The nice thing about problems with uniform distributions is that you can often avoid complicated integrals and rely on some pretty simple geometry if you establish the right kind of visual.
We can make our random variable be X and just have it represent the number of minutes after 7:00, so it can start at zero. The shape will then be a simple rectangle that runs horizontally from 0 to 20. Since the total area of the rectangle must equal 1, the height must equal 1/20. So the top side of the rectangle is 1/20 units high.
From here we just need to consider how the rectangle is divided up based on the constraints of the questions.
Question i: P(X < 7) would be the portion of the rectangle to the left of x = 7. So it is a rectangle of base length 7 and height 1/20. So its probability is 7 · 1/20 = 7/20 = 0.35.
Question ii: P(X > 12) would be the rectangle between x = 12 and x = 20. This is 8 · 1/20 = 8/20 = 2/5 = 0.4.
Question iii: P(X > 15 | X > 10) can be obtained by only considering the rectangle that is to the right of 10, which had an area of 1/2, and then seeing what portion of that rectangle is to the right of 15. The area between 15 and 20 is 5 · 1/20 = 5/20 = 1/4. Divide this by 1/2, and we get 1/2, or 0.5.
