Question

Find six rational numbers between $\frac{3}{5}$ and $\frac{4}{5}$

Answer 1

Find the rational number between the two rational numbers.

Rational no is a type of real numbers, which is in the form of $\raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.$ where $q$ not equal to $0$. Any fraction with non-zero denominators is a rational number.

Let $x$ be a rational no such that $x$ is less then the ${\frac{1}{6}}^{th}$ of the difference between the upper limit and lower limit.

$$\begin{gathered} \therefore x<\frac{1}{6}·\left(\frac{4}{5}-\frac{3}{5}\right) \\ \Rightarrow x<\frac{1}{6}·\frac{1}{5} \\ \Rightarrow x<\frac{1}{30} \end{gathered}$$

Say $x=\frac{1}{40}$.

From the property of rational number we know that sum of two rational number is a rational number.

So we can say the rational in between $\frac{3}{5}$ and $\frac{4}{5}$ are of form $\frac{3}{5}+n\frac{1}{40}$ where $n=1,2,...,6$

$\therefore$the rational numbers are

$\frac{3}{5}+\frac{1}{40},\frac{3}{5}+\frac{2}{40},\frac{3}{5}+\frac{3}{40},\frac{3}{5}+\frac{4}{40},\frac{3}{5}+\frac{5}{40},\frac{3}{5}+\frac{6}{40}.$

$=\frac{25}{40},\frac{26}{40},\frac{27}{40},\frac{28}{40},\frac{29}{40},\frac{30}{40}.$

Hence, the six rational numbers between $2$ and $3$ are $\frac{25}{40},\frac{26}{40},\frac{27}{40},\frac{28}{40},\frac{29}{40},\frac{30}{40}.$.