## 8. The Emir and His Chauffeur

Emir Ben Sidi Mohammed always drives from his palace to the airport at the same speed along a magnificent highway that runs through the desert. If his chauffeur increased the average speed of the car by $20 \mathrm{km} /$ h, the Emir would save 2 minutes, and if the chauffeur drove at a speed 20 km/h slower, the Emir would lose 3 minutes.

How far is the Emir's palace from the airport?
8. Let $I$ (km) be the unknown distance; $v$ (km/h) the speed of the emir's car, and finally, $t$ (h) the time it takes for the emir to travel from the palace to the airport. We obtain the following system of equations:

$$
v=I / t ; \quad v+20=I /(t-2 / 60) ; \quad v-20=I /(t+3 / 60) .
$$

Excluding the first equation, we find

$$
(v-20)(t-3 / 60)=v t, \quad(v+20)(t+3 / 60)=v t,
$$

or

$$
\begin{gathered}
-v / 30+20 t-2 / 3=0 \\
v / 20-20 t-1=0 .
\end{gathered}
$$

From the last two equations (add them!), we find: $v=100$, and $t=1 / 5=12$ min. Therefore, $I=v t=20$.

The airport and the emir's palace are separated by a distance of 20 km.