### Golden Thoughts

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

### Days and Dates

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

# Cubic Fractions

##### Stage: 3 and 4 Short Challenge Level:

As all the fractions are raised to the power 3, the expression which has the largest value is that with the largest fraction in the brackets.

Each of these fractions is a little larger than $1\frac{1}{2}$. Subtracting $1\frac{1}{2}$ from each in turn, we get the fractions $\frac{1}{14}$, $\frac{1}{6}$, $\frac{1}{4}$, $\frac{3}{10}$, $0$, the largest of which is $\frac{3}{10}$ (because $0$ < $\frac{1}{14}$ < $\frac{1}{6}$ < $\frac{1}{4}$ = $\frac{2\frac{1}{2}}{10}$ < $\frac{3}{10}$).

Hence $\left(\frac{9}{5}\right)^3$ is the largest.

This problem is taken from the UKMT Mathematical Challenges.

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