2.1.3: More about Constant of Proportionality
Lesson
Let's solve more problems involving proportional relationships using tables.
Exercise \(\PageIndex{1}\): Equal Measures
Use the numbers and units from the list to find as many equivalent measurements as you can. For example, you might write “30 minutes is \(\frac{1}{2}\) hour.”
You can use the numbers and units more than once.
\(\begin{array}{llll}{1}&{\frac{1}{2}}&{0.3}&{\text{centimeter}}\\{12}&{40}&{24}&{\text{meter}}\\{0.4}&{0.01}&{\frac{1}{10}}&{\text{hour}}\\{60}&{3\frac{1}{3}}&{6}&{\text{feet}}\\{50}&{30}&{2}&{\text{minute}}\\{}&{}&{\frac{2}{5}}&{\text{inch}}\end{array}\)
Exercise \(\PageIndex{2}\): Centimeters and Millimeters
There is a proportional relationship between any length measured in centimeters and the same length measured in millimeters.
There are two ways of thinking about this proportional relationship.
-
If you know the length of something in centimeters, you can calculate its length in millimeters.
- Complete the table.
-
What is the constant of proportionality?
length (cm) length (mm) 9 12.5 50 88.49 Table \(\PageIndex{1}\)
-
If you know the length of something in millimeters, you can calculate its length in centimeters.
- Complete the table.
-
What is the constant of proportionality?
length (mm) length (cm) 70 245 4 699.1 Table \(\PageIndex{2}\)
- How are these two constants of proportionality related to each other?
-
Complete each sentence:
- To convert from centimeters to millimeters, you can multiply by ________.
- To convert from millimeters to centimeters, you can divide by ________ or multiply by ________.
Are you ready for more?
- How many square millimeters are there in a square centimeter?
- How do you convert square centimeters to square millimeters? How do you convert the other way?
Exercise \(\PageIndex{3}\): Pittsburgh to Phoenix
On its way from New York to San Diego, a plane flew over Pittsburgh, Saint Louis, Albuquerque, and Phoenix traveling at a constant speed.
Complete the table as you answer the questions. Be prepared to explain your reasoning.
| segment | time | distance | speed |
|---|---|---|---|
| Pittsburgh to Saint Louis | 1 hour | 550 miles | |
| Saint Louis to Albuquerque | 1 hour 42 minutes | ||
| Albuquerque to Phoenix | 330 miles |
- What is the distance between Saint Louis and Albuquerque?
- How many minutes did it take to fly between Albuquerque and Phoenix?
- What is the proportional relationship represented by this table?
- Diego says the constant of proportionality is 550. Andre says the constant of proportionality is \(9\frac{1}{6}\). Do you agree with either of them? Explain your reasoning.
Summary
When something is traveling at a constant speed, there is a proportional relationship between the time it takes and the distance traveled. The table shows the distance traveled and elapsed time for a bug crawling on a sidewalk.
We can multiply any number in the first column by \(\frac{2}{3}\) to get the corresponding number in the second column. We can say that the elapsed time is proportional to the distance traveled, and the constant of proportionality is \(\frac{2}{3}\). This means that the bug’s pace is \(\frac{2}{3}\) seconds per centimeter.
This table represents the same situation, except the columns are switched.
We can multiply any number in the first column by \(\frac{3}{2}\) to get the corresponding number in the second column. We can say that the distance traveled is proportional to the elapsed time, and the constant of proportionality is \(\frac{3}{2}\). This means that the bug’s speed is \(\frac{3}{2}\) centimeters per second.
Notice that \(\frac{3}{2}\) is the reciprocal of \(\frac{2}{3}\). When two quantities are in a proportional relationship, there are two constants of proportionality, and they are always reciprocals of each other. When we represent a proportional relationship with a table, we say the quantity in the second column is proportional to the quantity in the first column, and the corresponding constant of proportionality is the number we multiply values in the first column to get the values in the second.
Glossary Entries
Definition: Constant of Proportionality
In a proportional relationship, the values for one quantity are each multiplied by the same number to get the values for the other quantity. This number is called the constant of proportionality.
In this example, the constant of proportionality is 3, because \(2\cdot 3=6\), \(3\cdot 3=9\), and \(5\cdot 3=15\). This means that there are 3 apples for every 1 orange in the fruit salad.
| number of oranges | number of apples |
|---|---|
| 2 | 6 |
| 3 | 9 |
| 5 | 15 |
Definition: Equivalent Ratios
Two ratios are equivalent if you can multiply each of the numbers in the first ratio by the same factor to get the numbers in the second ratio. For example, \(8:6\) is equivalent to \(4:3\), because \(8\cdot\frac{1}{2}=4\) and \(6\cdot\frac{1}{2}=3\).
A recipe for lemonade says to use 8 cups of water and 6 lemons. If we use 4 cups of water and 3 lemons, it will make half as much lemonade. Both recipes taste the same, because and are equivalent ratios.
| cups of water | number of lemons |
|---|---|
| 8 | 6 |
| 4 | 3 |
Definition: Proportional Relationship
In a proportional relationship, the values for one quantity are each multiplied by the same number to get the values for the other quantity.
For example, in this table every value of \(p\) is equal to 4 times the value of \(s\) on the same row.
We can write this relationship as \(p=4s\). This equation shows that \(s\) is proportional to \(p\).
| \(s\) | \(p\) |
|---|---|
| 2 | 8 |
| 3 | 12 |
| 5 | 20 |
| 10 | 40 |
Practice
Exercise \(\PageIndex{4}\)
Noah is running a portion of a marathon at a constant speed of 6 miles per hour.
Complete the table to predict how long it would take him to run different distances at that speed, and how far he would run in different time intervals.
| time in hours | miles traveled at 6 miles per hour |
|---|---|
| \(1\) | |
| \(\frac{1}{2}\) | |
| \(1\frac{1}{3}\) | |
| \(1\frac{1}{2}\) | |
| \(9\) | |
| \(4\frac{1}{2}\) |
Exercise \(\PageIndex{5}\)
One kilometer is 1000 meters.
1. Complete the tables. What is the interpretation of the constant of proportionality in each case?
| meters | kilometers |
|---|---|
| 1,000 | 1 |
| 250 | |
| 12 | |
| 1 |
The constant of proportionality tells us that:
| kilometers | meters |
|---|---|
| 1 | 1,000 |
| 5 | |
| 20 | |
| 0.3 |
The constant of proportionality tells us that:
2. What is the relationship between two constants of proportionality?
Exercise \(\PageIndex{6}\)
Jada and Lin are comparing inches and feet. Jada says that the constant of proportionality is 12. Lin says it is \(\frac{1}{12}\). Do you agree with either of them? Explain your reasoning.
Exercise \(\PageIndex{7}\)
The area of the Mojave desert is 25,000 square miles. A scale drawing of the Mojave desert has an area of 10 square inches. What is the scale of the map?
(From Unit 1.2.6)
Exercise \(\PageIndex{8}\)
Which of these scales is equivalent to the scale 1 cm to 5 km? Select all that apply.
- 3 cm to 15 km
- 1 mm to 150 km
- 5 cm to 1 km
- 5 mm to 2.5 km
- 1 mm to 500 m
(From Unit 1.2.5)
Exercise \(\PageIndex{9}\)
Which one of these pictures is not like the others? Explain what makes it different using ratios.
(From Unit 2.1.1)