Question

Find five rational numbers between $$\displaystyle\frac{3}{5}$$ and $$\displaystyle\frac{4}{5}$$.

Solution

Rational number is any number that can express in the form of $$\dfrac{p}{q}$$ of two integers, where '$$q$$' cannot be zero, so(i) The rational number between $$\dfrac{3}{5}$$ and $$\dfrac{4}{5}$$ is average of $$\dfrac{3}{5}$$ and $$\dfrac{4}{5}$$ $$\dfrac{1}{2}\left ( \dfrac{3}{5}+\dfrac{4}{5} \right )=\dfrac{1}{2}\left ( \dfrac{3+4}{5} \right )=\dfrac{7}{10}$$(ii) The second rational number between $$\dfrac{3}{5}$$ and $$\dfrac{4}{5}$$ can be calculated bye calculating the average of $$\dfrac{3}{5}$$ and $$\dfrac{7}{10}$$ $$\dfrac{1}{2}\left ( \dfrac{3}{5}+\dfrac{7}{10} \right )=\dfrac{1}{2}\left ( \dfrac{6+7}{10} \right )=\dfrac{13}{20}$$(iii) The third rational number between $$\dfrac{3}{5}$$ and $$\dfrac{4}{5}$$ can be calculated by calculating the average of $$\dfrac{7}{10}$$ and $$\dfrac{4}{5}$$ $$\dfrac{1}{2}\left ( \dfrac{7}{10}+\dfrac{4}{5} \right )=\dfrac{1}{2}\left ( \dfrac{7+8}{10} \right )=\dfrac{15}{20}$$(iv)  The fourth rational number between $$\dfrac{3}{5}$$ and $$\dfrac{4}{5}$$ can be calculated by calculating the average of $$\dfrac{3}{5}$$ and $$\dfrac{13}{20}$$ $$\dfrac{1}{2}\left ( \dfrac{3}{5}+\dfrac{13}{20} \right )=\dfrac{1}{2}\left ( \dfrac{12+13}{20} \right )=\dfrac{25}{40}$$(v) The fifth rational number between $$\dfrac{3}{5}$$ and $$\dfrac{4}{5}$$ can be calculated by calculating the average of $$\dfrac{13}{20}$$ and $$\dfrac{4}{5}$$ $$\dfrac{1}{2}\left ( \dfrac{13}{20}+\dfrac{4}{5} \right )=\dfrac{1}{2}\left ( \dfrac{13+16}{20} \right )=\dfrac{29}{40}$$Then five rational are $$\dfrac{7}{10}$$,$$\dfrac{13}{20}$$,$$\dfrac{15}{20}$$,$$\dfrac{27}{40}$$ and $$\dfrac{29}{40}$$MathematicsRS AgarwalStandard IX

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