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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 $$\dfrac{29}{40}$$

Mathematics
RS Agarwal
Standard IX

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