### 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.

### From All Corners

Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.

### Star Gazing

Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.

# Six Notes All Nice Ratios

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

We had a good collection of offers:

David from St.Albans, Adam from Ystalyfera, Adrianos, Julian from Wilson's School, and Andrei from Bucharest all sent in their chosen sets of fractions with their reasons.

Here's one set in a table : it shows, for example, that 3/4 is 15/16 of 4/5 (check that for yourself)

And here's the spreadsheet file I used to check the fraction sets which people sent in .(Excel file )

If you check some other possible solutions then you will find that Andrei has found the best set.