Proportions and Ratios

A ratio is a relationship between two values. For instance, a ratio of 1 pencil to 3 pens would imply that there are three times as many pens as pencils. For each pencil there are 3 pens, and this is expressed in a couple ways, like this: 1:3, or as a fraction like 1/3. There do not necessarily have to be those numbers of each, but a multiple of them. We could just as easily have 2 pencils and 6 pens, 10 pencils and 30 pens, or even half a pencil and one-and-a-half pens!

A proportion can be used to solve problems involving ratios. If we are told that the ratio of wheels to cars is 4:1, and that we have 12 wheels, how can we find the number of cars we could have? A simple proportion will do perfectly. We know that 4:1 is our given ratio and the new ratio with 12 wheels must be an equivalent fraction, so we can setup the problem like this, where x is our missing number of cars:

first proportion

To solve a proportion like this, we have to cross-multiply. This process involves multiplying the two extremes and then comparing that product with the product of the means. An extreme is the first number (4), and the last number (x), and a mean is the 1 or the 12.

mean extremes

To multiply the extremes we just do 4 * x = 4x. The product of the means is 1 * 12 = 12. The process is very simple if you remember it as cross-multiplying, because you multiply diagonally across the equal sign.


You should then take the two products, 12 and 4x, and put them on opposite sides of an equation like this: 12 = 4x. Solve for x by dividing each side by 4 and you discover that x = 3. Reading back over the problem we remember that x stood for the number of cars possible with 12 tires, and that is our answer.

It is possible to have many variations of proportions, and one you might see is a double-variable proportion. It looks something like this, but it easy to solve.


Using the same process as the first time, we cross multiply to get 16 * 1 = x * x. That can be simplified to 16 = x^2, which means x equals the square root of 16, which is 4 (or -4). You've now completed this lesson, so feel free to browse other pages of this site or search for more lessons on proportions.

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