Question

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

Answer 1

n = 5
n + 1 = 6
$x=\frac{3}{5},y=\frac{2}{3}$
$d=\frac{y-x}{n+1}=\frac{{\displaystyle \frac{2}{3}}-{\displaystyle \frac{3}{5}}}{6}=\frac{10-9}{90}=\frac{1}{90}$

Thus, rational numbers between $\frac{3}{5}\mathrm{and}\frac{2}{3}$ will be
$$\begin{gathered} \left(x+d\right),\left(x+2d\right),\left(x+3d\right),\left(x+4d\right),\left(x+5d\right) \\ =\left(\frac{3}{5}+\frac{1}{90}\right),\left(\frac{3}{5}+\frac{2}{90}\right),\left(\frac{3}{5}+\frac{3}{90}\right),\left(\frac{3}{5}+\frac{4}{90}\right),\left(\frac{3}{5}+\frac{5}{90}\right) \\ =\left(\frac{55}{90}\right),\left(\frac{56}{90}\right),\left(\frac{57}{90}\right),\left(\frac{58}{90}\right),\left(\frac{59}{90}\right) \\ =\left(\frac{11}{18}\right),\left(\frac{28}{45}\right),\left(\frac{19}{30}\right),\left(\frac{29}{45}\right),\left(\frac{59}{90}\right) \end{gathered}$$