Two cyclists set off simultaneously from points A and B towards each other at constant speeds.

Two cyclists set off simultaneously from points A and B towards each other at constant speeds. They met 1 hour later, and the first cyclist arrived at B 1.5 hours later than the second at A. Find how long each of them had been on the road.

The solution of the problem.

1. Let’s denote by x the speed of the first cyclist.

2. Let us denote by y the speed of the second cyclist.

3. How far has the first traveled from A to the meeting point?

x km / h * 1 h = x km.

4. What is the distance traveled by the second from B to the meeting point?

at km / h * 1 h = at km.

5. What is the distance from A to B?

x + y km.

6. How long did the first cyclist ride from A to B?

(x + y): x h.

7. How long did the second cyclist ride from B to A?

(x + y): y h.

8. Let’s compose and solve the equation.

(x + y): x – (x + y): y = 1.5;

y: x – x: y = 1.5;

x> 0; we introduce the variable z

z = y: x;

z ^ 2 – 1.5z – 1 = 0;

D = 6.25;

The equation has 2 roots z = -0.5 and z = 2.

Because x> 0 and y> 0, then 1 root z = 2 is suitable.

9. How long did the first cyclist ride from A to B?

(x + y): x = 1 + z = 3 hours.

10. How long did the second cyclist ride from B to A?

(x + y): y = 1 / z + 1 = 1.5 h.

Answer. The first cyclist traveled for 3 hours, the second traveled for 1.5 hours.