Proportion Calculator

1. What is a proportion?
2. How to solve proportions?
• Example calculation
3. What is constant of proportionality?

    What is a proportion?

From mathematics, a proportion is simply two ratios in an equation, for example 1/2 = 50/100, 75/100 = 3/4, 9/10 = 90/100. If one variable is a product of the other variable and a constant, the two variables are called directly proportional - in this case x/y is a constant ratio. If the product of two variables is a constant, the two are inversely proportional - in this case x·y is a constant.

Proportions are used in problems involving changing numbers while keeping a ratio constant. For example, if the price of a hamburger has risen by 10%, you might express this as a proportion: old price / 100 = new price / 110, so if you know the old price you can solve the proportion equation to find the new price. If the old price was $5, then $5/100 = x/110, then x = $5 / 100 * 110 = $5.5. While you can certainly do such calculations using our proportions calculator above, percentage math is easier using our percent calculator.

Proportions are also often used in unit conversion, where the difference between units of the imperial and metric system are proportionally constant. Scaling and resizing often require the calculation of proportions, such as if you know the desired width of an image, photo or video, you can figure out the required height to preserve the aspect ratio. Similarly, to read distances on a map you need to be able to solve proportions.

    How to solve proportions?

Solving proportional equations is fairly trivial, if you know the basic equation transformation laws - multiplying and dividing both sides by the same number is all that is required. Of course, with the help of our proportion calculator all the work is done for you.

    Example calculation

Say you have the proportion 4/5 = 12/x and need to find x. To solve for x, you need to first multiply both sides by x, resulting in x · 4/5 = 12. Then you divide both sides by 4/5, getting x = 12 / (4 / 5) = 12 / 4 * 5 = 3 * 5 = 15. Therefore, 4 is to 5 as 12 is to 15.

    What is constant of proportionality?

In solving proportions you can encounter the term "constant of proportionality", also known as the "unit rate" or "constant of proportional variation". It expresses the relationship of two variables (say x and y) when they are multiplicatively connected to a constant so that either their ratio or their product yields a constant. So we either have c = x / y or c = x · y where c is then the constant of proportionality between x and y.

With direct proportionality we have c = x / y which we can also express that as c / 1 = x / y and solve for c using the calculator above. If y = 5 for x = 20, then we have c / 1 = 20 / 5 hence c = 4. With inverse proportionality c = x · y which we can also express that as c / x = y / 1 and again solve for c. If y = 2 for x = 10, then we have c / 10 = 2 / 1 hence c = 20.

Common examples of direct proportionality include:

• the circumference of a circle and its diameter (the constant is known as π)
• the distance travelled by a moving object under constant speed is proportional to the time (the constant is the speed, you can explore this topic using our speed, distance & time calculator
• the relationship between the net force acting on an object and its acceleration. This relationship is governed by Newton's second law and the constant of proportionality is the object's mass.

Examples of proportionality varying inversely include:

• the number of people working on a given set task, if each has the same productivity, is inversely proportional to the time it will take to complete that task. The constant is the individual productivity - how long it would take a single worker to compelte the entire work.
• the number of identical pipes you need to fill the volume of a swimming pool in a given number of minutes. The constant is the time it takes a single pipe to fill in the pool. E.g. if you have 2 pipes (x) each with a debit of 1m³/s you can fill a 600m³ pool in 5 minutes (y). However, you only need 1 minute if you have 5 times the number of pipes (10 in total). The constant (c) in this case is 10, that is it takes 10 minutes for a single pipe to fill in the entire pool (calculation).

Our proportions calculator can be used to construct and check many more examples.