Question

The three rational numbers between 5 and 6 are $\dfrac{{21}}{4}$ , $\dfrac{{22}}{4}$ , $\dfrac{{23}}{4}$. Verify whether the statement is True or False?

Answer 1

Hint: A rational number is a sort of real number that has the form $\dfrac{p}{q}$ and $q$ does not equal to zero. A rational number is any fraction having non-zero denominators. We multiply both values by $\dfrac{x}{x}$, where $x$ is any real number, to get more and more rational numbers in between two rational numbers.

Complete step by step answer:
We have two rational numbers 5 and 6.
We can write 5 and 6 in the form of $\dfrac{p}{q}$ as
$5 = \dfrac{5}{1}$ and $6 = \dfrac{6}{1}$
We will multiply both values by $\dfrac{x}{x}$ , where x is any real number, to get more and more rational numbers in between two rational numbers because $\dfrac{x}{x}$ is 1 and on multiplying it gives same result.
Now,
We will take \[x = 4\] , since denominators are also 4.
And multiply both numerator and denominator by 4.
We get,
$\dfrac{5}{1} = \dfrac{{5 \times 4}}{{1 \times 4}}$
$ = \dfrac{{20}}{4}$
Similarly for 6
 $\dfrac{6}{1} = \dfrac{{6 \times 4}}{{1 \times 4}}$
$ = \dfrac{{24}}{4}$
Now, we can easily write three rational numbers between $\dfrac{{20}}{4}$ and $\dfrac{{24}}{4}$ are $\dfrac{{21}}{4}$ , $\dfrac{{22}}{4}$ , $\dfrac{{23}}{4}$ .
Hence, the given statement, that is three rational number between two rational 5 and 6 are $\dfrac{{21}}{4}$ , $\dfrac{{22}}{4}$, $\dfrac{{23}}{4}$ is True.

Note:
For finding a rational number between two rational numbers we can also use the formula $\dfrac{{a + b}}{c}$, where $c$ is greater than one and let the two rational numbers be a and b. In the above questions we can also assume x=5 we will get rational numbers But those rational numbers are not our requirement.