Proportions

A proportion is a special form of an algebra equation. It is used to compare two ratios or make equivalent fractions. A ratio is a comparison between two values. Such as the following:

1 apple: 3 oranges

This ratio compares apples to oranges. It means for every apple there are 3 oranges.

A proportion will help you solve problems like the one below.

Jane has a box of apples and oranges in the ratio of 2:3. If she has six apples, how many oranges does she have?

Before we begin to set up proportions for a word problem, we will concentrate on solving proportions. Remember, a proportion is a comparison between two ratios. The proportion shown below compares two ratios which are in the fraction form.

1 x

- = -

2 6

The four parts of the proportion are separated into two groups, the means and the extremes, based on their arrangement in the proportion. Reading from left-to-right and top-to-bottom, the extremes are the very first number, and the very last number. This can be remembered because they are at the extreme beginning and the extreme end. Reading from left-to-right and top-to-bottom, the means are the second and third numbers. Remembering that "mean" is a type of average may help you remember that the means of a proportion are "in the middle" when reading left-to-right, top-to-bottom. Both the means and the extremes are illustrated below.

The illustration of the means and extremes is shown.

Algebra properties tell us that the products of the means is equal to the product of the extremes. You should know that the fraction one-half is equal to two-fourths. This is shown as a proportion below.

1 2

- = -

2 4

Because they are equal, the product of the means is equal to the product of the extremes, this is shown below.

2 * 2 = 1 * 4

4 = 4

Without knowing it, you probably used this property to tell whether a fraction was equal to, greater than, or less than another fraction. This property is extremely useful when one of the means or one of the extremes is unknown(It is unknown if it is blank or contains a variable such as x). The proportion below shows a proportion with an unknown mean.

1 x

- = -

2 6

The problem is shown again for your reference.

1 x

- = -

2 6

To solve this, and find the value of x: write an equation, on the left side multiply the means, on the right side, multiply the extremes. Then solve the equation for x.

2 * x = 1 * 6

2x = 6

--- - Divide each side by 2

2 2

x = 3

Now you know that x is equal to 3. This means that

This time the variable is in a different position, but the same steps are used to solve it. Make an equation with the multiplication of the means on the left and the multiplication of the extremes on the right. Then solve it like we did below.

If you encounter a proportion that has one of its means or extremes left blank, or uses another symbol such as a question mark you can treat it as if it was a variable. Or you can replace the question mark or blank space with a variable such as x. See the example below.

Becomes:

5 * 90 = 9 * x

50 = x

50 = ?

A proportion with two of the same variable, can also be solved. Take the problem below for example.

When you encounter a situation like the above, a variable squared equals a number, you can do one of two things.

1. Find what number squared is equal to 25. Use our perfect squares chart for reference.

(or)

2. Change the problem to x = the squareroot of 25. The resulting number from either method will be equal to x and will be the answer.

Take a look at the problem below, notice it doesn't have a fraction on one side.

To solve this proportion, you have to change the whole number to a fraction, just as you did in math class, by putting it over a 1. The problem above would turn into the following:

It could then be solved like any other proportion.

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