5.2: Applications of Proportionality (2024)

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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.

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 $\PageIndex{1}$

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.

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 could 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 $\PageIndex{2}$

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 -- 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!

Solve Similar Figure Applications

When you shrink or enlarge a photo on a phone or tablet, figure out a distance on a map, or use a pattern to build a bookcase or sew a dress, you are working with similar figures. If two figures have exactly the same shape, but different sizes, they are said to be similar. One is a scale model of the other. All their corresponding angles have the same measures and their corresponding sides are in the same ratio.

Definition: SIMILAR FIGURES

Two figures are similar if the measures of their corresponding angles are equal and their corresponding sides are in the same ratio.

For example, the two triangles in Figure are similar. Each side of ΔABC is 4 times the length of the corresponding side of ΔXYZ.

5.2: Applications of Proportionality (1)

This is summed up in the Property of Similar Triangles.

Definition: PROPERTY OF SIMILAR TRIANGLES

If ΔABC is similar to ΔXYZ

To solve applications with similar figures we will follow the Problem-Solving Strategy for Geometry Applications we used earlier.

Definition: SOLVE GEOMETRY APPLICATIONS.

Read the problem and make all the words and ideas are understood. Draw the figure and label it with the given information.
Identify what we are looking for.
Name what we are looking for by choosing a variable to represent it.
Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
Solve the equation using good algebra techniques.
Check the answer in the problem and make sure it makes sense.
Answer the question with a complete sentence.

Example $\PageIndex{8}$

ΔABC is similar to ΔXYZ

Step 1. Read the problem. Draw the figure and label it with the given information.	Figure is given.
Step 2. Identify what we are looking for.	the length of the sides of similar triangles.
Step 3. Name the variables.	Let a= length of the third side of ΔABC. y= length of the third side of ΔXYZ.
Step 4. Translate.	Since the triangles are similar, the corresponding sides are proportional.
We need to write an equation that compares the side we are looking for to a known ratio. Since the side AB = 4 corresponds to the side XY = 3 we know $\dfrac{AB}{XY}=\dfrac{4}{3}$. So we write equations with $\dfrac{AB}{XY}$ to find the sides we are looking for. Be careful to match up corresponding sides correctly.	$\dfrac{AB}{XY}=\dfrac{BC}{YZ}=\dfrac{AC}{XZ}$.
Substitute.
Step 5. Solve the equation.

Step 6. Check.
Step 7. Answer the question.	The third side of ΔABC is 6 and the third side of ΔXYZ is 2.4.

TRy it $\PageIndex{15}$

ΔABC is similar to ΔXYZ. The lengths of two sides of each triangle are given in the figure.

Find the length of side a

Answer: 8

TRy it $\PageIndex{16}$

ΔABC is similar to ΔXYZ. The lengths of two sides of each triangle are given in the figure.

Answer: 22.5

The next example shows how similar triangles are used with maps.

Example $\PageIndex{9}$

On a map, San Francisco, Las Vegas, and Los Angeles form a triangle whose sides are shown in the figure below. If the actual distance from Los Angeles to Las Vegas is 270 miles find the distance from Los Angeles to San Francisco.

Read the problem. Draw the figures and label with the given information.	The figures are shown above.
Identify what we are looking for.	The actual distance from Los Angeles to San Francisco.
Name the variables.	Let x= distance from Los Angeles to San Francisco.
Translate into an equation. Since the triangles are similar, the corresponding sides are proportional. We'll make the numerators "miles" and the denominators "inches."

Solve the equation.
Check.
On the map, the distance from Los Angeles to San Francisco is more than the distance from Los Angeles to Las Vegas. Since 351 is more than 270 the answer makes sense.

Answer the question.	The distance from Los Angeles to San Francisco is 351 miles.

TRY IT $\PageIndex{17}$

On the map, Seattle, Portland, and Boise form a triangle whose sides are shown in the figure below. If the actual distance from Seattle to Boise is 400 miles, find the distance from Seattle to Portland.

Answer: 150 miles

Try it $\PageIndex{18}$

Using the map above, find the distance from Portland to Boise.

Answer: 350 miles

We can use similar figures to find heights that we cannot directly measure.

Example $\PageIndex{10}$

Tyler is 6 feet tall. Late one afternoon, his shadow was 8 feet long. At the same time, the shadow of a tree was 24 feet long. Find the height of the tree.

Read the problem and draw a figure.
We are looking for h, the height of the tree.
We will use similar triangles to write an equation.
The small triangle is similar to the large triangle.
Solve the proportion.
Simplify.
Check.
Tyler's height is less than his shadow's length so it makes sense that the tree's height is less than the length of its shadow.

Try it $\PageIndex{19}$

A telephone pole casts a shadow that is 50 feet long. Nearby, an 8 foot tall traffic sign casts a shadow that is 10 feet long. How tall is the telephone pole?

Answer: 40 feet

Try it $\PageIndex{20}$

A pine tree casts a shadow of 80 feet next to a 30-foot tall building which casts a 40 feet shadow. How tall is the pine tree?

Answer: 60 feet

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.

Your car uses 10 gallons of gas to go 300 miles.
- How many gallons of gas will you need to go 400 miles?
- Is this situation described by direct or inverse proportionality, and why? Give a one-sentence answer.
- What is the constant of proportionality in this situation, and how would you interpret it?
It takes 2 math professors a total of 6 hours to grade the exams for a large math class. Assume all professor grade at the same rate.
- How many professors would it take to grade the same exams in 4 hours?
- Is this situation described by direct or inverse proportionality, and why? Give a one-sentence answer.
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.)
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.)
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.
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.
Exercises $\PageIndex{1}$

Tonisha drove her car $320$ miles and used $12.5$ gallons of gas. At this rate, how far could she drive using $10$ gallons of gas?
Marcus worked $14$ hours and earned $ $210$. At the same rate of pay, how long would he have to work to earn $ $300$?
A picture of your author appearing on Jeopardy! that is $375$ pixels high and $475$ pixels wide needs to be reduced in size so that it is $150$ pixels high. If the height and width are kept proportional, what is the width of the picture after it has been reduced?
At a fast-food restaurant, a 22-ounce chocolate shake has 850 calories. How many calories are in their 12-ounce chocolate shake? Round your answer to nearest whole number.
Yurianna is going to Europe and wants to change $800 dollars into Euros. At the current exchange rate, $1 US is equal to 0.738 Euro. How many Euros will she have for her trip? Answer
7. $256$ miles

8. $20$ hours

9. $190$ pixels wide

10. 464 calories

11. 590.4 Euros

5.2: Applications of Proportionality (2024)

In this section, you will learn to:

Direct Proportionality

Definition: Directly Proportional

Example \(\PageIndex{1}\)

Solution

Definition: Constant of Proportionality

Example \(\PageIndex{1}\)

Solution

Inverse Proportionality

Example \(\PageIndex{2}\)

Definition: Inversely Proportional

Example \(\PageIndex{2}\)

Solution

Solve Similar Figure Applications

Definition: SIMILAR FIGURES

Definition: PROPERTY OF SIMILAR TRIANGLES

Definition: SOLVE GEOMETRY APPLICATIONS.

Example \(\PageIndex{8}\)

TRy it \(\PageIndex{15}\)

TRy it \(\PageIndex{16}\)

Example \(\PageIndex{9}\)

TRY IT \(\PageIndex{17}\)

Try it \(\PageIndex{18}\)

Example \(\PageIndex{10}\)

Try it \(\PageIndex{19}\)

Try it \(\PageIndex{20}\)

Exercises

Exercises \(\PageIndex{1}\)