I think an easier way to solve this problem is to realize that P=(1+2+3…+148+149+150) and Q=(1+2+3…+148). If you do P-Q that equals (1+2+3…148+149+150)-(1+2+3…+148). Distribute the negative and every value will cancel out except for 149 and 150 which equals 299. Hope this helps.

You found the wrong median. To find the median you have to have the sum of the first and the last number and divide them by 2. Thus, the median for the numbers between 1-150 is (1+150)/2=75.5; for the numbers 1-148 it is (1+148)/2=74.5.
The rest you do as you did. 150 x 75.5=11325; 148 x 74.5=11026; 11325-11026=299, therefore answer C.

There is generalize formula for sum of numbers from 1 to n => n(n+1)/2
For sum of 1 to 150, 150(150+1)/2 = 11325
For sum of 1 to 148, 148(148+1)/2 = 11026
So, Difference between sum of numbers = 299