Don’t really want an answer, but I don’t understand it so can someone explain this a little bit to get me going

Question

A parent has washed some nappies in a strong bleach solution and wishes to rinse them so that they contain as weak a bleach solution as possible. By wringing out, the nappies can be made to contain just half a litre of solution. Show that two thorough rinses, such that the solution strength is uniform, the first using 12 litres of water and the second using 8 litres of water, reduces the strength of the bleach solution to 1/425 of its original value If 20 litres of clean water is all that is available and the parent is prepared to do only two rinses, how best should the water be divided between the two rinses?

Answer 1

After the wash and wringing, the nappies contain $\frac 12$ litre of strong solution. In the first rinse you add $12$ litres of water, mix that with what was in the nappies to start, and wring them out. That leaves $\frac 12$ litre of the solution in the nappies. What is the concentration of that solution compared to the one they were washed in? The second step is the same with $12$ replaced by $8$. You are asked for the concentration after the second step compared to the start.

Answer 2

When you add $1/2 \,\mathrm{L}$ bleach solution to $12 \,\mathrm{L}$ of water (and mix thoroughly), you have a new solution of $25/2 \,\mathrm{L}$. The concentration of bleach is that solution is $\frac{1/2}{25/2} = 1/25$ of the original concentration and that is the concentration carried by the nappies from the first rinse to the second rinse. You should be able to determine the concentration of bleach after the second rinse by following a similar procedure. After the second rinse, you should determine that the nappies carry $1/2 \,\mathrm{L}$ of a solution at $1/425$ the concentration of the original nappies.

The question asks you to generalize. You have $20 \,\mathrm{L}$ of water which you may partition in any way you like into two rinses. Suppose you put quantity $x$ (in liters) into the first rinse. They you may put quantity $y$ water into the second rinse, with $y \leq 20 \,\mathrm{L} - x$. The question is, what values of $x$ and $y$ make the final bleach concentration the least?

Things to think about:

• Is it ever a good idea to use $y < 20 \,\mathrm{L} - x$?
• Can you compute the final concentration in terms of $x$ and $y$?
• Is there any way to eliminate one of these variables so that you are writing in terms of $x$ only or $y$ only?
• Can you minimize the resulting expression in terms of a single independent variable?