Carcass wrote:
|a|<|b|
Quantity A |
Quantity B |
ba+b−aa−b |
ab+a−bb−a |
A)The quantity in Column A is greater.
B)The quantity in Column B is greater.
C)The two quantities are equal.
D)The relationship cannot be determined from the information given.
I manipulated both sides to get to an answer:
ba+b−aa−b?ab+a−bb−ab−aa+b−aa−b?−bb−ab−aa+b?−bb−a+aa−bb−aa+b?−bb−a+(−1)−a(−1)(b−a)b−aa+b?−bb−a+−ab−ab−aa+b?−b−ab−ab−aa+b?(−1)(a+b)b−aHere I crossed multiplied. You usually wouldn't want to do this because you run the risk of changing the sign (the '?' in our algebra above). However, this is a unique situation, and can be better seen in the result of the cross multiplication:
(b−a)2?(−1)(a+b)2We don't have to worry about the cases when
b−a and
a+b are positive or negative since they are being squared, so they are always positive.
Given that
(a+b)2 is being multiplied by
−1, it must be the case that:
(b−a)2>(−1)(a+b)2So A is greater.The only case where this wouldn't be true is if they were equal, or Choice C. In other words:
b−a=a+b But quick algebra rules this possibility out:
b−a=a+b −a=a 0=2a 0=a Which would imply that
b=0, but we know that both cannot be
0 because of the initial restriction given in the question:
|a|<|b| (or by recognizing that the denominators in the fraction of both A and B cannot be 0).