### 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?

### Lower Bound

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

# Fractions of Fractions

##### Age 14 to 16 Short Challenge Level:

Answer: $\frac{27}{25}$

$$\frac{2}{3} \times \frac{5}{6} \times X = \frac{3}{4} \times \frac{4}{5} \times Y$$ $$\frac{5X}{9} = \frac{3Y}{5}$$ $$25X = 27Y$$ $$\frac{X}{Y} = \frac{27}{25}$$
This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.