Problem:
Two non-zero real numbers, a and b, satisfy ab=a−b. Find a possible value of ba​+ab​−ab.
Answer Choices:
A. −2
B. −21​
C. 31​
D. 21​
E. 2
Solution:
Find the common denominator and replace the ab in the numerator with a−b to get
ba​+ab​−ab=aba2+b2−(ab)2​=aba2+b2−(a−b)2​=aba2+b2−(a2−2ab+b2)​=ab2ab​=2​
\section*{OR}
Note that a=a/b−1 and b=1−b/a. It follows that ba​+ab​−ab=(a+1)+ (1−b)−(a−b)=2​.
The problems on this page are the property of the MAA's American Mathematics Competitions