Question

Find five rational numbers lying between $\frac{2}{5}\mathrm{and}\frac{3}{4}$.

Answer 1

Let:
x = $\frac{2}{5}$ , y = $\frac{3}{4}$ and n = 5
We know:
d = $\frac{y-x}{n+1}=\frac{{\displaystyle \frac{3}{4}}-{\displaystyle \frac{2}{5}}}{5+1}=\frac{{\displaystyle \frac{7}{20}}}{6}=\frac{7}{120}$

So, five rational numbers between x and y are:
(x + d), (x + 2d), (x + 3d), (x + 4d) and (x + 5d)

= $\left(\frac{2}{5}+\frac{7}{120}\right),\left(\frac{2}{5}+\frac{14}{120}\right),\left(\frac{2}{5}+\frac{21}{120}\right),\left(\frac{2}{5}+\frac{28}{120}\right)\mathrm{and}\left(\frac{2}{5}+\frac{35}{120}\right)$

= $\frac{55}{120},\frac{62}{120},\frac{69}{120},\frac{76}{120}\mathrm{and}\frac{83}{120}$