This problem could be solved through logic, algebraically, or by plugging in numbers. For all three solutions, our first task is to simplify y – x > x – y. Notice that it has like terms that can be combined – it would be very bad to neglect to simplify this before plowing ahead with the problem!
y – x > x – y
y > 2x – y
2y > 2x
y > x
So, y is greater than x.
The logic solution is certainly the fastest. Since all of the percent changes in Quantity A and Quantity B are changes through multiplication, order doesn’t matter. Thus, the 35% increase on both sides can be ignored – it is the same on both sides, and the order in which this occurs doesn’t matter.
Additionally, the order in which the other changes occur doesn’t matter. Also, the price p is a positive number that is the same on both sides, so it can be ignored as well.
All that’s left is: Quantity A decreases a smaller percent and increases a larger percent. Quantity B increases a smaller percent and decreases a larger percent. Quantity A is definitely greater.
Since y is greater than x, Quantity A is positive and Quantity B is negative.
Finally, plugging in numbers would also work. To make things easy, make p = 100, and make x and y easy percents, like 10 and 50, making sure y is greater than x.
Of course, we still had to simplify y – x > x – y in order to pick valid numbers, and this method is even faster if we realize we can ignore the 35% change on both sides.
See sample solution with p = 100, x = 10, and y = 50 below. (To decrease by 10%, multiply by 0.9. To increase by 50%, multiply by 1.5. To decrease by 50%, multiply by 0.5. To increase by 10%, multiply by 1.1).
100(0.9)(1.5) = 135
100(0.5)(1.1) = 55
Quantity A will be greater no matter what numbers you choose, provided that you make y > x.
The correct answer is A.
I was so close to the above conclusion, dammit. I had successfully deduced that Quantity B was the negative of Quantity A, but not that A was always positive and B was always negative. In my own explanation, I had even mentioned that it would be tempting to pick (A). I should have followed my instincts, I guess. But where did I go wrong in my math, such that (A) produced a negative result in my own calculations?