# It took David an hour to ride 20 km from his house to the nearest town. He then spent 40 minutes on the return journey. What was his average speed?

Sep 6, 2017

${\text{24 km h}}^{- 1}$

#### Explanation:

The average speed is simply the rate at which the distance travelled by David varies per unit of time.

$\text{average speed" = "distance covered"/"unit of time}$

In your case, you can take a unit of time to mean $1$ hour. Since you know that

$\text{1 h = 60 min}$

you can say that David needed

40 color(red)(cancel(color(black)("min"))) * "1 h"/(60color(red)(cancel(color(black)("min")))) = 2/3color(white)(.)"h"

to make the return trip.

Now, notice that on his way from his house to the town hall, David travels $\text{20 km}$ in exactly $1$ hour. This means that his average speed for the first part of the journey will be

${\text{average speed"_ 1 = "20 km"/"1 h" = "20 km h}}^{- 1}$

Since it takes less than an hour for David to complete the return trip, you can say that his average speed for the return trip will be higher $\to$ he will cover more distance per unit of time on his return trip.

More specifically, David will cover

1 color(red)(cancel(color(black)("h"))) * "20 km"/(2/3color(red)(cancel(color(black)("h")))) = "30 km"

in $1$ hour on his return trip, so his average speed will be

${\text{average speed"_2 = "30 km h}}^{- 1}$

So, you know the average speed for the first trip and the average speed for the return trip, so you can simply take the average of these two values, right? Wrong!

It is absolutely crucial to avoid going

$\textcolor{red}{\cancel{\textcolor{b l a c k}{{\text{average speed" = ("20 km h"^(-1) + "30 km h"^(-1))/2 = "25 km h}}^{- 1}}}}$

because you will get an incorrect answer $\to$ that's not how the average speed works! Instead, focus on the definition of average speed, which tells you that you must find the total distance covered by David per unit of time.

You know that you have

• $\text{total distance = 20 km + 20 km = 40 km}$
• $\text{total time" = "1 h" + 2/3color(white)(.)"h" = 5/3color(white)(.)"h}$

So if David covers $\text{40 km}$ in $\frac{5}{3}$ hours, how many kilometers does he cover in $1$ hour?

1 color(red)(cancel(color(black)("h"))) * "40 km"/(5/3color(red)(cancel(color(black)("h")))) = "24 km"

Therefore, you can say that David has an average speed of

"average speed" = color(darkgreen)(ul(color(black)("24 km h"^(-1))))

I'll leave the answer rounded to two sig figs, but don't forget that your values justify only one significant figure for the answer.

This is why the equation for average speed is given as

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

${\text{average speed" = "40 km"/(5/3color(white)(.)"h") = 40/(5/3)color(white)(.)"km"/"h" = "24 km h}}^{- 1}$