Find five rational numbers between $1$ and $2$.

We can approach this problem in at least two ways.

Recall that to find a rational number between $r$ and $s,$ you can add $r$ and $s$ and divide the sum by $2,$ that is $\frac{r+s}{2}$ lies between $r$ and $s .$ So, $\frac{3}{2}$ is a number between $1$ and $2 .$ You can proceed in this manner to find four more rational numbers between $1$ and $2 .$ These four numbers are $\frac{5}{4}, \frac{11}{8}, \frac{13}{8}$ and $\frac{7}{4}$.

Or

The other option is to find all the five rational numbers in one step. since we want five numbers, we write $1$ and $2$ as rational numbers with denominator $5+1$, i.e., $1=\frac{6}{6}$ and $2=\frac{12}{6} .$ Then you can check that $\frac{7}{6}, \frac{8}{6}, \frac{9}{6}, \frac{10}{6}$ and $\frac{11}{6}$ are all rational numbers between $1$ and $2 .$ So, the five numbers are $\frac{7}{6}, \frac{4}{3}, \frac{3}{2}, \frac{5}{3}$ and $\frac{11}{6}$.