Question

Find a rational number between the following:
$\left(a\right)\dfrac{-1}{3}$ and $\dfrac{2}{5}$
$\left(b\right)$4 and 5
$\left(c\right)$-2 and -1
$\left(d\right)\dfrac{1}{2}$ and $\dfrac{1}{4}$
$\left(e\right)\dfrac{1}{4}$ and $\dfrac{1}{3}$

Answer 1

Hint: We are given two points in each of the parts and we are required to find a rational number that exists between them. We will use the property of rational numbers that says that between every two rational numbers on the real line, there definitely exists a rational number between them. For that we might simply expand the real line by multiplying the numerator and denominator of the fractions if the given numbers appear as fractions or if they are integers then we can choose mid-point of both of them.

Complete step-by-step solution:
$\left(a\right)\dfrac{-1}{3}$ and $\dfrac{2}{5}$
These are two fractions. Observe that one quantity is negative and the other is positive. We know the structure of the real line, the negatives and the positives are separated by a 0 and 0 is a rational number. So, we can simply choose 0.
$\left(b\right)$4 and 5
We are given two integers. We know that for any two integers, the mid-point will lie in between them on the real line, so we choose the midpoint. And the mid-point is calculated by adding the two integers and then dividing them by 2. We do the following:
$\dfrac{4+5}{2}=\dfrac{9}{2}=4.5$
Hence, we have found a rational number 4.5 between 4 and 5.
$\left(c\right)$-2 and -1
Again we are given two integers. So, we find the midpoint and use the fact that mid-point lies between them.
$\dfrac{-1+\left(-2\right)}{2}=\dfrac{-3}{2}=-1.5$
Hence, we have found a rational number -1.5 between 4 and 5.
$\left(d\right)\dfrac{1}{2}$ and $\dfrac{1}{4}$
These are two fractions. And both of them are positive, so we simply try to make a common denominator and then choose an integer that lies between the numerators thus created. Here, we will create 8 in the denominator of both the fractions.
$\dfrac{1}{2}\times \dfrac{4}{4}=\dfrac{4}{8}$
$\dfrac{1}{4}\times \dfrac{2}{2}=\dfrac{2}{8}$
So, the fractions we have now are $\dfrac{4}{8}$ and $\dfrac{2}{8}$. We choose a rational between 2 and 4 which is 3.
So, a rational between $\dfrac{1}{2}$ and $\dfrac{1}{4}$ is $\dfrac{3}{8}$.
$\left(e\right)\dfrac{1}{4}$ and $\dfrac{1}{3}$
Again we try to convert the fractions with the same denominator. We will make the denominator 24 here.
$\dfrac{1}{3}\times \dfrac{8}{8}=\dfrac{8}{24}$
$\dfrac{1}{4}\times \dfrac{6}{6}=\dfrac{6}{24}$
So, the fractions we have now are $\dfrac{8}{24}$ and $\dfrac{6}{24}$. We choose a rational between 6 and 8 which is 7.
So, a rational between $\dfrac{1}{3}$ and $\dfrac{1}{4}$ is $\dfrac{7}{24}$.

Note: There is no such rule as to what rational to choose between two rational numbers. You can choose any of the infinite choices. Moreover, when you encounter fractions, you can convert the fractions into decimals and then choose a decimal between them. You can use this way if you are sure about making no calculation mistakes.