure or measurement to a proportional new size.