# John drove for two hours at the speed of 50 miles per hour (mph) and another x hours at the speed of 55 mph. If the average speed of the entire journey is 53 mph, which of the following could be used to find x?

Aug 4, 2015

$x = \text{3 hours}$

#### Explanation:

The idea here is that you need to work backwards from the definition of the average speed to determine how much time did John spend driving at 55 mph.

The average speed can be thought of as being the ratio between the total distance travelled and the total time needed to travel it.

$\text{average speed" = "total distance"/"total time}$

At the same time, distance can be expressed as the product between velocity (in this case, speed) and time.

So, if John drove for 2 hours at 50 mph, then he covered a distance of

${d}_{1} = 50 \text{miles"/color(red)(cancel(color(black)("h"))) * 2 color(red)(cancel(color(black)("h"))) = "100 miles}$

The second part of the total distance was travelled at 55 mph for x hours, so you can say that

${d}_{2} = 55 \text{miles"/color(red)(cancel(color(black)("h"))) * x color(red)(cancel(color(black)("h"))) = 55*x " miles}$

The total distance travelled is equal to

${d}_{\text{total}} = {d}_{1} + {d}_{2}$

${d}_{\text{total" = 100 + 55x" miles}}$

The total time needed was

${t}_{\text{total" = 2 + x" hours}}$

This means that the average speed is

$\overline{v} = \textcolor{b l u e}{\frac{100 + 55 x}{2 + x} = 53}$ $\to$ the equation that will lead you to $x$.

Solve this equation for $x$ to get

$53 \cdot \left(2 + x\right) = 100 + 55 x$

$106 + 53 x = 100 + 55 x$

$2 x = 6 \implies x = \frac{6}{2} = \textcolor{g r e e n}{\text{3 hours}}$