# A tourist calculated that if he walks to the railroad station with a speed of 4 mph, he’ll miss the train by half an hour, but if he walks with a speed of 5 mph, he’ll reach the station 6 minutes before the departure of the train ?

## What distance does the tourist have to cover?

Mar 26, 2017

The tourist has to walk $12$ miles.

#### Explanation:

We can use the formula:

"speed" = ("distance")/("time")

To create two equations we can use to solve this problem.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

We know that the tourist must reach the station in some time $t$ hours in order to catch the train. We also know that the station is some distance $d$ miles away.

The tourist says that if his speed is $4 \textcolor{w h i t e}{\text{-""mph}}$, then he will be 30 minutes late (which is 0.5 hours). This means that he will travel $d$ miles in $t + 0.5$ hours.

$4 = \frac{d}{t + 0.5}$

The tourist also says that if his speed is $5 \textcolor{w h i t e}{\text{-""mph}}$, then he will be 6 minutes early (which is 0.1 hours). This means that he will travel $d$ miles in $t - 0.1$ hours.

$5 = \frac{d}{t - 0.1}$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Now, we can take these two equations and use them to solve for $d$. But first, we will have to solve for $t$ and then plug in $t$ to find $d$.

We can multiply both equations by the denominator to get that:

$4 = \frac{d}{t + 0.5} \textcolor{w h i t e}{\text{XXXXX}} 5 = \frac{d}{t - 0.1}$

$4 \left(t + 0.5\right) = d \textcolor{w h i t e}{\text{XXXX}} 5 \left(t - 0.1\right) = d$

$4 t + 2 = d \textcolor{w h i t e}{\text{XXXXXX}} 5 t - 0.5 = d$

Now, we can set $4 t + 2$ equal to $5 t - 0.5$, since we know that they are both equal to $d$.

$\textcolor{w h i t e}{\text{X}} 4 t + 2 = 5 t - 0.5$
$4 t + 2.5 = 5 t$
$\textcolor{w h i t e}{\text{XXX}} 2.5 = t$

So the train leaves in $2.5$ hours. Now that we know that, we can figure out what the distance was. Remember that the tourist said he would get to the station $30$ minutes late if he walked at 4 mph. That means that he would get there in $3$ hours instead of $2.5$. So, we can write:

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

4 color(white)"-""mph" = d/(3 color(white)"-"""h")

$12 \textcolor{w h i t e}{\text{-""mi}} = d$

So the train station is 12 miles away.