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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.

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The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

Contact

A circular plate rolls in contact with the sides of a rectangular tray. How much of its circumference comes into contact with the sides of the tray when it rolls around one circuit?

Dilution Series Calculator

Age 14 to 16 Challenge Level:
 

This interactive diagram shows a dilution series made from four dilutions of an initial bacterial solution. At each stage of the dilution, a quantity of the culture is diluted with a quantity of the medium

Dilutions
?
cells/ml


Try to calculate the dilution factor for various combinations of initial numbers. Can you create a general rule for the overall dilution?  

Other problems

Investigating the Dilution Series