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.
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.
Answer: E.
The problems on this page are the property of the MAA's American Mathematics Competitions