Question

Find three rational numbers between $\frac{1}{5}$and$\frac{1}{3}$.

Answer 1

Finding the rational numbers :

To find the three rational numbers between any two rational numbers $aandb$ , we can use the formula.

$a+\frac{b-a}{n+1}$ , $a+\frac{2(b-a)}{n+1}anda+\frac{3(b-a)}{n+1}$

Given numbers are $\frac{1}{5}$ and $\frac{1}{3}$

Now, if $a=\frac{1}{5},b=\frac{1}{3}andn=3$ then the three rational numbers between $\frac{1}{5}$ and $\frac{1}{3}$ will be

Now,

$$\begin{gathered} \Rightarrow \frac{b-a}{n+1}=\frac{{\displaystyle \frac{1}{3}}-{\displaystyle \frac{1}{5}}}{3+1} \\ \Rightarrow \frac{b-a}{n+1}=\frac{{\displaystyle \frac{5-3}{15}}}{4} \\ \Rightarrow \frac{b-a}{n+1}=\frac{{\displaystyle \frac{2}{15}}}{4} \\ \Rightarrow \frac{b-a}{n+1}=\frac{{\displaystyle 2}}{15\times 4} \\ \Rightarrow \frac{b-a}{n+1}=\frac{{\displaystyle 2}}{60} \\ \Rightarrow \frac{b-a}{n+1}=\frac{{\displaystyle 1}}{30} \end{gathered}$$

we use the formula ,$a+\frac{b-a}{n+1}$ ,$a+\frac{2(b-a)}{n+1}$ and $a+\frac{3(b-a)}{n+1}$ for finding three rational numbers between $\frac{1}{5}$and$\frac{1}{3}$.

Now,

$$\begin{gathered} \Rightarrow \left(\frac{1}{5}+\frac{1}{30}\right),\left(\frac{1}{5}+\frac{2}{30}\right),\left(\frac{1}{5}+\frac{3}{30}\right) \\ \Rightarrow \left(\frac{6+1}{30}\right),\left(\frac{6+2}{30}\right),\left(\frac{6+3}{30}\right) \\ \Rightarrow \left(\frac{7}{30}\right),\left(\frac{8}{30}\right),\left(\frac{9}{30}\right) \\ \Rightarrow \left(\frac{7}{30}\right),\left(\frac{4}{15}\right),\left(\frac{3}{10}\right) \end{gathered}$$

Hence, The three rational numbers between $\frac{1}{5}$and$\frac{1}{3}$are $\frac{7}{30},\frac{4}{15},\frac{3}{10}$.