Question

Find three rational numbers lying between $\dfrac{3}{5}$ and $\dfrac{7}{8}$.

Answer 1

Hint:
Here we have to find the 3 rational numbers between $\dfrac{3}{5}$ and $\dfrac{7}{8}$. Rational number is defined as a number that can be expressed in form $\dfrac{p}{q}$, here $q$ cannot be zero. For that, we will find the average between the given numbers, which will be the rational numbers between the given numbers.

Complete step by step solution:
We know that rational numbers are any number that can be expressed in the form of $\dfrac{p}{q}$, where $q$ cannot be zero.
1) First rational number between $\dfrac{3}{5}$ and $\dfrac{7}{8}$can be calculated by finding average between them, which is
$\Rightarrow \dfrac{\dfrac{3}{5}+\dfrac{7}{8}}{2}$
On adding the fractional numbers in numerator, we get
$
  =\dfrac{\dfrac{24+35}{40}}{2} \\
  =\dfrac{\dfrac{59}{40}}{2} \\
$
On further simplification, we get
$=\dfrac{59}{80}$
Now, we have three numbers i.e. $\dfrac{3}{5}$, $\dfrac{59}{80}$ and $\dfrac{7}{8}$ so other remaining rational numbers can be calculated by taking average between $\dfrac{3}{5}$ and $\dfrac{59}{80}$, and between $\dfrac{59}{80}$ and $\dfrac{7}{8}$.
2) Second rational number between $\dfrac{3}{5}$ and $\dfrac{59}{80}$ can be calculated by finding the average between them.
$\Rightarrow \dfrac{\dfrac{3}{5}+\dfrac{59}{80}}{2}$
On adding the fractional numbers in numerator, we get
 $
   =\dfrac{\dfrac{48+59}{80}}{2} \\
  =\dfrac{\dfrac{107}{80}}{2} \\
$
On further simplification, we get
$=\dfrac{107}{160}$

3) The third rational number between $\dfrac{59}{80}$ and $\dfrac{7}{8}$ can be calculated by finding the average between them.
$\Rightarrow \dfrac{\dfrac{59}{80}+\dfrac{7}{8}}{2}$
On adding the fractional numbers in numerator, we get
 $
   =\dfrac{\dfrac{59}{80}+\dfrac{7}{8}}{2} \\
  =\dfrac{\dfrac{129}{80}}{2} \\
$
On further simplification, we get
$=\dfrac{129}{160}$

Thus, the 3 rational number between $\dfrac{3}{5}$ and $\dfrac{7}{8}$are $\dfrac{107}{160}$, $\dfrac{59}{80}$, and $\dfrac{129}{160}$.

Note:
Here we have used the average method to find the rational number between the given numbers. But we can also find the required rational number between the given numbers by just taking random numbers between the given numbers, which can be expressed in the form of $\dfrac{p}{q}$, where $q$ cannot be zero.