# 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:

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.

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.