Two bodies move evenly around a circle. If they move in different directions

Two bodies move evenly around a circle. If they move in different directions, then they meet every 2 minutes. If the bodies move in one direction, then the first body catches up with the second every 10 minutes. How many seconds faster the first body goes around the circle.

We will solve the problem through the equation.

Let x be the time it takes for the first body to traverse the circle.
Then x + 2 will be the time it takes for the second body to traverse the circle.
Let 2y be the circumference.
Let’s find the speed of the first: 2y: x.
Let’s find the speed of the second: 2y: (x + 2).
2y: x * 12 – 2y = 2y: (x + 2) * 12.
2y: x * 12 – 2y.

(24y – 2y * x): x = 24y: (x + 2).
(24y – 2y * x) * (x + 2) = 24y * x.
24y * x – 2y * x * x + 48y – 4y * x – 24y * x = 0.
-2y * x * x – 4y * x + 48y = 0.
x * x + 2x – 24 = 0. Solve the quadratic equation. The roots of the equation are -6 and 4.

x = 4 seconds. During this time, the first body traverses the circle.
Let’s find the time during which the second body passes the circle:
4 + 2 = 6 seconds.
Answer: 4 seconds, 6 seconds.