# Directly Proportional: Definition, Equation & Examples

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• 0:05 Relating Variables
• 1:10 Directly Proportional
• 3:05 Proportional Variables…
• 4:53 Proportional Variables…
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Lesson Transcript
Instructor: Neelam Mehta

Neelam has taught variety of math and science subjects. She has masters' degrees in Chemical Engineering and Instructional Technology.

In this lesson, you will learn the definition of the mathematical expression 'directly proportional' and be able to apply this expression to solving problems that related two variables.

## Relating Variables

In our world and everyday life, many of the human activities and their outcomes are related. How can we numerically explain the impact of our actions on the outcomes? Let's take a look at an example. Sam recently opened an ice cream shop. His weekly earning directly depends on the number of ice cream cups he sells throughout the week. In math, we say that Sam's earning is a function of the total number of ice cream cups sold. The amount of money Sam makes and the number of cups sold are two variables that are directly related.

In math, two or more variables can have many different kinds of relationships. These relationships are called variations. The three major variations that you are likely to come across are direct variation, inverse variation and joint variation.

In this lesson, our focus will be entirely on direct variation, also known as directly proportional. We have three objectives that we want to cover in this lesson:

• Define directly proportional
• Write an algebraic expression directly relating two variables in word problems
• Solve word problem using the proportional algebraic expression

## Definition: Directly Proportional

Let's say you are building airplanes of different sizes. The production cost of the airplane increases as the plane gets larger in size. Small size planes will have low cost, medium size will have medium cost, and large planes will have high cost. Let the size of the airplane equal to S and the cost equal to C. Notice that when S increases, C increases and when S decreases, so does C.

Variations describe how one quantity changes in relation to another. When one quantity increases constantly or decreases constantly with respect to another quantity then the two quantities are called directly proportional to each other. In the airplane example, we would say that the quantity C is directly proportional to S multiplied by a constant (k). We can write this formula as C = kS.

In above equation, C is the cost of construction, S is the size of airplane, and k is a constant number also known as the proportionality constant. If you were to plot the size of the plane on the x-axis and the cost on the y-axis, the slope of the line will be equal to the proportionality constant, k. When two quantities relate to each other by constant proportionality we get a linear function. Let's look at the graph of the cost of an airplane as a function of the size.

By looking at the equation of the line (y = 0.5x), we know that the slope of the line is 0.5. It means that, in this example, the proportionality constant is 0.5. The airplane production cost is directly proportional to the plane size multiplied by the constant 0.5.

Now that we understand what directly proportional means, we can look at some examples to see how we can apply this relationship variations to solve different types of problems.

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