You may also like

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.

At a Glance

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


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?

Investigating the Dilution Series

Age 14 to 16
Challenge Level
Imagine that you create a four-step dilution series, where the amounts of culture passed from one stage to the next and the amount of medium added at each stage are multiples of 10 between 10ml and 100ml each time. In each case there are initially 100,000 cells/ml; the final concentration is given in the diagram as a rounded number.
You can input your numbers into the dilution calculator.
Experiment with the dilution series to investigate these questions:
  1. What are the lowest/highest possible final concentrations?
  2. How could you create final concentrations of 10, 100, 160, 20, 125 and 1875 cells/ml?
  3. At each stage of dilution, how many different dilution factors are possible?
  4. Could you, or how would you, create an exact 1/11, a 1/17, and a 1/23 dilution?
  5. Could you, or how would you, create an exact 1/21 or 1/46 dilution?
  6. Explore impossible dilutions.



Dilution series are used in laboratories progressively to dilute a concentrated solution into a dilute solution. Diluting in the standard ratio 1:9 will create a solution of 10% of the concentration, otherwise called a 1/10 dilution. Successive 1/10 dilutions reduce the concentration by a factor of 10 each time, although other dilutions (as seen in this question) can be made by choosing other ratios.