43 (1285). From two cities $A$ and $B$, the distance between which is 180 km, a bus and a car set out towards each other at 6:20 AM. Their meeting occurred at 7:50 AM. If the bus had left 1 hour and 15 minutes earlier, and the car 15 minutes later, they would have met at 7:35 AM. What are the speeds of the bus and the car?
Solution. Let the speed of the bus be $v_{1}$ km/h, and the speed of the car be $v_{2}$ km/h. Since they met after 1.5 hours, we have the equation:

$$
1.5 v_{1} + 1.5 v_{2} = 180
$$

If the bus had left 1 hour and 15 minutes earlier, it would have been on the road for 2 hours and 30 minutes (7:35 - 5:05 = 2 hours 30 minutes).

If the car had left 15 minutes later, it would have been on the road for 1 hour (7:35 - 6:35 = 1 hour).

We get the equation:

$$
2.5 v_{1} + v_{2} = 180
$$

Thus, we have a system of two equations with two unknowns:

$$
\left\{\begin{array}{l}
1.5 v_{1} + 1.5 v_{2} = 180 \\
2.5 v_{1} + v_{2} = 180
\end{array}\right.
$$

from which $v_{1} = 40$ km/h, $v_{2} = 80$ km/h.

Answer: 40 km/h, 80 km/h.

44(1286). From city $A$ to city $B$, two buses left at 8:50. At the same time, a cyclist left from city $B$ to city $A$. One bus he met at 10:10, and the other at 10:50. The distance between the cities is 100 km. Find the speed of the cyclist if the speed of one bus is $1 \frac{5}{7}$ times the speed of the other.

Solution. Let the speed of one bus be $v_{1}$ km/h, the speed of the other bus $\frac{12}{7} v_{1}$ km/h, and the speed of the cyclist $v_{2}$ km/h.

Since the bus with the higher speed met the cyclist after $\frac{4}{3}$ hours (10:10 - 8:50 = 1 hour 20 minutes), and the other bus met the cyclist after 2 hours (10:50 - 8:50 = 2 hours), we have the system of equations:

$$
\left\{\begin{array}{l}
\frac{4}{3} \cdot \frac{12}{7} v_{1} + \frac{4}{3} v_{2} = 100 \\
2 v_{1} + 2 v_{2} = 100
\end{array}\right.
$$

Solving the system of equations, we find $v_{1} = 35$ km/h, $v_{2} = 15$ km/h. The speed of the second bus is $\frac{12}{7} v_{1} = 60$ km/h.

Answer: the speed of the cyclist is $15$ km/h.