Question

List five rational numbers between \[\dfrac{1}{2}\] and\[\dfrac{2}{3}\].

Answer 1

Hint: First we have to find the L.C.M. (least common multiple) of the denominators of the given fractions. And convert the given fractions into like fractions. Then multiply both numerator and denominator with the obtained L.C.M. to both convert like fractions. The numbers between the obtained fractions are the rational numbers between the given fractions. So, use this method to reach the solution of the given problem.

Complete step-by-step answer:

Since we have to find 5 rational numbers between \[\dfrac{1}{2}\] and\[\dfrac{2}{3}\], so first we find the L.C.M. of the denominators i.e., L.C.M. of 2 and 3 is \[2 \times 3 = 6\].
To convert the given fractions i.e., \[\dfrac{1}{2}\] and \[\dfrac{2}{3}\] in to like fractions we multiply their numerators and denominators with 3 and 2 respectively.
  \[ \Rightarrow \dfrac{1}{2} \times \dfrac{3}{3} = \dfrac{3}{6}{\text{ and }}\dfrac{2}{3} \times \dfrac{2}{2} = \dfrac{4}{6}\]
Now multiply numerator and denominators of the obtained fractions with the L.C.M. i.e., with 6.
\[ \Rightarrow \dfrac{3}{6} \times \dfrac{6}{6} = \dfrac{{18}}{{36}}{\text{ and }}\dfrac{4}{6} \times \dfrac{6}{6} = \dfrac{{24}}{{36}}\]
Consider the rational numbers between \[\dfrac{{18}}{{36}}{\text{ and }}\dfrac{{24}}{{36}}\]
\[ \Rightarrow \left( {\dfrac{{19}}{{36}}} \right),\left( {\dfrac{{20}}{{36}}} \right),\left( {\dfrac{{21}}{{36}}} \right),\left( {\dfrac{{22}}{{36}}} \right){\text{ and }}\left( {\dfrac{{23}}{{36}}} \right)\]
So, these are the rational numbers between \[\dfrac{1}{2}\] and\[\dfrac{2}{3}\].
Therefore, the 5 rational numbers between \[\dfrac{1}{2}\] and\[\dfrac{2}{3}\] is \[\left\{ {\dfrac{{19}}{{36}},\dfrac{{20}}{{36}},\dfrac{{21}}{{36}},\dfrac{{22}}{{36}},\dfrac{{23}}{{36}}} \right\}\]

Note: A rational number is a number that can be expressed as the quotient or fraction p/q of two integers, a numerator of p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number.