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.

17 Replies to “Thinking more about problem solving”

1. Miriam Richter says:

Carl,

I love all your reasoning but the answer is that it cannot be solved without more information. As a scientist there is no way I that I would make more than one, maybe two (if there was a gun to my head), assumptions. As a lawyer, if this were a statute, I would say that “fill” is not a term of art so the common understanding would be used – fill the jar all the way. Otherwise it would be described as not filled. Also, the question only asks about the French vanilla. Information regarding the second flavor is an extraneous distraction.

Lastly, as a former teacher, I would want to know context before I even tried to solve it – what grade level and what class is this from? Logic? Math? Arts & Crafts? Cooking?

Hmmmm.

2. Neil Ormos says:

It looked like a GCD problem to me from the start, even with the poorly specificied problem statement. Sometimes the problem will not be well specified because a technique for performing some task was taught in the class on the same day, and the teacher expects the context of the day’s lesson to be carried into the student’s work outside of class.

3. Katherine Koenig says:

Also, what’s the meaning of the term “mix”? The problem describes “chocolate cocoa mix” and “[F]rench vanilla cocoa mix,” which suggests we’re meant to assume that only one of chocolate and French vanilla can be included in each jar (“…only wants one flavor of mix in each jar”). However, what if we (ahem) mixed the cocoa and the French vanilla together to create a hybrid mix? That’s one flavor, and we’d have 80 teaspoons of it… and each teaspoon would include French vanilla cocoa, which would make it a “French vanilla cocoa mix.” So we’d have 80 teaspoons of “French vanilla cocoa mix” for which to find a jar, and not just 45 teaspoons. Then return to the problems of jar size, what’s “fill” mean, etc.

Funny how many ways you can approach this one!

1. Timothy G. Ackermann says:

Katherine, you are really right.
I was thinking along these lines too, but my answer was “zero” French vanilla cocoa mix.
The instructions say there should be only “one flavor of mix in each jar.” So the “one flavor” in “each jar” is your blend / hybrid. Thus, there is no jar with French vanilla cocoa mix–they are all blend.
And there could be a *really* large number of jars total–the desire was “She wants to fill as many jars as possible.”–just make them very very small.

4. Suzannah says:

This question is just a bad question all around and likely indicates that the person who constructed the question needs some education himself. One must know the minimum amount that can be placed in each jar. I also suspect that your fact pattern has a typo… that the 45 tsp is of French vanilla and one is combining the cocoa + French vanilla to result in French vanilla cocoa. Also it is unclear to me whether the amount of cocoa and the amount of French vanilla in each jar must be the same or if the requirement is that the same ratio of cocoa to French vanilla is provided in each jar. Anyway, I think the answer to what one was trying to ask is # of jars = x = 35/y where y = the minimum amount of French vanilla that can be placed in a jar, and assuming the same ratio of cocoa to French vanilla, and all the cocoa is to be distributed in the jars, the amount of cocoa in each jar will be 45/x… which essentially the ratio of cocoa to French vanilla will be 45:35 in each jar.

5. Dale says:

Personally, I want to buy vials that hold 1/1000 of a teaspoon. Then you can fill a *lot* of them! But because of “She wants to fill as many jars as possible.” I took it that the challenge was to interpret the problem in a way that maximizes the number of jars.

6. Newton says:

Even after Carl’s thoughtful analysis, I continue to believe that the “35 teaspoons of chocolate cocoa mix” is irrelevant.

I view this as a unit conversion problem – from (tsps of mix) to (number of jars filled)

I hope that (after considerable fuming and fussing and complaining about the teacher and the question) I might have suggested that my niece or nephew write:

Jars that contain only French vanilla cocoa mix contain no chocolate cocoa mix

number of jars filled with french vanilla cocoa mix after Jan does what she wants to do =
int ( (35 tsp) * (1 jar / (number of tsp in a jar) ) )”

Thereby providing an answer and politely informing the teacher that the question was not overly clear.

And remember, most _word_ problems in math, physics, chemistry,. etc. start with the unstated assumptions that
there is a shared, objective physical reality
such shared reality follows consistent rules that we fully understand
god does not miraculously recreate the entire universe an infinite number of times during each instant

I’ll post my answer once I’m done moving molecules of cocoa mix, one by one, into a practically limitless sea of jars.
But can anyone tell me the molar weight of “chocolate cocoa mix” and “french vanilla cocoa mix”?

8. Miriam Richter says:

Did any of us come up with that??????

Peter: I thought you were kidding!!! Do you have access to the cited lesson? I’m beyond curious at this point! What grade is this for?

PS: I can hear the lawyer jokes already!

9. Neil Ormos says:

Yes. Carl did. “And the answer turns out to be 9.” And Peter told us what grade the answer book was for: 6th.

10. Newton says:

the answer book helps a great deal!
Since the company that created the question “must” know the “correct” answer, I feel silly that I did not immediately see that either:
(i) The last “french vanilla” that I think I see in the problem is not actually there _and_ the jars hold more than 7.5 tsp and less than 8.8 tsp (calculating a tighter range is left as an exercise for the interested reader)
or
(ii) the “chocolate coco mix” is a distraction _and_ the last “french vanilla” is part of the problem _and_ each jar holds 5 tsp
or
(iii) Jan will not fill jars in accordance with her desires (the problem tells us what she wants, not what she did)

I feel so silly for not having seen that the first time I looked at the problem.

Extra Credit –
1. how would the answer vary if the 35 tsp and 45 tsp were for “fluffy” piles of mix and the mix was compressed and packed as tightly as possible into the jars using each of the following methods
(a) by hand,
(b) using a rolling pin or
(c) using a 20 ton hydraulic press and some very strong jars
2. how would the answer vary if each jar should be a different size?

11. Dan Feigelson says:

Carl, you’ve never studied Talmud, but this story problem, and your analysis of it, present the kind of thinking often shown in the Talmud itself and in the analysis thereof. Indeed, the problem is horribly worded, in that there are many unstated assumptions. To get at the answer that the person who wrote the problem wanted, the problem should have been stated:

If Jan has more than enough jars available so that she fills jars so that each jar is completely filled with either vanilla cocoa mix or chocolate cocoa mix, and each jar is the same size, and no mix is left un-jarred, what is the minimum number of jars that will be filled with vanilla cocoa mix, and how much cocoa mix does each jar hold? Assume you have any measuring implement of any size available to you.

But that’s not what it says. Per what’s written, I could use half-teaspoon jars and thus fill 90 of them with vanilla mix. If I had been given this problem in a high-school math class, I would have explained my assumptions and how these led to the answer I got, and if I wasn’t given full credit I would have raised hell with the teacher. If this had been a law school exam, we would have been expected to point out all the unstated assumptions and how if we tweak one of them (size of measuring implements available, size of jars available, number of jars available, permissibility vel non of mixing the mixes, etc.) that affects the outcome.

1. Neil Ormos says:

I’ve met him only online, but from distant observation, it wouldn’t surprise me at all if it turned out that Carl was learned on any particular topic… Midrash, art of the Yuraba peoples, whatever.

12. Miriam Richter says:

OK, so on rereading all the various ideas, I decided to see what was the minimum number of words that need to be changed to clarify the problem and make it solvable. I came up with -2:

Jan has 35 teaspoons of chocolate cocoa mix and 45 teaspoons of french vanilla cocoa mix. She wants to put the same amount (in whole tsps) of mix into each (5 tsp) 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?

Now this works (+5) but doesn’t address the lesson of finding the common denominator. To do that you need to remove words as follows:

Jan has 35 teaspoons of chocolate cocoa mix and 45 teaspoons of french vanilla cocoa mix. She wants to put the same amount, in whole tsps, of mix into each (5 tsp) 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?

This is minus 5 so we have a net change of -2 words!

13. Miriam Richter says:

correction in the last line change 5 to 7!

1. Suzannah says:

Seems to me the minimum words to clarify the question yet get the correct answer is 27 total words: Jan has 45 teaspoons of French vanilla cocoa mix. She wants to fill as many jars as possible with 5 tsp. How many jars will Jan fill? The original question is still, imho, messed up and so the so-called correct answer from the answer book cannot be said to be a correct answer. I mean where in the fact pattern is one told (1) the max capacity of the jars and that mix must be added in whole teaspoons, and/or (2) the max amount of mix that can be added to each jar? Absent these facts, ANY number (up to the number of French vanilla cocoa mix granules contained in 45 teaspoons) is a correct answer. Imho, poorly worded questions like this and answers that are asserted to be the “correct” answer teach kids to (a) just do as told/instructed and don’t think for themselves, and/or (b) confuse the problem-solving “thinkers” such that they don’t understand why and how the asserted “correct” answer is correct rather than their answer which is probably more correct. For example, but for my assumption that one was to “MAKE” the French vanilla cocoa mix by mixing chocolate cocoa mix and French vanilla mix, and that the “single flavor” meant that each jar must have the same ratio of the two mixes to give the “same” flavor French vanilla cocoa mix, my answer would have been right. Without making a mix and in the absence of facts #1 and #2 above, my original answer (without my accidentally mixing up the 35 and the 45 for the French vanilla cocoa mix) would be more correct than the asserted “correct” answer in the answer book: That is # of jars = x = 45/y where y = the minimum amount of French vanilla that can be placed in a jar. Thus, if y is 5 teaspoons, then yes, the answer is 9. If y is 15, then the answer would be 3, etc.

Anyway, this is the type of thing that irritated the heck out of me in school… and even in college, where, for example, I spent all night before a biophysical chemistry exam obsessed about why I kept getting a “wrong” answer according to the answers in the back of the textbook. I was obsessed about it and couldn’t study anything else because if I couldn’t understand this basic question and how to solve it, I was doomed with all the rest of the questions that were more complex and further expanded upon the basic premise. The test was first thing in the morning and so right before, I ran to see my prof and told him I just didn’t understand why the answer in the book is correct… The prof looked at it for a few minutes and then said “Well, because the answer in the textbook is wrong.” Mother Heifer.

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