### 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?

### 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?

# Ratios and Dilutions

##### Age 14 to 16Challenge Level

Scientists often require solutions which are diluted to a particular concentration. In this problem, you can explore the mathematics of simple dilutions.

Imagine you have a beaker containing a solution with a concentration of 100 000 cells per millilitre of liquid. You can transfer some of this solution into a second beaker, in multiples of 10ml, and add water in multiples of 10ml to dilute the solution.

If you diluted 100ml of the original solution with 100ml of water, what would be the concentration, in cells/ml, of your new solution?

Investigate other dilutions that can be made.
You could use this interactivity to check your ideas. The interactivity lets you perform a series of up to four dilutions. To perform a single dilution, transfer 0ml of water and 100ml of solution for the last three dilutions.

Here are some questions to consider:

• Can you make solutions which are half the strength of the original?
One third of the strength? One quarter? One fifth? ...
• What about fractions with a numerator greater than 1?
• Are there any concentrations you can make in more than one way?
• What can you say about the concentrations you can't make?

A series of dilutions can be performed, where a solution is diluted, and then the resulting solution is also diluted.

Find two dilutions which give a final concentration of:

• $50000$ cells/ml
• $33333.\dot3$ cells/ml
• $75000$ cells/ml
• $49000$ cells/ml
• $24000$ cells/ml
• $45000$ cells/ml
• $26666.\dot6$ cells/ml

How many different ways can you find to make a final concentration of $25000$ cells/ml?

You could use this interactivity to check your ideas. The interactivity lets you perform a series of up to four dilutions. To perform a series of two dilutions, transfer 0ml of water and 100ml of solution for the last two dilutions.

Find some concentrations which are impossible to create using two dilutions.
How can you convince yourself that they are not possible?
List the necessary criteria for deciding whether a concentration is possible or not.

You may wish to try the problems Investigating the Dilution Series and Exact Dilutions, which expand on the ideas in this problem.