Question

Find five rational number in between $\dfrac{3}{5}$ and $\dfrac{4}{5}$. \[\]

Answer 1

Hint: We recall rational numbers. We find $n$ rational numbers between $\dfrac{a}{b}$ and $\dfrac{c}{d}$ by first finding the equivalent fractions by converting the denominators $b,d$ to their least common multiple. If we do not get $n$ number of rational numbers, we multiply the least common multiple by multiplying $2,3,4,5,...$ to get $n$ rational numbers.

Complete answer:
We know that a rational number is always in the form $\dfrac{p}{q}$ where $p$ any integer is and $q$ is a non-zero integer.
We are asked to find 5 rational numbers between $\dfrac{3}{5}$ and $\dfrac{4}{5}$. We know how to compare rational numbers, we have to first find rational numbers which means we have to make the denominators equal to the least common multiple of the denominators. We see that in $\dfrac{3}{5},\dfrac{4}{5}$ the denominators are already equal.
If the denominators are equal we add in between the rational numbers by taking different integers in between the numerators that are 3 and 4 and take denominator 5. Since there are no integers in between 3 and 4 we multiply 2 to get the equivalent fractions as
\[\begin{align}
  & \dfrac{3}{5}=\dfrac{3\times 2}{5\times 2}=\dfrac{6}{10} \\
 & \dfrac{4}{5}=\dfrac{4\times 2}{5\times 2}=\dfrac{8}{10} \\
\end{align}\]
We see that there is only one integer 7 in between the numerators 6 and 8 when we multiply 2 and so we are going to get one fraction but we need 5. So we multiply 6 to get the equivalent fractions of $\dfrac{3}{5}$ and $\dfrac{4}{5}$ as,
\[\begin{align}
  & \dfrac{3}{5}=\dfrac{3\times 6}{5\times 6}=\dfrac{18}{30} \\
 & \dfrac{4}{5}=\dfrac{4\times 6}{5\times 6}=\dfrac{24}{30} \\
\end{align}\]
Now we see that there are 5 integers in between 18 and 24 which are 19, 20, and 21,22,23,24. We take them as numerators and take denominator 30 find the rational numbers in between $\dfrac{3}{5}$ and $\dfrac{4}{5}$ as,
\[\dfrac{19}{30},\dfrac{20}{30},\dfrac{21}{30},\dfrac{22}{30},\dfrac{23}{30}\]

Note:
We can directly find $n$ number fractions between $\dfrac{a}{d}$ and $\dfrac{b}{d}$ by multiplying $n+1$ in numerator and denominators to convert into equivalent fractions. We should note that there are infinite rational numbers between any two rational numbers. If the numerators are same and denominators are different the smallest denominator has the greater rational number and if the denominators are same the greater numerators has greater rational number.