### Rationals Between...

What fractions can you find between the square roots of 65 and 67?

### There's a Limit

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

### Not Continued Fractions

Which rational numbers cannot be written in the form x + 1/(y + 1/z) where x, y and z are integers?

# Fracmax

##### Age 14 to 16Challenge Level

Find the maximum of

$${1\over p} + {1\over q} + {1\over r}$$

where $p$, $q$ and $r$ are positive integers and

$${1\over p} + {1\over q} + {1\over r} < 1.$$

Prove that it is indeed a maximum.