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.