## 6. Invited Evening

To celebrate her husband being awarded the Legion of Honor, his wife, a well-known lady from the Paris region, decided to order cookies from the best pastry shop in Paris.

- Our evening, - she clarifies on the phone, - will start exactly at 6 PM. I want the fresh cookies to be delivered precisely at this time.
- Madam, - they respond, - the delivery vehicles leave at specific times, as soon as the cookies are baked. If there is not too much traffic, the vehicles can travel at an average speed of 60 km/h; then we will arrive at your place at 5:45 PM. But if there are traffic jams along the way, the average speed will drop to 20 km/h, and we will only arrive at 6:15 PM.

Try to determine at what average speed the delivery vehicle should travel to arrive at your place exactly at the start of the evening?
6. Let $v$ be the speed of interest (but currently unknown), $t$ the corresponding delivery time, and $d$ (km) the distance the car must travel. We obtain the following system of equations:

$$
\text { 1) } t=d / v, \text { 2) } t+1 / 4=d / 20, \text { 3) } t-1 / 4=d / 60 \text {. }
$$

From equations (2) and (3), it follows that

$$
20 t+5=d, \quad 60 t-15=d
$$

Therefore,

$$
40 t-20=0
$$

and, thus,

$$
t=1 / 2, \quad d=15 \text { and } v=30
$$

Thus, the car should move at a speed of 30 km/h.