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Proportions and Ratios

Definition of Ratio

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 have to be exactly 1 pencil and 3 pens, but some 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! In fact, that is how we will use ratios -- to represent the relationship between two numbers.

Definition of Proportion

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 in stock at the factory, how can we find the number of cars we can equip? A simple proportion will do perfectly. We know that 4:1 is our ratio, and the number of cars that match with those 12 wheels must follow the 4:1 ratio. We can setup the problem like this, where x is our missing number of cars:

$$ \frac{4}{1}=\frac{12}{x}$$

To solve a proportion like this, we will use a procedure called cross-multiplication. 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 make up a ratio

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.

$$ \frac{16}{x}=\frac{x}{1} $$

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.

Ratios and Proportions Calculator

Use the tool below to convert between fractions and decimal, or to take a given ratio expression and solve for the unknown value.

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