If \(\frac{t}{u}=\frac{x}{y}\)and \(\frac{t}{y}=\frac{u}{x}\) Where t, u, x, and y are Non-Zero Integers, Which of the Following is True?

Question: If  \(\frac{t}{u}=\frac{x}{y}\) and \(\frac{t}{y}=\frac{u}{x}\) where t, u, x, and y are non-zero integers, which of the following is true?

1. \(\frac{t}{u}=1\)

2. \(\frac{y}{x}=-1\)

100. \(t=-u\)

500. \(t=\pm{u}\)

“If \(\frac{t}{u}=\frac{x}{y}\) and \(\frac{t}{y}=\frac{u}{x}\) where t, u, x, and y are non-zero integers, which of the following is true?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. To solve GMAT Problem Solving questions a student must have knowledge about a good number of qualitative skills. The GMAT quant topics in the problem-solving part require calculative mathematical problems that should be solved with proper mathematical knowledge.

Answer:

First of all, we will write down all the given data from the question.

It is given that:

\(\frac{t}{u}=\frac{x}{y}\)----- (eqn 1)

and

\(\frac{t}{y}=\frac{u}{x}\)----- (eqn 2)

As we can see that \(\frac{t}{y}=\frac{u}{x}\) , so  this can be written as: \(\frac{t}{u}=\frac{y}{x}\)

Now, compare equation (1) and equation (2), we will get:

\(\frac{x}{y}=\frac{y}{x}\Rightarrow x^2=y^2\)

This means that: \(t^2=u^2\)and \(x^2=y^2\) , which gives |t| = |u| and |y| = |x|

So, |t| = |u|, means \(t=\pm{u}\)

and, |y| = |x|, means \(x=\pm{y}\)

Correct Answer: D

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