Question

Find three rational numbers between $-\frac{2}{5}$and$-\frac{1}{5}$.

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 ,$-\frac{2}{5}$ and $-\frac{1}{5}$

If $a=-\frac{2}{5},b=-\frac{1}{5}andn=3$ then the three rational numbers between $-\frac{2}{5}$ and $-\frac{1}{5}$ will be

Now,

$$\begin{gathered} \Rightarrow \frac{b-a}{n+1}=\frac{-{\displaystyle \frac{1}{5}}+{\displaystyle \frac{2}{5}}}{3+1} \\ \Rightarrow \frac{b-a}{n+1}=\frac{{\displaystyle \frac{-1+2}{5}}}{4} \\ \Rightarrow \frac{b-a}{n+1}=\frac{{\displaystyle \frac{1}{5}}}{4} \\ \Rightarrow \frac{b-a}{n+1}=\frac{{\displaystyle 1}}{5\times 4} \\ \Rightarrow \frac{b-a}{n+1}=\frac{{\displaystyle 1}}{20} \end{gathered}$$

Now,

we using the value of ,$\frac{b-a}{n+1}$ in formula for finding three rational numbers between $\frac{-2}{5}$and$\frac{-1}{5}$.

we get

$$\begin{gathered} \Rightarrow \left(-\frac{2}{5}+\frac{1}{20}\right),\left(-\frac{2}{5}+\frac{2}{20}\right),\left(-\frac{2}{5}+\frac{3}{20}\right) \\ \Rightarrow \left(\frac{-8+1}{20}\right),\left(\frac{-8+2}{20}\right),\left(\frac{-8+3}{20}\right) \\ \Rightarrow \left(\frac{-7}{20}\right),\left(\frac{-6}{20}\right),\left(\frac{-5}{20}\right) \\ \Rightarrow \left(\frac{-7}{20}\right),\left(\frac{-6}{20}\right),\left(\frac{-5}{20}\right) \end{gathered}$$

Hence, The three rational numbers between $-\frac{2}{5}$and$-\frac{1}{5}$are $\frac{-7}{20},\frac{-6}{20}and\frac{-5}{20}$.