Proportion Basics

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

Solving a Proportion

The illustration of the means and extremes is shown again for your reference.

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

Solving a Proportion

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

1             3
- is equal to -
2             6

Solving a Proportion

5   15
- = --
x   12

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.

x * 15 = 5 * 12

15x = 60 --- -- Divide each side by 15. 15 15
x = 4

Solving a proportion without a variable

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.

9   90
- = --
5   ?

Becomes: 9 90 - = -- 5 x
5 * 90 = 9 * x
450 = 9x --- -- 9 9
50 = x
50 = ?

Solving a proportion with two variables:

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

25   x
-- = -
x    1

x * x = 25 * 1
x2 = 25

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.


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

When a whole number is in place of a fraction:

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

-- = 3

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:

x    3
-- = -
12   1

It could then be solved like any other proportion.

Proportion Basics Resources

Proportion Practice Problems / Worksheet
20 problems to help you practice your proportion skills.

Proportion Calculator
Automatically solves and shows work for basic proportions.

Next Lesson:
Word Problem Basics

Introduces methods for solving integer word problems through equations.


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