Chapter 8.5, Problem 30PPSE

Solution

a.)

To calculate:The overall average miles per gallon if $50\%$ of the miles are on highway and $50\%$ around the town given that car gets $25$ mi/gal around town and $30$ mi/gal on the highway.

The overall average miles per gallon if $50\%$ of the miles are on highway and $50\%$ around the town given that car gets $25$ mi/gal around town is $27.27\text{ mpg}$ .

Given information:

The car gets $25$ mi/gal around town and $30$ mi/gal on the highway.

Formula Used:

Use the formula $\text{average mileage}=\frac{\text{total distance}}{\text{total number of gallons}}$ to calculate the overall average mileage.

Calculation:

Take $x$ be the total number of miles driven,

Then, $\frac{1}{2}x$ miles are driven on highway and $\frac{1}{2}x$ miles are driven around the town.

The number of gallons used on highway and around town is as follows using the formula $\text{Number of gallons}=\frac{\text{distance}}{\text{mileage}}$ .

On highway,

  $\begin{array}{c}\text{Number of gallons}=\frac{\frac{1}{2}x}{\text{30}}\\ =\frac{x}{60}\end{array}$

Around town,

  $\begin{array}{c}\text{Number of gallons}=\frac{\frac{1}{2}x}{\text{25}}\\ =\frac{x}{50}\end{array}$

Calculate average mileage using the formula $\text{average mileage}=\frac{\text{total distance}}{\text{total number of gallons}}$ .

  $\begin{array}{c}\text{average mileage}=\frac{x}{\frac{x}{60}+\frac{x}{50}}\\ =\frac{x}{\frac{11x}{300}}\\ =\frac{300}{11}\\ \approx 27.27\text{ mpg}\end{array}$

Thus, the overall average miles per gallon if $50\%$ of the miles are on highway and $50\%$ around the town given that car gets $25$ mi/gal around town is $27.27\text{ mpg}$ .

b.)

To calculate: The overall average miles per gallon if $60\%$ of the miles are on highway and $40\%$ around the town given that car gets $25$ mi/gal around town and $30$ mi/gal on the highway.

The overall average miles per gallon if $60\%$ of the miles are on highway and $40\%$ around the town given that car gets $25$ mi/gal around town is $27.78\text{ mpg}$ .

Given information:

The car gets $25$ mi/gal around town and $30$ mi/gal on the highway.

Formula Used:

Use the formula $\text{average mileage}=\frac{\text{total distance}}{\text{total number of gallons}}$ to calculate the overall average mileage.

Calculation:

Take $x$ be the total number of miles driven,

Then, $0.6x=\frac{3}{5}x$ miles are driven on highway and $0.4x=\frac{2}{5}x$ miles are driven around the town.

The number of gallons used on highway and around town is as follows using the formula $\text{Number of gallons}=\frac{\text{distance}}{\text{mileage}}$ .

On highway,

  $\begin{array}{c}\text{Number of gallons}=\frac{\frac{3}{5}x}{\text{30}}\\ =\frac{x}{50}\end{array}$

Around town,

  $\begin{array}{c}\text{Number of gallons}=\frac{\frac{2}{5}x}{\text{25}}\\ =\frac{2x}{125}\end{array}$

Calculate average mileage using the formula $\text{average mileage}=\frac{\text{total distance}}{\text{total number of gallons}}$ .

  $\begin{array}{c}\text{average mileage}=\frac{x}{\frac{x}{50}+\frac{2x}{125}}\\ =\frac{x}{\frac{9x}{250}}\\ =\frac{250}{9}\\ \approx 27.78\text{ mpg}\end{array}$

Thus, the overall average miles per gallon if $60\%$ of the miles are on highway and $40\%$ around the town given that car gets $25$ mi/gal around town is $27.78\text{ mpg}$ .