### There's a Limit

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

### Not Continued Fractions

Which rational numbers cannot be written in the form x + 1/(y + 1/z) where x, y and z are integers?

### Lower Bound

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

# Ratios and Dilutions

##### Age 14 to 16 Challenge 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? Use the interactivity below to see if you are right.

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Investigate other dilutions that can be made, using the interactivity to check your ideas.

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.

Use the interactivity below to investigate the concentrations which result from two dilutions. Try to predict what will happen before clicking "Get dilution" to check you are right.

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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?

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.