**Using slopes to compare unit rates :**

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

Slope = (5.5 - 2.75) / (2 - 1)

Slope = 2.75 / 1

Slope = 2.75 barrels/hour.

**Step 3 :**

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

Slope = Unit rate

Slope = rise / run

Slope = 10 / 4

Slope = 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, plug x = 1 in y = 375x

Slope = 375(1)

Slope = 375 miles/hour

**Step 2 :**

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

Slope = Unit rate

Slope = (850 - 425) / (2 - 1)

Slope = 425 / 1

Slope = 425 miles/hour.

**Step 3 :**

Compare the unit rates.

425 > 375

So, Plane B is flying faster.

After having gone through the stuff given above, we hope that the students would have understood "Using slopes to compare unit rates".

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