Question

Give three rationals between $\dfrac{1}{2}$ and $\dfrac{3}{4}$.

Answer 1

Hint:Rational numbers are the numbers which can be expressed in the form of p and q where $q \ne 0$ .Examples for rational numbers are prime and composite numbers, odd and even numbers, decimals and fractions. A number of rational numbers between two rational numbers can be located. Between any two rational numbers, countless rational numbers can be found.

Complete step by step answer:
First you need to know the following concepts: The first step in determining the rational numbers between two rational numbers is to check the value of the denominators. If the denominator values are the same, check the value of the numerators. If the numerators differ by a large value, then the rational numbers between the two rational numbers can be written in the increments of one for the numerator without altering the value of the denominator.

If the denominators are different then you need to make them the same.Equating the denominators can be done either by finding their LCM or by multiplying both the numerator and denominator of both the numbers with such natural numbers that the denominators of both the rational numbers become the same. Here in this question we have to give three rationales between $\dfrac{1}{2}$ and $\dfrac{3}{4}$.

Here it is clearly visible that the denominators of the two rational numbers are different. So we have to first make the denominators of the two rational numbers equal.
$\dfrac{1}{2} = \dfrac{{1 \times 2}}{{2 \times 2}} = \dfrac{2}{4}$ and $\dfrac{3}{4} = \dfrac{{3 \times 1}}{{4 \times 1}} = \dfrac{3}{4}$
We can see that there is no number between $2$ and $3$ . So we will make the denominators a little bit large as follows:
$\dfrac{1}{2} = \dfrac{{1\times 12}}{{2\times 12}} = \dfrac{{12}}{{24}}$ and $\dfrac{3}{4} = \dfrac{{3\times 6}}{{4 \times 6}} = \dfrac{{18}}{{24}}$
We changed the rationals to have equal denominators. So now rational numbers between $\dfrac{{12}}{{24}}$ and $\dfrac{{18}}{{24}}$ would be rational numbers between $\dfrac{1}{2}$ and $\dfrac{3}{4}$ because they are equivalent fractions.

Therefore rational numbers between $\dfrac{1}{2}$ and $\dfrac{3}{4}$ are:
$\dfrac{{13}}{{24}}$ , $\dfrac{{14}}{{24}}$ , $\dfrac{{15}}{{24}}$.

Note:To find the rational numbers between two rational numbers denominators should be made equal. Between any two rational numbers, countless rational numbers can be found. You can choose any of them on the basis of your requirements.