We can compare proportional relationships presented in different ways.

In this section, we are going to see, how slopes can be used to compare unit rates.

Example 1 :

The equation y = 2.75x represents the rate, in barrels per hour, that oil is pumped from Well A. The graph represents the rate that oil is pumped from Well B. Which well pumped oil at a faster rate?

Solution :

Step 1 :

Use the equation y = 2.75x to make a table for Well A’s pumping rate, in barrels per hour.

Step 2 :

Use the table to find the slope of the graph of Well A.

Slope  =  Unit rate

=  (5.5 - 2.75)/(2 - 1)

=  2.75/1

=  2.75 barrels/hour

Step 3 :

Use the graph to find the slope of the graph of Well B.

Slope  =  Unit rate

=  rise/run

=  10/4

=  2.5 barrels/hour

Step 4 :

Compare the unit rates.

2.75 > 2.5

So Well A’s rate, 2.75 barrels/hour, is faster.

Reflect :

Describe the relationships among the slope of the graph of Well A’s rate, the equation representing Well A’s rate, and the constant of proportionality.

The slope and the constant of proportionality equal the value 2.75 in the equation y = 2.75x.

Example 2 : 

The equation y = 375x represents the relationship between x, the time that a plane flies in hours, and y, the distance the plane flies in miles for Plane A. The table represents the relationship for Plane B. Find the slope of the graph for each plane and the plane’s rate of speed. Determine which plane is flying at a faster rate of speed.

Solution :

Step 1 :

Use the equation y = 375x to find the slope of the graph of Plane A.

Slope = Unit rate

Here, unit rate is the distance covered by the plane in one hour.

To find unit rate, substitute x = 1 in y = 375x.

Slope  =  375(1)

=  375 miles/hour

Step 2 :

Use the table to find the slope of the graph of Plane B.

Slope  =  Unit rate

=  (850 - 425)/(2 - 1)

=  425/1

=  425 miles/hour

Step 3 :

Compare the unit rates.

425 > 375

So, Plane B is flying faster.

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