Lesson 1: Ratios and Rates

Property

A ratio is commonly described as a pair of positive numbers, written $a : b$ and read as “$a$ to $b$.” When describing a ratio, the order of the quantities must be the same as the order of the numbers.

Examples

• In a garden, there are 5 rose bushes for every 2 lilac bushes. The ratio of roses to lilacs is 5:2.
• A pancake recipe requires 2 cups of flour for every 1 cup of milk. The ratio of flour to milk is 2:1.
• For every 3 games the team wins, they lose 1. The ratio of wins to losses is 3:1.

Explanation

A ratio is a recipe for comparing two amounts. It tells you, "for every this, you have that." The order is critical—a ratio of 2:3 is not the same as 3:2, just like putting on shoes then socks doesn't work!

Property

As a numeric quantity, a ratio is the same as a fraction.
But, operationally, they are quite different.
A fraction is a number—a point on the line.
A ratio $a : b$ expresses a relationship between two quantities: given a particular sample of the quantities $A$ and $B$, the number of $A$ is the same multiple of $a$ as the number of $B$ is of $b$.

Examples

• You completed $\frac{2}{3}$ of your homework. This fraction represents a single amount of finished work. The ratio of completed to uncompleted work is 2:1.
• The fraction $\frac{4}{5}$ is a number. The ratio 4:5 describes a relationship, such as having 4 sunny days for every 5 cloudy days.
• A pizza is cut into 8 slices. You eat 3 slices, which is $\frac{3}{8}$ of the pizza. The ratio of slices you ate to slices remaining is 3:5.

Explanation

A fraction, like $\frac{1}{2}$, is a single value representing part of a whole. A ratio, like 1:2, describes a relationship between two separate things. While they can be written similarly, they tell different kinds of stories.