# Two rational numbers between

Question:

Two rational numbers between $\frac{2}{3}$ and $\frac{5}{3}$ are

(a) $\frac{1}{6}$ and $\frac{2}{6}$

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

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

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

Solution:

We have,

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

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

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

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

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

Hence, the correct opion is (c).