Introduction:
In this problem, we are given that a fast train takes 3 hours less than a slow train to cover a distance of 600 km. We are also given that the speed of the slow train is 10 km/h less than that of the fast train. We need to find the speeds of both trains.

Step 1: Define Variables:
Let's assume the speed of the fast train is x km/h.
Since the speed of the slow train is 10 km/h less than that of the fast train, we can assume the speed of the slow train to be (x - 10) km/h.

Step 2: Calculate Time:
We know that time = distance / speed.
Therefore, the time taken by the fast train to cover the distance of 600 km is given by:
Time taken by fast train = 600 / x hours.

Similarly, the time taken by the slow train to cover the same distance is given by:
Time taken by slow train = 600 / (x - 10) hours.

Step 3: Formulate the Equation:
According to the problem, the fast train takes 3 hours less than the slow train to cover the distance of 600 km. Therefore, we can write the equation as:
Time taken by slow train - Time taken by fast train = 3.

Substituting the values of time from step 2, we get:
600 / (x - 10) - 600 / x = 3.

Step 4: Solve the Equation:
To solve the equation, we can cross multiply and simplify it:
600x - 600(x - 10) = 3x(x - 10).
600x - 600x + 6000 = 3x^2 - 30x.
3x^2 - 30x - 6000 = 0.

Now, we can solve this quadratic equation to find the values of x.

Step 5: Find the Speeds of the Trains:
Solving the quadratic equation, we find that x = 70 or x = -30/2.
Considering the positive value, the speed of the fast train is 70 km/h.
Substituting this value in the equation, we can find the speed of the slow train:
Speed of slow train = 70 - 10 = 60 km/h.

Conclusion:
Therefore, the speed of the fast train is 70 km/h and the speed of the slow train is 60 km/h.