Unit Rates with Fractions

Unit Rates with Complex Fractions

A unit rate tells how much of one quantity per 1 unit of another.

With fractions, divide the numerator fraction by the denominator fraction.

Example: A snail travels $\frac{3}{4}$ mile in $\frac{1}{2}$ hour. What is its speed?

$\text{Speed} = \frac{\frac{3}{4} \text{ mi}}{\frac{1}{2} \text{ hr}} = \frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \times \frac{2}{1} = \frac{6}{4} = 1\frac{1}{2} \text{ mph}$

Constant of Proportionality

In a proportional relationship $y = kx$, the constant of proportionality $k$ is the unit rate.

| $x$ | $y$ | $\frac{y}{x}$ | |-----|-----|------| | 2 | 6 | 3 | | 4 | 12 | 3 | | 5 | 15 | 3 |

$k = 3$, so $y = 3x$

Identifying Proportional Relationships

A relationship is proportional if:

1. The graph is a straight line through the origin
2. The ratio $\frac{y}{x}$ is constant for all pairs
3. It can be written as $y = kx$

Graphs of Proportional Relationships

• Always pass through $(0, 0)$
• The unit rate $k$ is the slope of the line
• Steeper line = larger unit rate

Real world: Recipes, maps, currency exchange, and speed are all proportional relationships!