Given a two-digit number, make a three digit number by putting a 6 as the rightmost digit. then, add 6 to the resulting three digit number and remove the rightmost digit to obtain another two digit number. if the result was 76, what is the original two digit number?

By working backwards, 
you get 76 as your answer because you have to remove the rightmost digit, right? so the three-digit number is 

76x 
(x stands for the rightmost digit, because we don't know what the rightmost digit is) 

we are told we have to add six to the three digit number to get 76x, so we have to subtract 6 from 76x: 
76x - 6 = ??? 

since we assume that the original two-digit number is NOT 76, and any number from 760 to 765 inclusive minus 6 is logically in the 750-range, we can conclude that the original two-digit number is 75.