# Question 5b908

Feb 2, 2017

The distance is 24 miles.

#### Explanation:

The basic rule to remember in all problems involving distance, rate and time is:
$d = r \cdot t$ (distance = rate x time)
Beyond that, you need to clearly identify any variables in the equations you write.

We seek the distance in this case, however it is often easier to use a variable for one of the rates (either for the train or driving) or one of the times. This avoids have to change $d = r t$ into a fraction.

Let $r =$ the rate when he is driving (in miles/hr)
Thus r+6= the rate of the train (in miles/hr)

$d = r \cdot t$ for driving becomes $d = r \cdot 48$
$d = r \cdot t$ for the train becomes $d = \left(r + 6\right) \cdot 40$
But the distance between home and work is the same in both cases!
So $48 r = 40 \left(r + 6\right)$
$48 r = 40 r + 240$ using the distributive property
$8 r = 240$
$r = \frac{240}{8} = 30$ miles/hr

Be careful, though, because the question did not ask for a rate! It asked for the distance.
We have the rate in miles/hour, but we have the time in minutes.
48 minutes = $\frac{48}{60}$ hours
$30 \cdot \left(\frac{48}{60}\right) = 24$ miles using rate * time =distance