[Pri20150303SJM] Race to the Bottom

Question

Thinking / Planning

Noting that Sarah spent all her money in both scenarios and that 22´(18´something) = 18´(22´something) , I am going to first guess that Sarah takes 18 days and 22 days to spend all her money in the 1^st scenario and the 2^nd scenario respectively. [ Heuristics: H07. Use guess and check & H08. Make suppositions ]  Then I try to adjust my educated guess.

Johari spent more money in the second scenario.  The difference in his spending among the two cases is 6, so we need

                10´(number of days₂) – 12´(number of days₁) = 6

Solution

If Sarah takes 18 days and 22 days respectively to spend all her money, then

the difference in Johari’s spending among the two cases is

                10´(22) – 12´(18) = 4

´³/₂:         10´(33) – 12´(27) = 6

Ans (a): Sarah has $22´27 = $594.
Ans (b): Johary has $(21+12´27) = $345.

Commentary

If the amounts of money involved were in the trillions, we would have thought that Sarah and Johari are certain countries, wouldn't we?

Anyway, a parent from a Facebook parent-support group posed this question asking for a simple solution without using ratios.  This question seems to be the equivalent of a system of simultaneous equations in four variables.  A few of us tried various methods to solve it, but all were quite complicated.  I knew this can be solved using algebra, but struggled for some time to give a simple solution.

I guess all solutions (even the one above) would have some notion of “ratio” hidden in it.  The reason is: every time you multiply or divide by something, there is actually a ratio involved.  But I hope this solution is “easy” enough.