8. Rate Tables, Proportions, and Missing Values

When a rate is constant, the relationship is proportional. That means multiplying one quantity by the same factor gives the other.

Example: A car travels 60 miles each hour.

\[ \begin{array}{c|c} \text{hours} & \text{miles}\\ 1 & 60\\ 2 & 120\\ 3 & 180\\ 4 & 240 \end{array} \]

To find a missing value, use the unit rate or set up a proportion.

Example: 5 pounds of apples cost 8 dollars. How much do 12 pounds cost?
\[ \text{unit price}=8/5=1.60\text{ dollars per pound}. \]
\[ 12(1.60)=19.20\text{ dollars}. \]

Proportion method:
\[ \frac{8}{5}=\frac{x}{12}. \]
Cross multiply:
\[ 5x=96,\qquad x=19.20. \]

Case study - recipe: 3 cups of flour make 2 loaves. Cups per loaf:
\[ 3/2=1.5. \]
For 5 loaves:
\[ 1.5(5)=7.5\text{ cups}. \]

Common mistake: adding instead of multiplying. If 2 loaves need 3 cups, 5 loaves is not simply 6 cups; use cups per loaf.