EDIT: Taken from the CB! FAQ on Gamefaqs

Go down the stairs and through the south hall, then you should stumble onto a
few rooms. There are locked treasure chests here. They are coded in Al Bhed
though, and the Al Bhed Primer Vol. XX. might help. It’s right here in the same
room. To open the chest, you need to match up words that appear, with the Al
Bhed translation. Primers or no Primers, I'll give the answers to you all.

#1 - Center Bottom Word
#2 - Right Center Word
#3 - The Center word

There. Now get your fabulous prize, a sphere for the ability grid. Now exit the
room and go down the stairs. Go through the right-hand path after saving.
Search the floor for Al Bhed Primer Vol. XXI.. Enter the blue lighted door to
enter a fight. The chests in the room require Al Bhed translation. So if you
got it then:

1st Number: Add both numbers
2nd Number: Subtract 2nd from the 1st
3rd Number: Multiply Numbers 1 and 2
4th Number: Add both numbers

This won't make since unless you have picked up all 21 Al Bhed Primers up to
this point. The other chest asks you question you need to answer them
correctly. They are displayed in Al Bhed, but if you have not gotten all the
Primers:

Question 1 : 3rd Option
Question 2 : 4th Option
Question 3 : 2nd Option
Question 4 : 1st Option