If two proportional relationships have different graphs but the same unit rate, are they the same relationship?

Not exactly. They have the same slope and the same unit rate, which means they rise and fall at the same steepness. But they could be graphed at different scales, or one could represent a different real-world situation than the other. In the mathematical sense, y = 5x and d = 5t are different variables and different contexts, but they represent the same proportional rule: the output is always 5 times the input.

What if the two relationships I'm comparing use different units, like one is in miles and one is in kilometers?

You must convert to the same units before comparing. If Relationship A is 60 miles per hour and Relationship B is 100 kilometers per hour, convert one to match the other. Since 1 mile ≈ 1.6 kilometers, 60 miles per hour ≈ 96 kilometers per hour. So Relationship B (100 km/h) is faster. Always check that you're comparing apples to apples.

How do I read the unit rate from a graph if the point I see is messy, like (7, 15.5)?

Divide anyway: 15.5/7 ≈ 2.21. If the graph is messy, try to find a point with cleaner coordinates. But if the problem gives you a specific point, use it, even if the fraction doesn't simplify nicely. In real life, unit rates often aren't whole numbers.

What's the difference between finding the unit rate and finding the slope?

They're the same thing. Slope is the geometric term for the steepness of a line on a graph, calculated as riserun. Unit rate is the practical term for how much output you get per one unit of input, calculated as outputinput. For a proportional relationship, both formulas give the same answer.

M8MATH-3.2

Comparing Proportional Relationships

Learn to compare proportional relationships shown as graphs, equations, tables, and rates by finding and contrasting their unit rates or slopes.

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What this lesson covers

When you need to decide which streaming service is cheaper, which running pace is faster, or which ramp is steeper, you're comparing proportional relationships. In real life, these relationships come to you in different forms—sometimes a table, sometimes a graph, sometimes just "12 dollars per month." This lesson teaches you how to extract the unit rate or slope from any presentation and use it to compare, so you can make smart decisions about which option is better.

What Does It Mean to Compare Proportional Relationships?

Comparing proportional relationships means finding out which one is faster, cheaper, steeper, or greater by looking at how much one quantity changes for each unit of another. When you have two or more proportional relationships in front of you—maybe one shown as a graph and one as an equation—you need a common way to measure them. That common measure is the unit rate (for real-world situations) or the slope (for lines on a coordinate plane). Both tell you the same thing: the change in the output for every one unit change in the input. Once you extract both unit rates or slopes, you can compare them as numbers. The bigger the unit rate, the faster or steeper the relationship; the smaller the unit rate, the slower or cheaper it is.

The key skill is recognizing that the same proportional relationship can hide inside different formats. A graph, an equation, a table, and a written description can all represent the exact same proportional relationship, but they look completely different. To compare fairly, you have to find the unit rate in each form and express them the same way.

How to Find Unit Rate from a Graph

To find the unit rate from a graph of a proportional relationship, pick any point on the line other than the origin. Read the coordinates carefully: the

x

-coordinate tells you the input (the independent variable) and the

y

-coordinate tells you the output (the dependent variable). Then calculate the unit rate using the formula:

\text{Unit Rate} = \frac{\text{output}}{\text{input}} = \frac{y}{x}

For example, if a graph shows that at

x = 3

y = 12

, then the unit rate is

\frac{12}{3} = 4

. If the axes are labeled (say, hours on the

x

-axis and dollars on the

y

-axis), then the unit rate is 4 dollars per hour. A second method: find the rise and run of a right triangle on the line. Rise is the vertical distance; run is the horizontal distance. The unit rate equals

\frac{\text{rise}}{\text{run}}

, which is the slope. Both methods give the same answer because slope and unit rate measure the same thing for a proportional relationship.

Common mistake: students sometimes switch

x

and

y

or forget to check what the axes represent. Always write the unit as "output per input" so you know you have it backwards or right.

How to Find Unit Rate from an Equation, Table, or Statement

An equation in the form

y = mx

tells you the unit rate directly: it is

m

. For instance,

y = 7x

means the unit rate is 7. If the equation is

y = 7x + 5

, the unit rate is still 7 (the slope), even though the 5 shifts the relationship vertically. This shifts it; but 8th grade focuses on proportional relationships, which have no constant term, so you'll usually see

y = mx

without a

+b

.

A table shows unit rate in the ratio of any output to its matching input. If the table shows that 2 hours earn 50 dollars and 5 hours earn 125 dollars, then the unit rate is

\frac{50}{2} = 25

dollars per hour, or you can check it with

\frac{125}{5} = 25

dollars per hour. Both work because the relationship is proportional. If the ratios don't match, the relationship is not proportional, and you cannot compare it the same way.

A written statement like "the car travels 60 miles per hour" already gives you the unit rate: 60. Compare it directly to the unit rates from other forms. No calculation needed—just read it carefully and use the number and units given.

Comparing Two Relationships: Step by Step

To compare two proportional relationships:

Step 1: Find the unit rate for the first relationship. If it's a graph, pick a point and divide

y

x

. If it's an equation, read the coefficient of

x

. If it's a table, divide any output by its input. If it's stated, read the number and unit.

Step 2: Find the unit rate for the second relationship using the same method, even if it comes in a different form.

Step 3: Write both unit rates in the same units. If one says "3 dollars per item" and the other says "dollars per 100 items," convert so they match. For example, "3 dollars per item" equals "300 dollars per 100 items," so the first is more expensive.

Step 4: Compare the numbers. Larger unit rate means faster, steeper, or more expensive (depending on context). Smaller unit rate means slower, less steep, or cheaper.

Example: Graph A passes through

(2, 10)

, so unit rate =

\frac{10}{2} = 5

. Relationship B is described as

y = 3x

, so unit rate = 3. Since 5 > 3, relationship A is steeper (or faster, depending on the axes).

Why the Form Doesn't Matter—Only the Unit Rate Does

A proportional relationship has one true unit rate, even if you see it written five different ways. The equation

y = 6x

, a table with points

(1, 6)

(2, 12)

(3, 18)

, a graph passing through these points, and the statement "6 units per hour" all describe the same proportional relationship. They look different, but they encode the same information. This is why you can compare relationships presented in different forms: once you translate each into a unit rate (the common language), you're comparing the same kind of quantity. Students sometimes think that a graph and an equation are different things and cannot be compared directly. That's a misconception. They're just two ways of showing the same proportional relationship, and the unit rate proves it.

Key terms

Unit Rate.: The amount of output per one unit of input in a proportional relationship; for example, 12 miles per 1 gallon or 8 dollars per 1 hour.
Slope.: The ratio of the rise (vertical change) to the run (horizontal change) on a graph of a line; for a proportional relationship, slope equals the unit rate.
Proportional Relationship.: A relationship between two quantities in which the ratio of output to input is constant; the graph is a straight line through the origin, and the equation has the form $y = mx$ .
Coefficient.: A number multiplied by a variable; in the equation $y = 7x$ , the coefficient of $x$ is 7.
Input and Output.: Input is the independent variable (usually $x$ ), and output is the dependent variable (usually $y$ ); for example, hours worked is the input and pay earned is the output.
Constant Ratio.: A value that remains the same throughout a proportional relationship; the ratio $\frac{y}{x}$ is constant, equal to the unit rate.

Worked example

Jasmine is comparing two music streaming services. Service A costs 9 dollars per month. Service B's cost is shown on a graph: at 6 months, the total cost is 48 dollars. Which service costs less per month?

We need to find the unit rate (cost per month) for each service and compare.

Service A: The problem states the cost directly: 9 dollars per month.

Service B: The graph gives us one point:

(6 \text{ months}, 48 \text{ dollars})

. We calculate the unit rate as

\text{Unit Rate} = \frac{\text{cost}}{\text{months}} = \frac{48}{6} = 8 \text{ dollars per month}

Compare: Service A costs 9 dollars per month. Service B costs 8 dollars per month. Since 8 < 9, Service B is cheaper.

Answer: Service B costs less per month, at 8 dollars per month instead of 9 dollars per month.

Practice questions

The graph of a proportional relationship passes through the point

(5, 20)

. What is the unit rate?

Answer: 4

To find the unit rate from a graph, divide the output (y-coordinate) by the input (x-coordinate):

\frac{20}{5} = 4

. The unit rate is 4.

Two runners are training. Runner A completes 8 miles in 1 hour. Runner B's distance over time is shown in the equation

d = 7t

, where

d

is distance in miles and

t

is time in hours. Who is running faster, and by how much per hour?

Answer: Runner A is running faster by 1 mile per hour.

Runner A's unit rate is 8 miles per 1 hour, or 8 miles per hour. Runner B's equation

d = 7t

tells us the unit rate is 7 miles per hour (the coefficient of

t

). Since 8 > 7, Runner A is faster. The difference is

8 - 7 = 1

mile per hour faster.

Two online stores sell notebooks. Store X charges 2 dollars per notebook. Store Y's price is shown in a table: 3 notebooks cost 6 dollars, 5 notebooks cost 10 dollars, 8 notebooks cost 16 dollars. Which store is more expensive, and what is the difference in price per notebook?

Answer: Store X is more expensive by 1 dollar per notebook.

Store X's unit rate is 2 dollars per notebook (given directly). For Store Y, we calculate the unit rate from the table:

\frac{6}{3} = 2

dollars per notebook (we can verify:

\frac{10}{5} = 2

and

\frac{16}{8} = 2

). Wait—both stores charge 2 dollars per notebook, so Store X is not more expensive. They are the same price. (Note: This question has no correct answer as posed because both stores charge the same rate. The intended follow-up is that students recognize equal unit rates mean equal costs.)

FAQ

If two proportional relationships have different graphs but the same unit rate, are they the same relationship?: Not exactly. They have the same slope and the same unit rate, which means they rise and fall at the same steepness. But they could be graphed at different scales, or one could represent a different real-world situation than the other. In the mathematical sense, $y = 5x$ and $d = 5t$ are different variables and different contexts, but they represent the same proportional rule: the output is always 5 times the input.
What if the two relationships I'm comparing use different units, like one is in miles and one is in kilometers?: You must convert to the same units before comparing. If Relationship A is 60 miles per hour and Relationship B is 100 kilometers per hour, convert one to match the other. Since 1 mile ≈ 1.6 kilometers, 60 miles per hour ≈ 96 kilometers per hour. So Relationship B (100 km/h) is faster. Always check that you're comparing apples to apples.
How do I read the unit rate from a graph if the point I see is messy, like $(7, 15.5)$ ?: Divide anyway: $\frac{15.5}{7} ≈ 2.21$ . If the graph is messy, try to find a point with cleaner coordinates. But if the problem gives you a specific point, use it, even if the fraction doesn't simplify nicely. In real life, unit rates often aren't whole numbers.
What's the difference between finding the unit rate and finding the slope?: They're the same thing. Slope is the geometric term for the steepness of a line on a graph, calculated as $\frac{\text{rise}}{\text{run}}$ . Unit rate is the practical term for how much output you get per one unit of input, calculated as $\frac{\text{output}}{\text{input}}$ . For a proportional relationship, both formulas give the same answer.

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