A ratio is a mathematical comparison of two numbers, based on division. For example, suppose you bring 2 scarves and 3 caps with you on a ski vacation. Here are a few ways to express the ratio of scarves to caps
[tex]2:3 \: \: \: \: 2 \: to \: 3 \: \: \frac{2}{3} [/tex]
The simplest way to work with a ratio is to turn it into a fraction. Be sure to keep the order the same: The first number goes on top of the fraction, and the second number goes on the bottom.
In practice, a ratio is most useful when used to set up a proportion — that is, an equation involving two ratios. Typically, a proportion looks like a word equation, as follows:
[tex] \frac{scarves}{caps} = \frac{2}{3} [/tex]
For example, suppose you know that both you and your friend Andrew brought the same proportion of scarves to caps. If you also know that Andrew brought 8 scarves, you can use this proportion to find out how many caps he brought. Just increase the terms of the fraction
[tex] \frac{2}{3} [/tex]
so that the numerator becomes 8. Do this in two steps:
[tex] \frac{scarves}{caps} = \frac{2 \times 4}{3 \times 4} \\ \\ \frac{scarves}{caps} = \frac{8}{12} [/tex]
As you can see, the ratio 8:12 is equivalent to the ratio 2:3 because the fractions
[tex] \frac{2}{3} \: \: and \: \: \frac{8}{12} [/tex]
are equal. Therefore, Andrew brought 12 caps
