2.1: Types of Proportionality

In the previous section, we studied proportions, and used them to solve problems involving ratios. In this section, we continue our study of proportions, and investigate two different types of proportionality.

In this section, you will learn to:

• Recognize direct proportionality relationships, and use them to answer questions involving direct proportionality
• Compute and interpret the constant of proportionality in context
• Recognize inverse proportionality relationships, and use them to answer questions involving inverse proportionality

There are two main types of proportionality. We will learn about the more common one first.

Direct Proportionality

Definition: Directly Proportional

Two quantities are directly proportional if, as one quantity increases, the other quantity also increases at the same rate.

Direct proportionality describes all of the proportion problems we've seen before. Here is another example that shows how direct proportionality works, and introduces the next important notion.

Example \(\PageIndex{1}\)

At an hourly wage job, you work for \(5\) hours, and get paid \(\$83.75\). How much money will you earn if you work \(7\) hours? (Assume that you are not making overtime pay, or any other sort of special pay rate.)

Solution

This is a situation described by direct proportionality. Since this is an hourly wage job, and we're told that there is no overtime pay or other special pay rate, we can safely assume that we make the same amount per hour in this job. In other words, the rate of pay will be the same no matter how many hours we work. That means that if the number of hours worked increases, the amount you're paid increases at the same rate, no matter the number of hours worked.

This is a situation described by direct proportionality. Since this is an hourly wage job, and we're told that there is no overtime pay or other special pay rate, we can safely assume that we make the same amount per hour in this job. In other words, the rate of pay will be the same no matter how many hours we work. That means that if the number of hours worked increases, the amount you're paid increases at the same rate, no matter the number of hours worked.

We can solve this using a proportion, using the same techniques as in the last section. We will set up the following proportion equation: \[\frac{\$83.75}{5 \text{ hours}} = \frac{\$x}{7 \text{ hours}}\]

Notice that we have picked a variable, \(x\), to denote the answer we are trying to find -- the number of dollars earned for working \(7\) hours. Also notice how we've labeled our units, and have made sure that the corresponding quantities are together. If you do this every time -- label your units, and make sure corresponding quantities stay together -- you can solve any direct proportion problem.

Now for the process that will actually help us solve for \(x\). First, rewrite the equation without labels: \[\frac{83.75}{5} = \frac{x}{7}\]

Next, apply Cross Multiplication \[5x = 83.75 \times 7\]

Next, simplify the right side (using a calculator): \[5x = 586.25\]

Finally, apply Division undoes Multiplication to find \(x\): \[x = \frac{586.25}{5} = 117.25\]

That means that if you work \(7\) hours, you will make \(\$117.25\). Think for a moment to see if that's reasonable: it's more, but not too much more, than you made working for \(5\) hours. So, it seems like a sensible answer.

The approach above works just fine to find the desired answer. However, what if you wanted to know how much you'd make working for \(3\) hours? Or \(4\) hours? Or \(10\) hours? You could just reproduce the work above each time, you wanted. But you may also find the following approach quicker:

Definition: Constant of Proportionality

In a situation involving directly proportional quantities, the constant of proportionality is the common ratio that describes the comparison of any two corresponding quantities. In other words, it is the constant rate of change between the two quantities.

Let's see how to find a constant of proportionality, and how to interpret it.

Example \(\PageIndex{1}\) Revisited

You're in the same situation as the previous example: you work for \(5\) hours, and earn \($83.75\). What is the constant of proportionality in this example, and what does it mean in context?

Solution

The constant of proportionality is the common ratio that describes the comparison of any two corresponding quantities. In this situation, we actually have two sets of corresponding quantities, one of which we found in the previous example. You know that you'll make \($83.75\) working for \(5\) hours, and we calculated above that you'll make \(\$117.25\) if you work for \(7\) hours. Let's look at these two ratios:

\[\frac{\$83.75}{5 \text{ hours}} \quad \text{ and } \quad \frac{\$117.25}{7 \text{ hours}}\]

Both of these ratios are fractions, and we can simply divide the top by the bottom to reduce them to a single number. Using a calculator, we can see that

\[\frac{\$83.75}{5 \text{ hours}} = \$83.75 \div 5 \text{ hours} = \$16.75 \text{ per hour}\]

and

\[\frac{\$117.25}{7 \text{ hours}} = \$117.25 \div 7 \text{ hours} = \$16.75 \text{ per hour}\]

These are the same answer! Do you see why? We originally found our value for \(x\) in the previous example by setting the two ratios equal. So, they must give the same answer when divided.

This shared rate — \(\$16.75\) per hour — is the constant of proportionality in this situation. It is the shared value of all ratios described by this problem, where the top of the ratio is money earned, and the bottom is hours worked.

What does this constant of proportionality mean in this situation? In this case, the constant of proportionality is your hourly pay rate. In other words, it's how much you make per hour.

A few more comments on the example above: now that you know this rate, it's quite simple to find how much money you'll make if you work \(3, 4,\) or \(10\) hours. You just multiply your hourly rate by the number of hours worked. For example, if you worked \(4\) hours, you could calculate:

\[\underset{\text{hours}}{4} \times \$16.75 \text{ per hour} = \$67.00\]

This means you would make \(\$67.00\) working for \(4\) hours. Notice that if we reverse the process — in other words, if we try to extract the constant of proportionality knowing that we make \(\$67.00\) in \(4\) hours, we get:

\[\frac{\$67.00}{4 \text{ hours}} = \$67.00 \div 4 \text{ hours} = \$16.75 \text{ per hour}\]

It's the same rate we found before. This is why it's called a constant of proportionality: it stays the same, even as the corresponding quantities change.

Constants of proportionality will change in meaning depending on the context of the problem. For example, you might ask: If \(5\) people eat a total of \(10\) slices of pizza, how many slices does each person eat? In this case, the constant of proportionality could be found by: \[\frac{10 \text{ slices}}{5 \text{ people}} = 2 \text{ slices per person}\]

In this case, a correct interpretation would be: "The constant of proportionality is \(2\) slices per person, which means that each person eats \(2\) slices of pizza." When asked for an interpretation, you should write a sentence similar to the previous -- your goal is the explain the meaning of the constant of proportionality in context of the situation. You will need to read carefully and use critical thinking to deduce a meaningful interpretation. As with many questions in this class, there are multiple good answers to these types of questions!

Inverse Proportionality

Let's start this section with a question to illustrate the main concept. Before reading ahead, try to answer this question on your own:

Example \(\PageIndex{2}\)

Suppose it takes \(6\) sanitation workers, all working simultaneously, \(4\) hours to pick up the trash and recycling in a given neighborhood. How many sanitation workers would it take to pick up the trash and recycling in the same neighborhood in \(3\) hours? (Assume that all sanitation workers work at the same rate, and can work independently.)

Did you try to answer the question yourself? What did you come up with? If you're like many students who carefully read the previous sections, you might have written this down:

\[\frac{6 \text{ sanitation workers}}{4 \text{ hours}} = \frac{x \text{ sanitation workers}}{3 \text{ hours}}\]

Then you'd apply Cross Multiplication to get:

\[4x = 18\]

and then you'd use Division which undoes Multiplication to get

\[x = \frac{18}{4} = 4.5\]

Now, the numerical answer \(4.5\) is a bit nonsensical, because it's talking about a number of people. So you would round up to \(5\), and say "It would take 5 sanitation workers to pick up the trash and recycling in \(3\) hours."

But wait a second: this does not make sense! Think about it: if it takes \(6\) workers \(4\) hours to accomplish this task, shouldn't it take \(5\) workers more time than \(4\) hours? After all, there is the same amount of work to be done, but fewer people to do it! So the answer "5 workers" cannot possibly be correct. We expect a number of workers that is larger than 6 to get the task done in a shorter amount of time.

We can learn two things from the previous discussion:

• It is important to evaluate whether or not an answer to a question makes sense in context by asking: What sort of answer would I expect to get? Does my answer seem reasonable?
• Not all problems can be solved using direct proportionality!

The good news is that this type of problem can be solved in a relatively simple way. We define the main concept in this section to see how these problems work.

Definition: Inversely Proportional

Two quantities are inversely proportional if, as one quantity increases, the other quantity decreases at the same rate.

Note how similar this definition is to the previous definition of direct proportionality. The only difference here is that one quantity increases while the other decreases:

Direct Proportionality	Inverse Proportionality
As one quantity increases	As one quantity increases
the other quantity increases	the other quantity decreases

In order to determine what type of problem you're working on, you'll need to think critically about the quantities involved, and use clues from your experience and the context of the problem to determine how the quantities are related. Things like the previous problem — when a group of people are working together to accomplish a specific task — are one of the primary examples of inverse proportionality. Let's see the same example again, and this time find the correct answer.

Example \(\PageIndex{2}\) Revisited

Suppose it takes \(6\) sanitation workers, all working simultaneously, \(4\) hours to pick up the trash and recycling in a given neighborhood. How many sanitation workers would it take to pick up the trash and recycling in the same neighborhood in \(3\) hours? (Assume that all sanitation workers work at the same rate, and can work independently.)

Solution

The way to approach this is to find the number of worker hours needed to accomplish the task of picking up the trash and recycling in this neighborhood. A worker hour is defined to be an hour of work done by a worker, and that number will remain constant no matter the number of workers used.

To find the number of worker hours needed for this particular neighborhood, we simply multiply the known number of workers by the known number of hours:
\[\underset{\text{workers}}{6} \times \underset{\text{hours}}{4} = \underset{\text{worker hours}}{24}\]

That means that it will require \(24\) worker hours to pick up the trash and recycling in this neighborhood.

To find the number of workers needed to pick up the trash and recycling in \(3\) hours, we divide the number of worker hours by the number of hours to find the number of workers:

\[\frac{24 \text{ worker hours}}{3 \text{ hours}} = \underset{\text{worker hours}}{24} \div \underset{\text{hours}}{3} = \underset{\text{workers}}{8}\]

This means it will take \(8\) workers \(3\) hours to pick up the trash and recycling. This makes sense, since it's larger than \(6\), which was the number of workers needed to accomplish the task in \(4\) hours.

All inverse proportionality problems work this way — multiply the two known corresponding quantities, and then divide to find the answer. As always, label your units, and check to see if your answers make sense!

Exercises

Make sure that when you are asked to interpret something, you write a complete sentence describing the meaning of your numerical answer in the context of the problem.

1. Your car uses 10 gallons of gas to go 300 miles.
   1. Is this situation described by direct or inverse proportionality, and why? Give a one-sentence answer.
   2. How many gallons of gas will you need to go 400 miles? Round to the nearest tenth. 
   3. What is the constant of proportionality in this situation, and how would you interpret it?
2. It takes 2 math professors a total of 6 hours to grade the exams for a large math class. Assume all professors grade at the same rate.
   1. Is this situation described by direct or inverse proportionality, and why? Give a one-sentence answer.
   2. How many professors would it take to grade the same exams in 4 hours?
3. A family drinks 2 gallons of milk every 9 days. How many gallons of milk will they use in 2 weeks? Be careful with units here! (Round to one decimal place.)
4. At a rate of 30 miles per hour, a certain trip takes 2 hours. How long would the same trip take at 40 miles per hour? (Round to one decimal place or give a fractional answer.)
5. A group of 10 musicians can play a song in 6 minutes. How long would it take for a group of 20 musicians to play the same song? [Hint: Evaluate whether your answer makes sense!!!]
6. Think of a real-world example of direct proportionality that is different than ones we've covered in this section. Give a 2-3 sentence description of the quantities involved, and why you think they are directly proportional. If you use a source, please cite it by providing a URL.
7. Think of a real-world example of inverse proportionality that is different than ones we've covered in this section. Give a 2-3 sentence description of the quantities involved, and why you think they are inversely proportional. If you use a source, please cite it by providing a URL.