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?
Which rational numbers cannot be written in the form x + 1/(y + 1/z) where x, y and z are integers?
A mother wants to share a sum of money by giving each of her children in turn a lump sum plus a fraction of the remainder. How can she do this in order to share the money out equally?
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. Full Screen Version
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. Full Screen Version
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.