Given information:
- The motorcyclist rode the first half of his way at a constant speed.
- He was delayed for 5 minutes and to make up for the lost time, he increased his speed by 10 km/h.
- The total path covered by him is equal to 50 km.

Let's solve the problem step by step:

1. Define the variables:
Let's assume the initial speed of the motorcyclist as 'x' km/h.

2. Calculate the time taken for the first half of the way:
Since the motorcyclist covered the first half of the way at a constant speed, the time taken can be calculated using the formula:

Time = Distance / Speed

The distance covered in the first half of the way is 50 km / 2 = 25 km.

So, the time taken for the first half of the way is 25 km / x km/h.

3. Calculate the time taken for the second half of the way:
To make up for the lost time, the motorcyclist increased his speed by 10 km/h.

Therefore, the speed for the second half of the way is (x + 10) km/h.

The distance covered in the second half of the way is also 25 km.

So, the time taken for the second half of the way is 25 km / (x + 10) km/h.

4. Calculate the total time taken:
The total time taken is the sum of the time taken for the first half and the time taken for the second half.

Total time taken = Time taken for the first half + Time taken for the second half

Total time taken = 25 km / x km/h + 25 km / (x + 10) km/h

5. Convert the delay in minutes to hours:
The motorcyclist was delayed for 5 minutes. To include this delay in our calculation, we need to convert it to hours.

5 minutes = 5/60 hours = 1/12 hours

6. Account for the delay in the total time taken:
To make up for the lost time, the motorcyclist increased his speed for the second half of the way. Therefore, the total time taken should be reduced by the time of delay.

Total time taken - Delay = 25 km / x km/h + 25 km / (x + 10) km/h - 1/12 hours

7. Calculate the total path covered:
The total path covered is equal to 50 km.

Total path covered = Distance for the first half + Distance for the second half

Total path covered = 25 km + 25 km = 50 km

8. Set up the equation:
Since the total path covered is equal to 50 km, we can set up the equation:

Total path covered = Distance for the first half + Distance for the second half

50 km = 25 km + 25 km

9. Solve the equation:
Now, let's substitute the expression for the total time taken - Delay in the equation and solve for 'x':

50 km = 25 km + 25 km

Simplifying the equation, we get:

50 km = 50 km

Thus, the equation is true for all values of 'x'.

10. Conclusion:
From the equation, we can see that the initial speed of the motorcyclist, 'x', can be any value as long as the total path covered