# Practice GMAT Problem Solving Question

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Assume that x and y are positive integers such that (x/y) > 1. Which of the following must be less than 1?

Correct Answer:

**E**- 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.)
- 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: x^{3}/y^{4}

C: x^{3}y

D: x(y^{3})

E: y/x - 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. - 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.
- 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: 4^{3}/2^{4}=64/16=4, which is not less than 1.

C: 4^{3}*2=64*2=128, which is not less than 1.

D: 4(2^{3})=4*8=32, which is not less than 1.

We already eliminated A, so this confirms that E must be the correct answer.

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