Explanation

This question could be described through algebra, but it's well suited for analysis by cases. Let's try a case with two two-digit numbers, 17 and 21. If their tens digits are switched, we get 27 and 11. The difference between the first pair is 4 and the difference between the second pair is 16, so the difference has changed by and hence this case is invalid. We can try 17 and 25, which differ by 8. When we flip, they are 27 and 15, which differ by 12. The difference has changed by , so this is a valid pair of numbers.

Now that we understand the rule, we want to find the biggest possible original difference, which will be between 76 and 90, per the answer choices. Keeping the units digits from our previous case, how about 97 and 15? They differ by 82. When we flip, they are 17 and 95, and they differ by 78. The difference has changed by , so this is a valid case. The original difference was 82, so we can conclude that (C) could be the answer and that (A) and (B) are definitely out. We can try to construct an initial difference of 90 or 94. These cases are impossible, because the smallest two-digit number, 10, would push the other number to 100 or above, and it would no longer be a two-digit number.

The correct answer is (C).