# A bus takes 12 hours to travel the same distance that an airplane travels in 1.5 hours. The average speed of the airplane is 120 km/hour faster than five times the average speed of the bus. What is the average speed of the bus?

Sep 5, 2015

The speeds are:
Bus: 40 km/h
Plane: 320 km/h

#### Explanation:

Let $D$ be the distance covered by both vehicles, $B$ the speed of the bus, $P$ the speed of the plane.

From the first sentence we can figure out, that:

$B = \frac{D}{12}$ and $P = \frac{D}{1.5}$. (1)

The second sentence shows us the relation between $P$ and $B$

$P = 120 + 5 B$ (2)

If we substitute $B$ and $P$ from (1) in (2) we get the equation with $D$ only, so we can calculate $D$

$\frac{D}{1.5} = 5 \cdot \frac{D}{12} + 120$

We multiply both sides by 12.

$8 D = 5 D + 1440$
$3 D = 1440$
$D = 480$

The distance is 480 km.
Now we can calculate both speeds:

$B = \frac{480}{12} = 40$ km/h

$P = \frac{480}{1.5} = 320$ km/h