What is the difference between ratios and rates?

Both rates and ratios are a comparison of two numbers.

A rate is simply a specific type of ratio.

The difference is that a rate is a comparison of two numbers with different units, whereas a ratio compares two numbers with the same unit.

For example, in a room full of students, there are 10 boys and 5 girls. This means the ratio of boys to girls is 10:5.

If we simplify the ratio, we see that the ratio of boys to girls is 2:1, since #10 -: 5 = 2# and #5 -: 5 = 1#. Thus, there are 2 boys in the room for every 1 girl.

Let's say we'd like to buy a soda for each student in the classroom. The local pizza place offers a group discount on soda purchases: $10 for 20 sodas.

Since we only need 15 sodas, we're wondering how much each soda costs at the discounted rate. To find out, we'll set up two rates, because we have 2 different units, dollars and number of sodas:

If #(20 " sodas")/($10)#, then #(1 " soda")/(x)#

We cross multiply for #x#:

#20x = $10#

#x= 10-:20#

#x = 1/2#

Since our unit is $, we'll convert #1/2# to dollars. One half of a dollar is $0.50.

Thus, the discounted rate for the sodas is $.50/1 soda.

Here is a helpful video on simplifying rates and ratios: