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Rationals Between...

Age 14 to 16 Challenge Level:

Charlie and Alison are exploring fractions and surds.
They are looking for fractions with different denominators that lie between $\sqrt{65}$ and $\sqrt{67}$.

Can you find some fractions that lie between $\sqrt{65}$ and $\sqrt{67}$?

Charlie and Alison found that for some denominators, there is no fraction between $\sqrt{65}$ and $\sqrt{67}$. Click to reveal their thoughts.

Charlie said:

$\sqrt{65}$ is approximately $8.06$, and $\sqrt{67}$ is approximately $8.18$.
Fractions with a denominator of $4$ end in $0$ or $.25$ or $.5$ or $.75$ so there is no fraction with a denominator of $4$ between $\sqrt{65}$ and $\sqrt{67}$.

Alison agreed with Charlie but thought about it in a slightly different way:

I'm looking for a fraction $\frac{p}{q}$ where $\sqrt{65}<\frac{p}{q}<\sqrt{67}$.
This means that $65<\frac{p^2}{q^2}<67$,
or $65q^2<{p^2}<67q^2$.

Suppose $q=4$.

$32^2=1024$, and $33^2=1089$, so there is no perfect square between $1040$ and $1072$.

Therefore, $q\neq4$, so there is no fraction with a denominator of $4$ between $\sqrt{65}$ and $\sqrt{67}$.

Can you find other denominators where there is no fraction in the interval?
How will you know when you have found them all?