This is my answer to the question considering times when Kyle's end result would be the same either way:

Firstly, I took into consideration what the denominators were of each fraction, 12 and 20. I then thought that if the fraction were equal when simplified, there would be a matching mean and median. From this information I began to search for common factors of 12 and 20. (12: 1,2,3,4,6,12. 20: 1,2,4,5,10,20)

Common Factors: 1, 2, 4.

I now used these common factors as simplified denominators. Let's try two as we know we can make these fractions equal 50%.

10/20 and 6/12.

If you work the means and medians out, you will find that they are both 50%.

Mean:
50% + 50% = 100%
100% divided by 2 = 50%

Median:
Halfway between 50% and 50% equals 50%.

Now let's try to make the fractions equal 100%. They will be able to simplify to 1/1.

20/20 and 12/12.

Mean:
100% + 100% = 200%
200% divided by 2 = 100%

Median:
Halfway between 100% and 100% equals 100%.

Now let's try to make the fractions equal 25%. They will be able to simplify to 1/4.

5/20 and 3/12.

Mean:
25% + 25% = 50%
50% divided by 2 = 25%

Median:
Halfway between 25% and 25% equals 25%.

Answers so far: [25% and 25%], [50% and 50%], [100%,100%]

With this knowledge we can add [75% and 75%] to the list.