# The time required to drive a certain distance varies inversely as the speed. If it takes 4 hours to drive the distance at 40 mph, how long will it take to drive the distance at 50 mph?

Aug 6, 2015

It will take $\text{3.2 hours}$.

#### Explanation:

You can solve this problem by using the fact that speed and time have an inverse relationship, which means that when one increases, the other decreases, and vice versa.

In other words, speed is directly proportional to the inverse of the time

$v \propto \frac{1}{t}$

You can use the rule of three to find the time needed to travel that distance at 50 mph - remember to use the inverse of time!

$\text{40 mph" -> 1/4 "hours}$

$\text{50 mph" -> 1/x "hours}$

Now cross-multiply to get

$50 \cdot \frac{1}{4} = 40 \cdot \frac{1}{x}$

$x = \left(\text{4 hours" * 40color(red)cancelcolor(black)("mph"))/(50color(red)cancelcolor(black)("mph")) = color(green)("3.2 hours}\right)$

Alternatively, you can use the fact that distance is defined as the product between speed and time

$d = v \cdot t$

SInce the distance is the same in both cases, you can write

{: (d = 40 * 4), (d = 50 * x) :}} implies 40 * 4 = 50 * x

Once again,

$x = \frac{40 \cdot 4}{50} = \text{3.2 h}$