Question

Find two rational numbers between $\frac{1}{2}$and$\frac{3}{4}$.

Answer 1

Finding the rational numbers :

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

$a+\frac{b-a}{n+1}$ and $a+\frac{2(b-a)}{n+1}$

Given numbers are ,$\frac{1}{2}$ and $\frac{3}{4}$

Now, if $a=\frac{1}{2},b=\frac{3}{4}andn=2$ then the two rational numbers between $\frac{1}{2}$ and $\frac{3}{4}$ will be

Now,

$$\begin{gathered} a+\frac{b-a}{n+1}=\frac{1}{2}+\frac{{\displaystyle \frac{3}{4}}-{\displaystyle \frac{1}{2}}}{2+1} \\ \Rightarrow \frac{1}{2}+\frac{{\displaystyle \frac{3-2}{4}}}{3} \\ \Rightarrow \frac{1}{2}+\frac{{\displaystyle \frac{1}{4}}}{3} \\ \Rightarrow \frac{1}{2}+\frac{{\displaystyle 1}}{12} \\ \Rightarrow \frac{7}{12} \end{gathered}$$

Now,

we use the formula ,$a+\frac{2(b-a)}{n+1}$ for finding another rational numbers between $\frac{1}{2}$and$\frac{3}{4}$.

Now,

$$\begin{gathered} \Rightarrow \frac{1}{2}+\frac{2\left({\displaystyle \frac{3}{4}-\frac{1}{2}}\right)}{2+1} \\ \Rightarrow \frac{1}{2}+2\left(\frac{1}{12}\right) \\ \Rightarrow \frac{6+2}{12} \\ \Rightarrow \frac{8}{12} \\ \Rightarrow \frac{2}{3} \end{gathered}$$

Hence, The two rational numbers between $\frac{1}{2}$and$\frac{3}{4}$are $\frac{7}{12}and\frac{2}{3}$.