Which of the following is greatest?

First of all, let me reiterate whiplash2411's point:
You can immediately rule out the two square terms as they will be smaller than their respective square root terms.
When positive fractions less than 1 are squared, they become smaller.
Square of 1/2 is 1/4 which is smaller than 1/2.

So only 3 options to consider
a. 7/12
c. 9/13
d. 17/20
Many ways to quickly deal with it. Some have already been discussed above. Some I will take here:

Method 1:
Compare 7/12 and 17/20
I see that 17 is more than double of 7 but 20 is less than double of 12. So 17/20 is greater than 7/12
or look at it this way 7/12 = 14/24.
14/24 is less than 17/20 definitely because 17/20 has greater numerator and smaller denominator.

Compare 9/13 and 17/20
17 is almost double of 9 but 20 is quite less than the double of 13. So 17/20 is greater than 9/13.
So 17/20 is the greatest.

Method 2:
Compare 7/12 and 9/13. If I increase the numerator and denominator by the same number in a positive fraction less than 1, the fraction increases. e.g. 7/12 < \(\frac{7 +1}{12 + 1}\) i.e.
7/12 < 8/13. We know 8/13 is definitely less than 9/13. (Denominator same, numerator greater)

Now compare 9/13 and 17/20.
9/13 < \(\frac{9 +8}{13 + 8}\) i.e. 17/21.
We know 17/21 is definitely less than 17/20 (Numerator same, denominator smaller)
So 17/20 is the greatest.

Formal method:
Compare 7/12 and 9/13. Is 7*13 (=91) greater or 9*12 (=108) greater?
Since 9*12 is greater so 9/13 is the greater fraction. (Essentially you are making the denominator same 12*13 in this case but you don't need to calculate it. You just need to compare the numerators you will get)

Now compare 9/13 and 17/20. Is 9*20 (=180) greater or 17*13 (don't want to calculate but it is definitely much greater than 170, even greater than 200) so 17/20 is the greatest.