Question

Find three rational numbers lying between $\frac{3}{5}\mathrm{and}\frac{7}{8}$. How many rational numbers can be determined between these two numbers?

Answer 1

$x=\frac{3}{5}\mathrm{and}y=\frac{7}{8}$
n = 3
$d=\frac{\left(y-x\right)}{n+1}=\frac{{\displaystyle \frac{7}{8}}-{\displaystyle \frac{3}{5}}}{3+1}=\frac{11}{40}\times \frac{1}{4}=\frac{11}{160}$
Rational numbers between $x=\frac{3}{5}\mathrm{and}y=\frac{7}{8}$ will be
$$\begin{gathered} \left(x+d\right),\left(x+2d\right),...,\left(x+nd\right) \\ \Rightarrow \left(\frac{3}{5}+\frac{11}{160}\right),\left(\frac{3}{5}+2\times \frac{11}{160}\right),\left(\frac{3}{5}+3\times \frac{11}{160}\right) \\ \Rightarrow \left(\frac{107}{160}\right),\left(\frac{118}{160}\right),\left(\frac{129}{160}\right) \\ \Rightarrow \left(\frac{107}{160}\right),\left(\frac{59}{80}\right),\left(\frac{129}{160}\right) \end{gathered}$$
There are infinitely many rational numbers between two given rational numbers.
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