Proportion
A proportion is a statement that says that two ratios are equal. They can be used in many everyday situations like comparing sizes, cooking, calculating percentages, and more. Proportions can be written as equivalent fractions or as equal ratios.
Example
| Equivalent fraction | : |  |
= |  |
| Equal ratio | : | 1:2 | = | 7:14 |
When we say that the ratios in a proportion are equal, we mean that we can multiply or divide one ratio by some constant to result in the other. So, for the example above, we can either multiply all the terms in the ratio, 1:2, by 7, or divide 7:14 by 7.
1:2 = (1×7):(2×7) = 7:14
7:14 = (7÷7):(14÷7) = 1:2
From the above, we can see that the ratios are related by a factor of 7, so they are proportional.
Terms of a proportion
| There are four terms of a proportion, often written as: | |
|
|
| The terms in the proportion are a, b, c, and d. In a proportion, "extreme" and "mean" are used to refer to specific terms in the proportion. Referencing the above, a and d are extremes, and b and c are means. |
or a:b = c:dProportionality
If we know that two ratios are proportional, we know that the product of their means is equal to the product of their extremes. So, recalling the ratio of 1:2 = 7:14 from above that the extremes are a and d, or 1 and 14, and the means are b and c, or 2 and 7:
| (1) |  |
= |  |
| (2) | |
= | |
| (3) |  |
= |  |
| (4) |  |
= | |
In other words, given two ratios, we know that:
ad = bc
This fact allows us to use some basic algebra to solve certain problems.
Using proportions to solve problems
There are many different examples of problems we can solve using proportions. To solve a problem using proportions, we need to know 3 of the 4 values in the proportion. We can then use the fact we discovered above, that the product of the means is equal to the product of the extremes of a proportion (ad = bc), to solve the problem.
Examples
1.
Remember that ad = bc, so:
1d = 48
So d = 48 and our proportion is:
Now let's look at a problem that we could encounter in real life.
2. Ally is baking some cookies and has a recipe for 24 cookies. One of the items in the recipe is butter. One stick of butter in the recipe makes 24 cookies. In other words, the ratio is 1 stick of butter: 24 cookies. Ally decides that she only wants to make a dozen cookies. Assuming that she changes all of the other proportions as well, if she wants only 12 cookies, how many sticks of butter does she need to use?
Multiply 1 × 12 and 24 × c to get:
12 = 24c
Solving for c,
This means that Ally would need 
, or half a stick of butter, to make 12 cookies.
It is also possible to use proportions alongside percentages, since percentages can be written as fractions, or ratios.
3. The original price of a shirt is $30. There is a 15% sale on the shirt. Use proportions to find the price of the shirt after the 15% discount.
We set this problem up first by converting the 15% to a ratio:
Then we set this equal to the ratio of the price:
So, 15% of $30 is $4.50, and the final price of the shirt is:
$30 - $4.50 = $25.50
See also equivalent ratios, scale.