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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 $q=4$.
$65\times16<{p^2}<67\times16$
$1040<{p^2}<1072$
$32^2=1024$, 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?
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 $q=4$.
$65\times16<{p^2}<67\times16$
$1040<{p^2}<1072$
$32^2=1024$, 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?
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