Which of the following expressions is correct?

Let's compare the pairs one by one.

• Comparing $\frac{1}{4}\mathrm{and}\frac{2}{3}$

$\frac{1}{4}=\frac{1\times 3}{4\times 3}=\frac{3}{12},$

$\frac{2}{3}=\frac{2\times 4}{3\times 4}=\frac{8}{12}$

$\therefore \frac{3}{12}<\frac{8}{12}$$\Rightarrow \frac{1}{4}<\frac{2}{3}$

Which means $\frac{1}{4}>\frac{2}{3}$ is wrong.

• Comparing $\frac{1}{5}\mathrm{and}\frac{2}{3}$

$\frac{1}{5}=\frac{1\times 3}{5\times 3}=\frac{3}{15},$

$\frac{2}{3}=\frac{2\times 5}{3\times 5}=\frac{10}{15}$

$\therefore \frac{3}{15}<\frac{10}{15}$$\Rightarrow \frac{1}{5}<\frac{2}{3}$

Which means $\frac{1}{5}>\frac{2}{3}$ is wrong.

• Comparing $\frac{1}{2}\mathrm{and}\frac{5}{12}$
  $\frac{1}{2}=\frac{1\times 6}{2\times 6}=\frac{6}{12},$$\frac{5}{12}$

$\therefore \frac{6}{12}>\frac{5}{12}$

$\Rightarrow \frac{1}{2}>\frac{5}{12}$

Which means $\frac{1}{2}<\frac{5}{12}$ is wrong.

• Comparing $\frac{4}{12}\mathrm{and}\frac{5}{12}$

$\frac{1}{3}=\frac{1\times 4}{3\times 4}=\frac{4}{12},\frac{5}{12}$

$\therefore \frac{4}{12}<\frac{5}{12}$

$\Rightarrow \frac{1}{3}<\frac{5}{12}$

Which means $\frac{1}{3}<\frac{5}{12}$ is correct.

Hence, only $\frac{1}{3}<\frac{5}{12}$ is correct.