Just 1-2-3

$\large \dfrac{ { \color{#3D99F6} 1} } { {\color{#20A900}\frac{2}{3}} }= \frac{3}{2}$ $\text{ and }\ \dfrac{ {\color{#3D99F6}\frac{ 1 } { 2 }} } { {\color{#20A900}3}}=\frac{1}{6}$

By comparison $\frac{3}{2}$ is greater.

Let's ask the more general question: Which one is bigger, $e_1 = \frac a{\frac b c}\ \ \ \text{or}\ \ \ \ e_2 = \frac{\frac a b }c?$ I will assume that $a,b,c$ are positive real numbers.

First, we can rewrite $e_1 = \frac{ac}{b} = c\cdot \frac a b;\ \ \ \ \ \ e_2 = \frac 1 c \cdot \frac a b.$

Comparing the two, we see that $e_1 > e_2\ \ \ \ \ \text{precisely when}\ \ \ \ \ c > \frac 1 c.$ Interestingly, the comparison only depends on the value of $c$ ... Clearly, the last inequality is true precisely if $c > 1$ . Therefore we conclude

• $e_1$ is greater if $c > 1$ ,

• $e_2$ is greater if $c < 1$ ,

• $e_1 = e_2$ if $c = 1$ .

In the current problem, $c = 3$ , so that the first option is true: the $\boxed{\text{first}}$ expression is greater.

It might be easier for some people to conceptualise this problem by substituting the appropriate fractions with their respective decimal number.

$\large\dfrac{\color{#3D99F6}1}{\color{#20A900}\frac{2}{3}} = \dfrac{\color{#3D99F6}1}{\color{#20A900}0.\dot{6}} > 1 > \dfrac{\color{#3D99F6}0.5}{\color{#20A900}3} = \dfrac{\color{#3D99F6}\frac{1}{2}}{\color{#20A900}3}$

[EDIT: May 8, 2017]

Also, not for want of playing devil's advocate, but rather for a potential new understanding; Consider that the problem has been written using the LaTeX language - Toggle LaTeX in the [...] menu, or hover your cursor over the problem, and you will see that it hasn't simply been structured in a '1/2/3 or 1/2/3 ?' comparison form.

It is structured as follows (without the colors, etc.):

Which is larger, \dfrac{1}{\frac{2}{3}} or \dfrac{\frac{1}{2}}{3} ?

Once we understand how the structure of a language presents itself, we can then determine the meaning of what is presented to us when someone else uses it.

One way to divide any number (n) by a fraction is to multiply n by the reciprocal of the fraction. 1 / (2/3) = 1 x (3/2) = 1.5 clearly larger than (1/2) / 3, or 1/6.

1 divided by something smaller than 1 is greater than 1, 1/2 divided by something greater than one is smaller than 1/2, so the answer is 1/2/3 is greater (since the result is greater than 1 and the other is smaller than 1/2)

First one:- $\large \dfrac{ { \color{#3D99F6} 1} } { {\color{#20A900}\frac{2}{3}} }$

Second one:- $\large \dfrac{ {\color{#3D99F6}\frac{ 1 } { 2 }} } { {\color{#20A900}3}}= \large \frac{1}{6}$

Clearly we can see that the denominator is smaller in the first one therefore the first one is the answer.