Identifying and Representing Relationships

1.3.3 Identifying and Representing Relationships

1.3.3 • Identifying and Representing Relationships

Activity

The table represents the relationship between the base length and the height of some parallelograms. Both measurements are in inches.

1	48
2	24
3	16
4	12
6	8

Pairs of factors for the integer 48.

What is the relationship between the base length and the height of these parallelograms?

Compare your answer:

As the base length increases by 1, the height decreases, but not by a steady amount.
Multiplying the base length and the height gives 48. The area of each parallelogram is 48 square inches.
$b \cdot h = 48$ , or $\frac{48}{b} = h$ (or equivalent).
The height varies inversely as $b$ .

Visitors to a carnival are invited to guess the number of beans in a jar. The person who guesses the correct number wins $300. If multiple people guess correctly, the prize will be divided evenly among them.

What is the relationship between the number of people who guess correctly and the amount of money each person will receive?

Compare your answer:

1	300
2	150
3	100
5	60
15	20

This table presents pairs of integer values (X, Y) that demonstrate an inverse relationship, where their product (X * Y) consistently equals 300.

The amount in dollars received by each winner is 300 divided by the number of winners.
$a = \frac{300}{n}$ , or $a \cdot n = 300$ (or equivalent), where $a$ is the dollar amount a winner receives and $n$ is the number of winners.
The amount in dollars, $a$ , varies inversely as $n$ .

A $\frac{1}{2}$ -gallon jug of milk can fill 8 cups, while 32 fluid ounces of milk can fill 4 cups.

What is the relationship between the number of gallons and ounces? If you get stuck, try creating a table.

Compare your answer:

If 4 cups contain 32 fluid ounces, then 8 cups must contain 64 fluid ounces. Because there are 16 cups in 1 gallon and 16 is twice as much as 8, there must be 128 fluid ounces in a gallon.

$\frac{1}{2}$	8	64
1	16	128

This table presents a series of initial values and their corresponding products when multiplied first by 16 and then by 8.

Multiplying the number of gallons by 128 gives the number of fluid ounces.
$f = 8 \cdot 8 \cdot 2 \cdot g$ , or $f = 128 g$ (or equivalent), where $f$ is the number of fluid ounces and $g$ is the number of gallons.
The number of fluid ounces, $f$ , varies directly as the number of gallons, $g$ .

Building Character: Social Intelligence

Illustration of four diverse people, each inside a puzzle piece, interacting and waving. Various gears, leaves, and small icons surround them, representing teamwork and collaboration.

Social intelligence is the ability to connect with other people. Think about your current sense of social intelligence. Are the following statements true for you??

I have a lot of relationships that are mutually beneficial, enjoyable, and supportive.
Most of the time, I can tell how other people feel and have a good idea about how to respond appropriately.

Don’t worry if none of these statements are true for you. Developing this trait takes time. Your first step starts today!

Self Check

Number of guests, $g$	1	2	3	12
Number of cupcakes per guest, $n$	24	12	8	2

What is one way to write the relationship between the number of guests and the number of cupcakes per guest from the table above?

$n = 24 - g$
$n=\frac{g}{24}$
$n=\frac{24}{g}$
$n= 24g$

Additional Resources

Finding Relationships between Quantities

There are times when the relationship between quantities may not be obvious. In some cases, the relationship between quantities might take a bit of work to figure out, by doing calculations several times or by looking for a pattern.

Here are two examples.

Example 1

A plane departed from New Orleans and is heading to San Diego. The table shows its distance from New Orleans, $x$ , and its distance from San Diego, $y$ , at some points along the way.

miles from New Orleans	miles from San Diego
100	1,500
300	1,300
500	1,100
	1,020
900	700
1,450
$x$	$y$

What is the relationship between the two distances?

Compare your answer: Your answer may vary, but here are some samples.

The distances represent one path—the distance from New Orleans to San Diego. The two distances sum to 1600 miles. $x + y = 160$

Do you see any patterns in how each quantity is changing?

Compare your answer: Your answer may vary, but here is a sample.

As the $x$ -values increase, the $y$ -values decrease. As the $x$ -values increase by 200 miles, the $y$ -values decrease by 200 miles. The quantities, $x$ and $y$ , sum to 1600 even after changing.

What is the value of $x$ when $y$ = 1020?

580 miles

What is the value of $y$ when $x$ = 1450?

150.

Example 2

A company decides to donate $50,000 to charity. It will select up to 20 charitable organizations, as nominated by its employees. Each selected organization will receive an equal donation amount.

What is the relationship between the number of students, $s$ , and the dollar amount each student will receive, $d$ ? To begin, let's examine some specific values to help uncover the pattern.

If 5 organizations are selected, how much will each charity receive?

$10,000

If 10 organizations are selected, how much will each charity receive?

$5,000

If 20 organizations are selected, how much will each charity receive?

$2,500

Do you notice a pattern here? 10,000 is $\frac{50, 000}{5}$ , 5,000 is $\frac{50, 000}{10}$ , and 2,500 is $\frac{50, 000}{20}$ .

We can generalize that the amount each organization receives is 50,000 divided by the number of selected organizations, or $d = \frac{50, 000}{n}$ .

Try it

Finding Relationships between Quantities

A local business is going to hand out $20,000 in scholarships to students at local high schools.

What is the relationship between the number of students, $s$ , and the dollar amount each student will receive, $d$ ?

If 2 students are selected, what is the amount of the scholarship they will receive?

10,000

If there are 5 students selected, what is the amount of the scholarship they will receive?

4,000

If there are 10 students selected, what is the amount of the scholarship they will receive?

2,000

If there are 20 students selected, what is the amount of the scholarship they will receive?

1,000

What equation can be used to model the relationship between the number of students, $s$ , receiving scholarships and the dollar amount, $d$ , they receive?

Compare your answer:

Your answer may vary, but here is a sample. The amount each student receives is $20,000 divided by the number of students or, $d = \frac{20000}{s}$ . The dollar amount each student receives, $d$ , varies inversely as the number of students, $s$ .