# Practice GMAT Problem Solving Question

Assume that x and y are positive integers such that (x/y) > 1. Which of the following must be less than 1?
 A) x(y-1) B) x3(y-4) C) x3(y) D) x(y3) E) (x-1)y
1. Begin with the inequality (x/y)>1. We can rewrite this by multiplying both sides by y to yield x > y. Note that since x and y are stated to be positive, the direction of the inequality does not change. (If we did not know that x and y were positive, we could not multiply y or x since we would not know whether we need to flip the inequality sign.)
2. Now go through the possible answers, simplifying them so that the answer is more apparent. Remember that taking a number to a negative power is the same as dividing by the number with a positive power:
A: x/y
B: x3/y4
C: x3y
D: x(y3)
E: y/x
3. Now consider each choice, starting with the simplest two: A and E.
A is simply a repeat of the original inequality (x/y) > 1, so this value is clearly not less than 1.
4. Considering choice E: if you divide y by x and x>y, this results in any proper fraction: for example 1/2, 2/5, or 9/11. Any proper fraction is less than 1, resulting in E being the correct choice.In other words, choice E guarantees that we will be dealing with a fraction whose numerator is smaller than its denominator, meaning a smaller number will be divided by a larger number. This always results in a value less than one.
5. Just to be sure, you can evaluate the other answers, disproving them by showing a single counterexample in each case. The values x=4 and y=2 happen to work for all of these choices:
B: 43/24=64/16=4, which is not less than 1.
C: 43*2=64*2=128, which is not less than 1.
D: 4(23)=4*8=32, which is not less than 1.
We already eliminated A, so this confirms that E must be the correct answer.