Question

Two rational numbers between $\frac{2}{3}\mathrm{and}\frac{{\displaystyle 5}}{3}$ are
(a) $\frac{1}{6}\mathrm{and}\frac{2}{6}$

(b) $\frac{1}{2}\mathrm{and}\frac{2}{1}$

(c) $\frac{5}{6}\mathrm{and}\frac{7}{6}$

(d) $\frac{2}{3}\mathrm{and}\frac{4}{3}$

Answer 1

We have,

$\frac{2}{3}=\frac{2\times 2}{3\times 2}=\frac{4}{6}\mathrm{and}\frac{{\displaystyle 5}}{3}=\frac{5\times 2}{3\times 2}=\frac{10}{6}$

And, $\frac{1}{2}=\frac{1\times 3}{2\times 3}=\frac{3}{6}\mathrm{and}\frac{2}{1}=\frac{2\times 6}{1\times 6}=\frac{12}{6}$

Also, $\frac{2}{3}=\frac{2\times 2}{3\times 2}=\frac{4}{6}\mathrm{and}\frac{{\displaystyle 4}}{3}=\frac{4\times 2}{3\times 2}=\frac{8}{6}$

Since, $\frac{1}{6}<\frac{2}{6}<\frac{{\displaystyle 3}}{{\displaystyle 6}}\left(\frac{1}{2}\right)<\frac{4}{6}\left(=\frac{2}{3}\right)<\frac{5}{6}<\frac{7}{6}<\frac{{\displaystyle 8}}{{\displaystyle 6}}\left(=\frac{4}{3}\right)<\frac{10}{6}\left(=\frac{5}{3}\right)<\frac{{\displaystyle 12}}{{\displaystyle 6}}\left(=\frac{2}{1}\right)$

So, the two rational numbers between $\frac{2}{3}\mathrm{and}\frac{{\displaystyle 5}}{3}$ are $\frac{5}{6}\mathrm{and}\frac{7}{6}$.

Hence, the correct opion is (c).