Problem Analysis:
Let's assume the speed of the slow train as x km/hr. According to the given condition, the speed of the fast train is 10 km/hr more than the speed of the slow train, so the speed of the fast train is (x + 10) km/hr. We know that the distance traveled by both trains is the same, which is 600 km.

Calculating Time:
Let's calculate the time taken by the slow train to cover 600 km. We can use the formula: Time = Distance / Speed.
For the slow train, Time = 600 / x hours.

Now, let's calculate the time taken by the fast train to cover 600 km. Again, using the formula: Time = Distance / Speed.
For the fast train, Time = 600 / (x + 10) hours.

Given Time Difference:
The fast train takes 3 hours less than the slow train, so we have the equation:
600 / x - 600 / (x + 10) = 3.

Solving the Equation:
To solve the equation, we can simplify it by taking a common denominator:
[(600 * (x + 10)) - (600 * x)] / (x * (x + 10)) = 3.

Simplifying further:
600x + 6000 - 600x = 3x^2 + 30x.
6000 = 3x^2 + 30x.

Rearranging the equation to form a quadratic equation:
3x^2 + 30x - 6000 = 0.

We can divide the equation by 3 to simplify it further:
x^2 + 10x - 2000 = 0.

Using the Quadratic Formula:
Using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a), where a = 1, b = 10, and c = -2000, we can find the values of x.

x = (-10 ± √(10^2 - 4 * 1 * -2000)) / (2 * 1).

Calculating the discriminant:
x = (-10 ± √(100 + 8000)) / 2.

x = (-10 ± √8100) / 2.

x = (-10 ± 90) / 2.

Simplifying further:
x = (90 - 10) / 2 or x = (-90 - 10) / 2.

x = 80 / 2 or x = -100 / 2.

x = 40 or x = -50.

Since speed cannot be negative, the speed of the slow train is 40 km/hr.
The speed of the fast train is (40 + 10) = 50 km/hr.

Conclusion:
Therefore, the speed of the slow train is 40 km/hr, and the speed of the fast train is 50 km/hr.