what whole number and rational number to the nearest half is 97−−√ between?

Answer 1

To find the whole number and rational number to the nearest half that is between $\sqrt{97}$, we can compare $\sqrt{97}$ to the squares of the numbers near it to determine which whole number $\sqrt{97}$ is closest to.

We can make a table of some perfect squares: $$ \begin{array}{c|c} x & x^2 \\ \hline 9 & 81 \\ 10 & 100 \\ 11 & 121 \\ \end{array} $$We see that $9^2=81<97<100=10^2$. Therefore, we know that $\sqrt{97}$ is between $9$ and $10$.

To find the rational number to the nearest half that is between $\sqrt{97}$, we need to compare $\sqrt{97}$ to the squares of the numbers halfway between $9$ and $10$.

We can make a similar table of some perfect squares: $$ \begin{array}{c|c} x & x^2 \\ \hline 9\frac{1}{2} & 90\frac{1}{4} \\ \\ 9\frac{3}{4} & 95\frac{7}{16} \\ \\ 9\frac{7}{8} & 96\frac{49}{64} \\ \\ 9\frac{15}{16} & 97\frac{225}{256} \\ \\ 10 & 100 \\ \end{array} $$We see that $\displaystyle 9\frac{7}{8} = \frac{79}{8}$ is largest square less than $97$. Therefore, $\sqrt{97}$ is between $9\frac{7}{8}$ and $10$, which simplified is $\boxed{\frac{79}{8}}$.