Proportional Relationships

Step 01 of 03

A unit rate tells you "how much per ONE." Find it by dividing the total by the count.

Step 02 of 03

Whole-number example: 12 cookies cost $4.80. Unit rate = $\$4.80/12=\$0.40$ per cookie.

Fraction example: $\frac{3}{4}$ cup of flour for $\frac{1}{2}$ loaf. Per loaf: $\frac{3}{4}÷\frac{1}{2}=\frac{3}{4}\cdot 2=\frac{3}{2}$ cups.

Step 03 of 03

Compare two rates by computing each as "per one" of the same unit.

Step 01 of 03

Two quantities are proportional if their ratio is constant. Equivalently, doubling one doubles the other.

Step 02 of 03

Test a table. Compute $y/x$ for each row.

Constant ratio of 3 — proportional.

Step 03 of 03

Counter-example. Pairs $(1,5),(2,7),(3,9)$ have ratios 5, 3.5, 3 — different. NOT proportional. (It IS linear, with slope 2 and y-intercept 3, but the line doesn't pass through the origin.)

Step 01 of 03

If $y=kx$, the number $k$ is the constant of proportionality — the ratio of y to x.

Step 02 of 03

Find $k$. Pick any (x, y) pair from the table or graph and compute $k=y/x$.

Step 03 of 03

Interpret $k$ in context. "Cost vs. pounds": $k$ = price per pound. "Distance vs. time": $k$ = speed. Always read the units to get its meaning.

If 5 lbs of apples cost $7.50, how much do 12 lbs cost?

Step 01 of 03

Find the unit rate.

Step 02 of 03

Multiply by the new quantity.

Step 03 of 03

Cross-multiplication shortcut. Set up a proportion $\frac{5}{7.50}=\frac{12}{x}$, cross-multiply: $5x=90$, $x=18$.