Ratios tell you how any two parts of a whole relate to each other. For example, you might have a ratio that compares how many boys are in your class versus how many girls are in your class, or a ratio in a recipe that tells you how the amount of oil compares to the amount of sugar. Once you know how the two numbers in a ratio relate to each other, you can use that information to calculate how the ratio relates to the real world.
A Quick Review of Ratios
It might help to think of ratios as fractions, for two reasons. First, you can actually write ratios as fractions; 1:10 and 1/10 are the same thing. Second, just like in fractions, the order you write numbers in for a ratio matters.
Let's say you're comparing the ratio of salt to sugar in a recipe that calls for 1 part salt to 10 parts sugar. You write the numbers in same order as the items the numbers represent. So, since salt comes first, you'd write the "1" for 1 part salt first, followed by the "10" for 10 parts sugar. That gives you a ratio of 1 to 10, 1:10 or 1/10.
Now imagine that you were to switch the numbers around, letting your ratio of salt to sugar be 10:1. Suddenly, you have 10 parts of salt for every 1 part of sugar. Whatever you're making with a 10:1 ratio is going to taste very different than if you'd used a 1:10 ratio!
Finally, just like fractions, ratios are ideally given in their simplest terms. But they don't always start out that way. So just as a fraction of 3/30 can be simplified to 1/10, a ratio of 3:30 (or 4:40, 5:50, 6:60 and so on) can be simplified to 1:10.
Solving for Missing Parts in a Ratio
You might be able to tell how to solve a 1:10 ratio by simple examination: For every 1 part you have of the first thing, you'll have 10 parts of the second thing. But you can also solve this ratio using the technique of cross-multiplication, which you can then apply to more difficult ratios.
As an example, imagine that you've been told there is a 1:10 ratio of left-handed to right-handed students in your class. If there are three left-handed students, how many right-handed students are there?
Set up the Problem
Solve for x
You're actually given two ratios in the example problem: The first, 1/10, is the known ratio of left-handed to right-handed students in class. The second ratio also represents the number of left-handed to right-handed students in class, but you're missing an element. Write the two ratios out as equal to each other, with the variable x acting as a placeholder for the missing element. So to continue the example, you have:
1/10 = 3/x
Multiply the numerator of the first fraction by the denominator of the second fraction, and set this equal to the numerator of the second fraction times the denominator of the first fraction. Set the two products as equal to each other. Continuing the example, this gives you:
1(x) = 3(10)
With a more difficult problem, you'd now have to solve for x. But in this case, simplifying the equation is all you have to do to get a value for x:
x = 30
Your missing quantity is 30; you might have to look back at the original problem to remind yourself that this represents the number of right-handed students in class. So if there are 3 left-handed students in class, there are also 30 right-handed students.