Ratios Part 1: What's a Ratio?

Have you ever cooked, measured, drawn, or played music?    

 Spices in teaspoons. Ratios can help you figure out how much of different spices to add to recipes.  A harmonica rests on sheet music.

Images by Sarah Pflug and Samantha Hurley

If so, chances are good that you have used ratios. Ratios indicate the relationship between two numbers, and you can find them all around you. You will use percentages, which are a specific type of ratio out of 100, to calculate tips, tax, and discounts as well as interpret statistics you might see in polls or on the news. In school, you might use ratios to compare coefficients across equations, convert units, analyze data, and find proportions of shapes. Because they represent relationships between numbers, ratios are everywhere!

Ratios are read as, “a value to another value.” Ratios can be written as a fraction (with division separating the numbers) or with a colon separating the numbers.

Example

2 to 1        3 to 5       7 to 8       5 to 1

  \({\frac{2}{1}}\)              \({\frac{3}{5}}\)               \({\frac{7}{8}}\)             \({\frac{5}{1}}\)

  2:1             3:5             7:8            5:1

Ratios can also include more than two numbers.

Example

1 apple to 2 oranges to 5 lemons → 1:2:5 

Ratios are so useful because they can be scaled up and down by multiplying or dividing the top and bottom of a fraction by the same number, which is like multiplying or dividing by 1. This maintains the existing relationship and can allow you to find other pairs of numbers that have the same relationship.

If two ratios are equivalent when they are fully reduced (meaning they represent the same relationship), they create a proportion. Pairs of numbers that represent the same ratio are proportional. The number by which you can multiply all parts of one ratio to get another, equivalent ratio is called the constant of proportionality.

A pizza cut into six slices.

Image by Sarah Pflug

Example:

Let’s say we’re going to make pizza and are using the following measurements:

\({\frac{1}{2}}\) cups of warm water to 2 \({\frac{1}{4}}\) teaspoons of yeast to 4 cups of flour (we will also add sugar and salt). The recipe makes 2 pizzas.

First, let’s say we want to make 4 pizzas. If our recipe makes 2, we need to double those measurements — in other words, multiply our recipe by a constant of proportionality of 2. Multiplying each of the required ingredients by 2, we need 3 cups of warm water, 4 \({\frac{1}{2}}\) teaspoons of yeast, and 8 cups of flour.

Now let’s use the same recipe, but this time we only want to make 1 pizza. We need to halve the recipe, or multiply it by a constant of proportionality of \({\frac{1}{2}}\) Halving each of the required ingredients shows that we need \({\frac{3}{4}}\) cups of warm water, 1 \({\frac{1}{8}}\) teaspoons of yeast, and 2 cups of flour.