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6th-8th Grade Math: Practice & Review55 chapters | 469 lessons

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Lesson Transcript

Instructor:
*Damien Howard*

Damien has a master's degree in physics and has taught physics lab to college students.

Constants of proportionality might sound like something complicated, but chances are, you've encountered them before. In this lesson, we'll explore the constant of proportionality and how to find it.

One Saturday morning, you find yourself at the local grocery store helping out with a little shopping for your family. You've been asked to go pick up a can of tomatoes. When you get to the aisle, you see the brand you want on sale: two for $3.00. In your head, you divide three by two and know the tomato can will cost $1.50. What you just did, without even knowing it, is find the constant of proportionality for the cost of a tomato can.

The **constant of proportionality** is the ratio between two directly proportional quantities. In our tomato example, that ratio is $3.00/2, which equals $1.50. Two quantities are **directly proportional** when they increase and decrease at the same rate. When you buy more or less, the total price of the tomato cans in this example goes up or down correspondingly.

Even though the amount of tomato cans bought and the total price can change, the price for a single tomato can, our constant of proportionality, remains the same. This is why we call the constant of proportionality a constant, a number with a fixed value.

In this lesson, we're going to look at multiple ways to identify the constant of proportionality. We'll see how to identify it in equations, word problems, diagrams, tables, and graphs.

Earlier we mentioned that the ratio for the constant of proportionality in our example was $3.00/2. We can use this ratio as an example to find a general equation for any constant of proportionality. The constant of proportionality (*k*) equals the total price (*y*) divided by the number of cans (*x*).

*k* = *y*/*x*

Or, the total price equals the constant of proportionality multiplied by the number of cans:

*y* = *k**x*

The second formula is a common alternate form of the constant of proportionality formula. While working with the equations, just replace the total price and number of cans with whatever two other directly proportional quantities you have in your problem.

When trying to find the constant of proportionality, you're not always going to begin with an equation. This is especially true when it comes to speech. In a math class, we replicate verbal descriptions with word problems. Let's look at an example of this.

Chris mows 3 acres of his parent's farmland in 12 hours. Over the next several days, he mows 8 more acres in a total of 32 hours. How many hours does Chris spend per acre mowed?

Here, the constant of proportionality ratio we're trying to find is how many hours it takes to mow an acre. In other words, hours divided by acres mowed. We have two situations to test: 3 acres in 12 hours and 8 acres in 32 hours.

*k* = *y*/*x*

1) *k* = 12/3 = 4 hours per acre

2) *k* = 32/8 = 4 hours per acre

We can see it takes Chris 4 hours to mow a single acre. This is our constant because no matter how many acres he mows, or how much time he spends mowing, it will always take 4 hours to mow a single acre.

We just used a word problem to find the constant of proportionality, but in other situations, you might be asked to find one through a visual diagram or a table.

Diagrams and tables like the ones below work exactly the same way for finding the constant of proportionality. If we compare the two, we can see that each column of the table is like one image in the diagram.

Let's use this diagram and table to find the number of pieces of candy you get per dollar. We have three different sizes of candy boxes. Each image in the diagram and each column in the table tells us the price of a box and how many pieces of candy are in it. If there's a constant of proportionality, we should get the same number of pieces of candy per dollar for each of the three boxes.

*k* = *y* / *x*

1) *k* = 40 / 2.00 = 20 pieces per dollar

2) *k* = 80 / 4.00 = 20 pieces per dollar

3) *k* = 120 / 6.00 = 20 pieces per dollar

Regardless of the size of the box you purchase, you're always getting 20 pieces of candy per dollar.

The final way we can find a constant of proportionality is by using a line graph. A line graph that has a constant of proportionality in it will be a straight line that passes through the origin, e.g., the (0,0) point.

To find the constant of proportionality here let's look at the equation of a line for a straight graph that passes through the origin.

*y* = *m**x*

*y* is your vertical y-value for any given point on the line, *x* is your horizontal x-value for any given point on the line, and *m* is the slope. Does this equation look familiar to you? It looks exactly like the alternate form of the equation for the constant of proportionality we went over earlier in the lesson.

*y* = *k**x*

In a graph like this, the constant of proportionality is equal to the graph's slope. Our graph has a slope, and therefore a constant of proportionality of 25 miles per hour.

We call the ratio between two directly proportional quantities the **constant of proportionality**. When two quantities are **directly proportional**, they increase and decrease at the same rate. While these two quantities may increase or decrease, the constant of proportionality always remains the same. This is why we call it a constant.

The constant of proportionality (*k*) ratio is shown in this formula:

*k* = *y* / *x*

Here, *y* and *x* are two quantities that are directly proportional to each other. Often we see the constant of proportionality equation in this form as well:

*y* = *k**x*

We can use these equations to find the constant of proportionality in word problems, diagrams, tables, and graphs. Finally, it's worth noting that when looking at a straight-line graph that goes through the origin, (0,0) point, the constant of proportionality will always be equal to that graph's slope.

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6th-8th Grade Math: Practice & Review55 chapters | 469 lessons

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