How to write and simplify ratios, unit rates and unit prices, what a proportion is, solving proportions with cross products, and using proportions with scale drawings.
Ratios
A smoothie recipe uses \(2\) cups of strawberries for every \(3\) cups of yogurt. That comparison is a ratio.
Order matters: strawberries to yogurt is \(2 : 3\), but yogurt to strawberries is \(3 : 2\).
Like fractions, ratios can be simplified. A class with \(12\) girls and \(16\) boys has a girls-to-boys ratio of \(12 : 16 = 3 : 4\). For every \(3\) girls there are \(4\) boys.
When both quantities are measured, use the same units first. The ratio of \(8\) inches to \(2\) feet is not \(8 : 2\). Since \(2\) feet is \(24\) inches, it's \(8 : 24 = 1 : 3\).
Rates and unit rates
A rate is a ratio between quantities with different units, like miles and hours. A unit rate is a rate per one unit of the second quantity.
Unit prices help you compare deals at the store.
Proportions
\(\frac{2}{3} = \frac{8}{12}\) is a proportion: both ratios simplify to \(\frac{2}{3}\).
Check: in \(\frac{2}{3} = \frac{8}{12}\),\(2 \cdot 12 = 24\) and \(3 \cdot 8 = 24\).
Solving a proportion
When one number in a proportion is unknown, call it \(x\), set the cross products equal and divide.
Scale drawings and maps
A map scale is a ratio. "\(1\) inch \(: 20\) miles" means every inch on the map stands for \(20\) real miles.
Practice
Quick check
Which ratio is equivalent to \(6 : 10\)?
\(6 : 10\) simplifies to \(3 : 5\), and so does \(9 : 15\).
Quick check
If \(\frac{3}{8} = \frac{x}{40}\), what is \(x\)?
Cross products: \(8x = 120\), so \(x = 15\). Or notice that \(40 = 8 \cdot 5\), so \(x = 3 \cdot 5\).
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