Lesson 1: Proportional Relationships with Fractions

Property

Given two quantities $x$ and $y$, they are said to be proportional if, whenever we multiply one by a factor $r$, the other is multiplied by the same factor, $r$.
If quantities $y$ and $x$ are in proportion, then the unit rate of $y$ with respect to $x$ is the amount of $y$ that corresponds to one unit of $x$.
If $m$ is the unit rate, then for any value of $x$, the corresponding $y$ value is $mx$.
This unit rate is also called the constant of proportionality, leading to the equation $y = mx$.

Examples

• The number of feet ($F$) is proportional to the number of miles ($M$). Since there are 5280 feet in a mile, the unit rate is 5280 feet per mile. The relationship is $F = 5280M$.
• In a recipe, the ratio of flour to sugar is 3:2. This is a proportional relationship. For every 3 cups of flour, you need 2 cups of sugar. The constant of proportionality of sugar to flour is $\frac{2}{3}$.
• A car travels 195 miles in 3 hours at a constant speed. The unit rate is $\frac{195}{3} = 65$ miles per hour. The distance $d$ traveled in $t$ hours is given by $d = 65t$.

Explanation

This means two quantities change together at a constant rate. If you double one, the other doubles. The unit rate, or constant of proportionality $m$, is the magic number that connects them in the formula $y = mx$.

Property

The scale factor is the ratio of the lengths in the new figure to the corresponding lengths in the original figure.

Examples

• A map has a scale factor of $\frac{1}{10000}$. A road that is 3 cm long on the map is $3 \times 10000 = 30000$ cm, or 300 meters, in real life.
• A triangle with a base of 8 inches is enlarged to a similar triangle with a base of 20 inches. The scale factor is $\frac{20}{8} = 2.5$.
• To reduce a 12-foot wall to fit on a blueprint with a scale factor of $\frac{1}{48}$, its length on the blueprint is $12 \times \frac{1}{48} = \frac{1}{4}$ foot, or 3 inches.

Explanation

The scale factor is the number you multiply by to change the size of a figure. A scale factor greater than 1 makes the figure bigger (an enlargement), while a factor between 0 and 1 makes it smaller (a reduction).