Question

Find six rational numbers between $3$ and $4$

Answer 1

Find the rational number between two number.

Rational no is a type of real numbers, which is in the form of $\raisebox{1ex}{$p$}\!\left/ \!\raisebox{-1ex}{$q$}\right.$ where $q$ not equal to $0$. Any fraction with non-zero denominators is a rational number.

Let $x$ be a rational common difference.

So $x$ be a rational number such that $x$ is less than ${\frac{1}{6}}^{th}$of the difference between the upper limit and the lower limit.

$$\begin{gathered} \therefore x<\frac{1}{6}·\left(4-3\right) \\ \Rightarrow x<\frac{1}{6}·1 \\ \Rightarrow x<\frac{1}{6} \end{gathered}$$

Say $x=\frac{1}{7}$

From the property of rational number we know that sum of two rational number is a rational number.

So we can say the rational in between $3$ and $4$ are of form $3+n\frac{1}{7}$ where $n=1,2,...,6$

$\therefore$the rational numbers are $3+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$7$}\right.,3+\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$7$}\right.,3+\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$7$}\right.,3+\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$7$}\right.,3+\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$7$}\right.,3+\raisebox{1ex}{$6$}\!\left/ \!\raisebox{-1ex}{$7$}\right.$

$=\frac{22}{7},\frac{23}{7},\frac{24}{7},\frac{25}{7},\frac{26}{7},\frac{27}{7}$

Hence, the six rational numbers between $3$ and $4$ are $\frac{22}{7},\frac{23}{7},\frac{24}{7},\frac{25}{7},\frac{26}{7},\frac{27}{7}$.