A colleague of mine was wrestling with a homework problem that had been given to her schoolchild:
Jan has 35 teaspoons of chocolate cocoa mix and 45 teaspoons of french vanilla cocoa mix. She wants to put the same amount of mix into each jar, and she only wants one flavor of mix in each jar. She wants to fill as many jars as possible. how many jars of french vanilla cocoa mix will Jan fill?
A first difficulty with this problem, in my view, is that the writer of the problem failed to capitalize “French”.
Setting aside that difficulty, let’s now turn to the seemingly unlimited number of possible unstated assumptions in this problem.
A first unstated assumption is that apparently the sole measuring implement available to Jan is a teaspoon. The writer of the problem never quite comes out and says that Jan has any measuring implements at all! Maybe what Jan has is a first bag labeled “contents 35 teaspoons” and a second bag labeled “contents 45 teaspoons” in which case it is not at all clear how Jan can proceed at all! Or maybe Jan has a measuring flask that has markings for one, two, three, or four teaspoons. The writer of the problem does not say. But from context we can take a stab in the dark that what the writer is hinting around at is that the sole measuring implement available to Jan is a teaspoon. So far, so good.
The next thing we must puzzle out is how many jars Jan has available for this enterprise. When I first approached the problem, I sort of imagined maybe Jan only had available whatever the minimal number of jars was that would permit satisfying the requirement stated (each jar has the same amount of mix as any other jar). If so, then the answer is, Jan will fill one jar of French vanilla cocoa mix, because one jar will have 35 teaspoons full of one kind of mix and the other jar will have 35 teaspoons full of the other kind of mix.
But then it was pointed out to me by another colleague that maybe there was yet another unstated assumption. Maybe the writer of the problem is also assuming that no matter how many jars Jan might want to have, there are that many jars available or more. Jan has a limitless supply of jars (maybe)!
Okay, and again we toss into the word salad of unstated assumptions that the only measuring implement available to Jan is the teaspoon.
Then I guess the answer is … 45 jars. Jan will put one teaspoon of chocolate cocoa mix into each of 35 previously empty jars, and will put one teaspoon of French vanilla cocoa mix into each of 45 previously empty jars, ending up with 80 filled jars. Each jar will have one teaspoon of cocoa mix in it, meaning that each jar will have the same amount of mix in it as any other jar. This completely satisfies all of the requirements stated in the problem. To be clear, this approach does satisfy the requirement that Jan has put the same amount of mix into each jar. It also satisfies the requirement that Jan only wants one flavor of mix in each jar. It certainly satisfies the requirement that Jan wants to fill as many jars as possible. 80 jars is really a lot of jars! If she wanted to fill as many jars as possible, 80 is surely the right way to go.
But this sounds like a crazy way to go. I am not sure what Jan was planning to do with the jars. Put labels on them and try to sell them in a farmer’s market? Customers would think it quite weird for a jar to be so close to being empty, with a mere teaspoon of cocoa lying in the bottom of the jar. But again, maybe each jar is only slightly larger than one teaspoonful in size, in which case this is a perfect way to go. Jan finishes with 80 jars, each of which is nearly full.
Okay so probably floating around here is an unstated assumption that each jar is quite a bit bigger than one teaspoonful in size, so that filling it with just one teaspoonful will seem odd to whoever looks at the jar later.
Oh we can also back up one more step. The writer never said it, but from context I guess we can also assume that each jar is not smaller than a single teaspoonful in size.
We then return to the unstated assumption that the only measuring implement available is a single teaspoon. Which is almost certainly factually false. The jars themselves are measuring implements. Each jar, likely as not, is the same size as the other jars. Why not simply pour some chocolate cocoa mix into a first jar until it is full, and then pour more chocolate cocoa mix into a second jar until it is full, and continue until you do not have enough chocolate cocoa mix to fill another jar? And then start over again, this time filling jars with French vanilla cocoa mix, filling one jar after another until you do not have enough French vanilla cocoa mix to fill another jar? Go back to the place where the writer said “she wants to fill as many jars as possible.” The only way to fill a jar is … wait for it … to fill the jar. Right. So what you do is … fill the jars. And you keep doing it until you fill as many jars as you are able to fill.
A first problem with this approach (actually filling jars) is that we do not actually know whether we can safely assume that each jar is exactly the same size as each of the other jars. A second problem with this approach is that maybe each jar is actually bigger than 35 teaspoons, in which case it is not possible to “fill” even one jar with the chocolate cocoa mix. A third problem with this approach is that even if we do manage to fill some jars with chocolate cocoa mix and even if we do manage to fill some jars with French vanilla cocoa mix, it is a very unsatisfying approach because the approach sort of assumes that we will be discarding some unusable small portion of the chocolate cocoa mix as well as some unusable small portion of the French vanilla cocoa mix. (Maybe we will mix up the leftovers later and make our own cocoa and drink it so that it will not go completely to waste.) A fourth problem with this approach is that nobody has told us how big the jars are so that if we were to use the jars as our measuring implements, we will not actually arrive at any particular number of jars that would permit us to state a numerical answer to the problem.
Okay so probably we can bracket the writer’s unstated assumption about jar size. Probably the writer assumes each jar is quite a bit smaller than 35 teaspoonfuls in size, but is nonetheless quite a bit larger than just one teaspoonful in size.
We go back and re-read the question, trying to guess if the writer of the question was somehow hamfistedly failing to make clear yet another unstated requirement. “She wants to fill as many jars as possible.” Maybe the word “fill” was being asked to do some heavy lifting here? Maybe the writer was trying (but failing) to say something like “she wants to get each jar as full as possible.” Now that’s more like it. At the farmer’s market, the would-be purchaser of the jar might react better if the jar were not quite so empty as to have only one teaspoonful of cocoa lying in the bottom of the jar.
So we return then to the (unstated) assumption that the only measuring implement to Jan is a teaspoon. And we load up the word “fill” with double or triple duty, assuming that the writer was actually trying to say something like “get each jar as full as we can, while nonetheless not straying from the requirement that each jar has the same amount in it as the other jars, and while somehow arriving at the largest number of jars getting sort-of at least partially filled.” Meaning that we don’t actually “fill” the jars in the sense of the jars ending up being “full”. But we try to fill them more than just tossing a single embarrassing teaspoonful into the bottom of the jar. Right. Layered on top of this being the assumption that no matter how many jars Jan needs, that number of jars or more will somehow be available for this activity.
If indeed this is what the writer was hamfistedly trying to communicate, then what it adds up to is what geeks call a Diophantine analysis problem (Wikipedia article). When we say “a Diophantine analysis” problem we mean “a problem where the only answers that are permitted are numbers that are integers”. We mean a problem where we are talking about things or items or stuff that is quantized. Here it turns out the question posed by the writer, when reworded as an old-fashioned arithmetic question, is “what is the greatest common factor of 35 and 45?” The answer of course is “5”. So you put 5 units of the chocolate stuff into jars one by one, until you have put it into 7 jars. And you put 5 units of the French vanilla stuff into jars one by one, until you have put it into 9 jars. The units here are teaspoonfuls. And the answer turns out to be 9.
The answer is sort of satisfying because we do not end up having to discard some of the cocoa. And it is sort of satisfying because we do not end up with oddly embarrassing nearly empty jars. But it is very unsatisfying because we had to make a large number of assumptions about what the writer of the problem might or might not have been trying to say.
Would you have been able to figure out this problem if a niece or nephew had brought it home from school and dropped it into your lap? Please post a comment below.