1. ## average speeds

On a drive from your ranch to Austin, you wish to average 64 mph. The distance from your ranch to Austin is 80 miles. However, at 40 miles (half way), you find you have averaged only 48 mph.

What average speed must you maintain in the remaining distance in order to have an overall average speed of 64 mph?

2. Hello, Mr_Green!

We can talk our way through this one . . .

On a drive from your ranch to Austin, you wish to average 64 mph.
The distance from your ranch to Austin is 80 miles.
However, at 40 miles (half way), you find you have averaged only 48 mph.

What average speed must you maintain in the remaining distance
in order to have an overall average speed of 64 mph?
We will use: .$\displaystyle \text{Distance }\:=\:\text{Speed} \times \text{Time}\quad\Rightarrow\quad T \:=\:\frac{D}{S}\quad\Rightarrow\quad S \:=\:\frac{D}{T}$

You want to drive 80 miles at an average speed of 64 mph.
. . You must make the trip in: .$\displaystyle \frac{80\text{ miles}}{64\text{ mph}} \,=\,\frac{5}{4}\text{ hours} \:=\:75\text{ minutes}$

You already drove 40 miles at 48 mph.
. . So you've spent: .$\displaystyle \frac{40\text{ miles}}{48\text{ mph}} \:=\:\frac{5}{6}\text{ hours} \:=\:50\text{ minutes.}$

This leaves: .$\displaystyle 75 - 50 \:=\:25$ minutes to drive the remaining 40 miles.

Your speed must be: .$\displaystyle \frac{40\text{ miles}}{\frac{25}{60}\text{ hours}} \;=\;{\color{blue}96\text{ mph}}$

Good luck!