how to find constant of proportionality

How to Identify the Constant of Proportionality

The constant of proportionality is the ratio that relates two given values in what is known as a proportional relationship. Other names for the constant of proportionality include the constant ratio, constant rate, unit rate, constant of variation, or even the rate of change.

In this article, we'll show you how to find the constant of proportionality and graph its equation through reviewing a sample word problem.

The Constant of Proportionality in Real Life

Take two things that we know are directly proportional in our everyday lives, such as the amount you pay for gas and the amount of gas you receive. We know that the more gas you pump, the more money you have to pay at the gas station.

This means money and gas are two proportional quantities that relate to each other through a linear equation. For example, we could use y = price in dollars, x = gas in gallons, and a constant of proportionality k to represent the amount of money you have to pay at the gas station: y = kx. In other words, the price you pay is in direct proportion to the gallons pumped.

Using the equation, you can calculate the final price by calculating k times x, where k is the constant value of gas per gallon.

Graphing a Proportional Relationship

You can easily graph a proportional relationship as a straight line on a coordinate plane. The reason it is a straight line is because it is directly proportional — the constant of proportionality is the slope. The slope or increase is constant because the proportionate change along both the x and y axes never changes.

Let's look at our example of paying for gas.

If gas is $3.50 per gallon, the y-value represents the total cost, while the x-value represents the number of gallons that you pump. The linear equation is y = 3.50x. You can see that for every extra gallon along the x-axis, the line goes up 3.50 spaces along the y axis!

Using the coordinates (4, 14) as an example, we can see that 4 gallons of gas would cost $14 because in the equation y = 3.50x, we see that y = 3.50(4) = 14.

constant of proportionality: graph

Image credit: Desmos

Finding the Constant of Proportionality in the Real World

The constant of proportionality goes by many other names, but you can remember it as the constant value that defines the proportional relationship between two quantities. It is also called the constant ratio, constant rate, unit rate, constant of variation, or rate of change.

If you pay attention, you can see proportional relationships all around you — from the amount you pay for produce to how you measure baking ingredients. So, you'll go far if you understand the constant of proportionality!