Given a fraction with $ x $ is a real number, judge the scope of $ x $.

Question

From an ACT Math test:

Suppose that $ x $ is a real number and $ \frac { 4 x } { 6 x ^ 2 } $ is a rational number. Which of the following statements about $ x $ must be true?

1. $ x $ is rational
2. $ x $ is irrational
3. $ x = 1 $
4. $ x = \frac 2 3 $
5. $ x = \frac 3 2 $

The answer says it must be a rational number, but how about an irrational number, say, $ \frac 4 3 $, which can also satisfy $ \frac { 4 x } { 6 x ^ 2 } $ a rational number?

Answer 1

Simply $\dfrac{4x}{6x^2}=\dfrac{4}{6x}$ can be irrational, but doesn't have to -- for example let's take $x=\dfrac{10}{3}.$

Then we have

$\dfrac{4}{6x}=\dfrac{2}{3x}=\dfrac{2}{3\cdot\frac{10}{3}}=\dfrac{2}{10}.$

It isn't irrational number definitely.

By the way the question says which statement must be true.

Answer 2

A rational number is any number that can be expressed as a ratio of two integers $\frac{p}{q}$ with $q\neq 0$. That is $\mathbb Q:= \{\frac{p}{q}|p\in\mathbb Z,q\in\mathbb Z, q\neq0\}.$ Moreover $\frac{4}{3}$ is a rational number.

Since $x$ is a positive real number we have $$\frac{4x}{6x^2}=\frac{4}{6x}=\frac{2}{3x}$$

which we are told is a rational number. Then $\frac{2}{3x}$ is of the form $\frac{p}{q}$, where $p=2$ and $q=3x$ (and we know $x\neq 0$).

So statement $A$ must be true.

Answer 3

A number is rational if it can be written as a fraction with integer numerator and denominator. $\frac43$ is not irrational, it's an improper fraction. If $\frac{4x}{6x^2}=\frac{2}{3x}=\frac ab$, then $x=\frac{2b}{3a}$. If $a$ and $b$ are integers, so are $2b$ and $3a$. Therefore, $x$ is rational.