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# xy is different form zero

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xy is different form zero [#permalink]  21 Feb 2019, 00:08
Expert's post
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Question Stats:

83% (00:17) correct 16% (00:00) wrong based on 18 sessions
$$xy \neq 0$$

 Quantity A Quantity B $$\frac{x - y}{x}$$ $$\frac{x - y}{y}$$

A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.
[Reveal] Spoiler: OA

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Re: xy is different form zero [#permalink]  28 Feb 2019, 19:53
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Expert's post
This is one for picking numbers. But here's an easy answer. All we have to do is make y = x, then both quantities are 0 and are thus equal. But when they're different numbers, the two quantities will be different. That's enough to show that the answer is D.

If we pick two easy positive numbers, like x = 2 and y = 1, then Quantity A = $$\frac{1}{2}$$ and Quantity B = 1. So Quantity B is bigger.

But if we make one of them negative, like x = 2 and y = -1, then Quantity A = $$\frac{3}{2}$$ and Quantity B = -3. So Quantity A is bigger.
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Re: xy is different form zero [#permalink]  14 May 2019, 08:38
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Expert's post
Carcass wrote:
$$xy \neq 0$$

 Quantity A Quantity B $$\frac{x - y}{x}$$ $$\frac{x - y}{y}$$

A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.

Here's an often-tested fraction property: $$\frac{a-b}{c}=\frac{a}{c}-\frac{b}{c}$$

So, when we apply the above property, we can rewrite each quantity as follows:
QUANTITY A: $$\frac{x}{x}-\frac{y}{x}$$
QUANTITY B: $$\frac{x}{y}-\frac{y}{y}$$

Simplify:
QUANTITY A: $$1-\frac{y}{x}$$
QUANTITY B: $$\frac{x}{y}-1$$

Add 1 to both quantities AND add y/x to both quantities to get:
QUANTITY A: $$1+1$$
QUANTITY B: $$\frac{x}{y}+\frac{y}{x}$$

Simplify:
QUANTITY A: $$2$$
QUANTITY B: $$\frac{x}{y}+\frac{y}{x}$$

At this point, we may be able to see that we can make the fractions as big as we want (e.g., x/y = 500/1) or we can make them as small as we want (e.g., x/y = -500/1)
So, Quantity B can be greater than 2 or less than 2.

If you're not convinced, let's test some values

case i: x = 100 and y = 1
We get:
QUANTITY A: $$2$$
QUANTITY B: $$\frac{100}{1}+\frac{1}{100}=100\frac{1}{100}$$
In this case, Quantity B is greater

case ii: x = -100 and y = 1
We get:
QUANTITY A: $$2$$
QUANTITY B: $$\frac{-100}{1}+\frac{1}{100}=-99\frac{99}{100}$$
In this case, Quantity A is greater

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Re: xy is different form zero   [#permalink] 14 May 2019, 08:38
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# xy is different form zero

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