To find a rational number x between two rational numbers \(\frac{a}{b}\) and \(\frac{c}{d},\) we use
\(\text{x}=\frac{1}{2}(\frac{a}{b}+\frac{c}{d})\)
Therefore,
to find rational number x (let) between \(\frac{2}{3}\) and \(\frac{3}{4}\)
\(\text{x}=\frac{1}{2}(\frac{2}{3}+\frac{3}{4})\)
\(\Rightarrow\) \(\text{x}=\frac{1}{2}(\frac{8+9}{12})\)
\(\Rightarrow\) \(\text{x}=\frac{1}{2}\times\frac{17}{12}\)
\(\Rightarrow\) \(\text{x}=\frac{17}{24}\)
Now if we find a rational number between \(\frac{2}{3}\) and \(\frac{17}{24}\) it will also be between \(\frac{2}{3}\) and \(\frac{3}{4}\)since \(\frac{17}{24}\) lies between \(\frac{2}{3}\) and \(\frac{3}{4}\)
Therefore,
to find rational number y (let) between \(\frac{2}{3}\) and \(\frac{17}{24}\)
\(\text{y}=\frac{1}{2}(\frac{2}{3}+\frac{17}{24})\)
\(\Rightarrow\) \(\text{y}=\frac{1}{2}(\frac{16+17}{24})\)
\(\Rightarrow\) \(\text{y}=\frac{1}{2}\times\frac{33}{24}\)
\(\Rightarrow\) \(\text{y}=\frac{33}{48}\)
Now if we find a rational number between \(\frac{17}{24}\) and \(\frac{3}{4}\) it will also be between \(\frac{2}{3}\) and \(\frac{3}{4}\)since \(\frac{17}{24}\) lies between \(\frac{2}{3}\) and \(\frac{3}{4}\)
Therefore,
to find rational number z (let) between \(\frac{17}{24}\) and \(\frac{3}{4}\)
\(\text{z}=\frac{1}{2}(\frac{17}{24}+\frac{3}{4})\)
\(\Rightarrow\) \(\text{z}=\frac{1}{2}(\frac{17+18}{24})\)
\(\Rightarrow\) \(\text{x}=\frac{1}{2}\times\frac{35}{24}\)
\(\Rightarrow\) \(\text{z}=\frac{35}{48}\)
