## The ten day week, now resolved!

This post is by James

Last week, I published a simple question:  if the week were ten days long, and we got three days off, instead of two, would be get more or less days off than we currently do, with the seven days week with two days off?

Of course, there can only be one right answer, but it turns out there are lots of different ways of getting there!  Here is my personal solution:

Let us consider the smallest number which is a multiple of both ten and seven.  In our case, that is 70.  Now this means that 70 days contains either 7 ten-day weeks, or 10 seven-day weeks.  Since we get two days off per week in the seven-day week, this means that the seven-day week gets 20 days off in seventy days.  The ten-day week has three days off per week, but only seven weeks, so we get 21 days off with the ten-day week.  Thus we conclude that the ten-day week gives us slightly more time off.

There are other ways of doing this.  My father, for instance, when he was working it out, phrased the problem differently.  His logic was that in a seven-day week, we get one day off for every 3 and a half days of the week.  Thus to get three days off, at the same rate, the week would have to be 10 and a half days long:  seven days for the first two days off, and three-and-a-half days for the third day off.

Because 10 and a half is slightly bigger than ten, the ten-day week actually gives us more time off.

If this all seems a bit complicated, hopefully the picture will help.

Both of these are perfectly good ways to solve the problem!  They both give the right answer, and we can extend both of them to solve other problems (like how much more time you would get off each year with the ten-day week.  It’s about five days.)

This is relevant for math education because the original problem contains the exact same math as the dry-as-dust problem:

$\text{Circle the bigger fraction: }$

$\frac{3}{10}\;\;\;\;\;\; \frac{2}{7}$

Yes, our original perfectly sensible and interesting problem can be turned into a seemingly pointless problem which makes kids wonder why anybody ever bothers with the stuff!  We bother, of course, because in the hands of somebody who understands the math, the dry-as-dust description also happens to be reliable and fast.  But without understanding the problems behind the math, is it any wonder students think that math is all about “circle the bigger fraction,” rather than exploring fun and exciting what-ifs?

The other reason we use the dry math is because it works!  It always works, and it always works the same way.  For instance, if somebody is cooking and comes across a bizarre recipe which calls for 2/7 of a cup of sugar and the user only has 3/10 of a cup, do they have enough sugar?  Mathematically, actually solving the problem uses the exact same math as this!  It’s not obvious, but it is true.  And once you understand both problems, you can see it.  But without motivation, who actually gets that far?  Very few.  And so education needs the context before people engage.

Now I’m curious.  Can anyone else thinks of anyone contexts which result in this problem?  The stranger, the better!