Gifted Issues Discussion homepage Forums General Discussion DS' approach to a test question

I was going to post this under Bad Homework Questions, because I had myself convinced that it was a question without a single answer, but I finally figured out where I was going wrong on it.

However, DS had such an offbeat approach to solving it that I wanted to ask if any mathy people here can tell me why it works or if it's a massive coincidence.

We've been going through the practice SAT on their website, and this was one of the questions:

There are 25 trays on a table in the cafeteria. Each tray contains a cup only, a plate only, or both a cup and a plate. If 15 of the trays contain cups and 21 of the trays contain plates, how many contain both a cup and a plate?

DS looked at it and immediately said 11. I asked him how he got that, and he said that cups are 60%, plates are 84%, which is 144%, so you have to subtract 44% to get back to 100%, and 44/100 equals 11/25, so the answer is 11.

Is there any sense to that?

But it doesn't specify maximum cups or plates? There could possibly 25 of each?

Well, 11 is clearly the correct answer. He didn't get to the answer the same way I did but his way seems to work. To test his method just plug in different numbers to the same problem.

For example, I picked random numbers and did the same problem with 50 trays, 42 with plates, 21 with cups. Using my method, you get 13 with cups and plates. Take his method and cups are 42% and plates are 84% which is 126. 26/100=13/50. Really a pretty clever little method.

The only thing I'd say is that this method gets a little messier if you have numbers that aren't quickly and easily converted to percentages. Notice my numbers as well as the original ones lend themselves to his method pretty well. If there were 33 total trays with 27 having plates and 15 having cups, I'd find his method much less convenient than a more standard approach.

When we add the number of cups (15) to the number of plates (21) we can look at it as counting all trays with cups only, all trays with plates only, and double the number of trays with both cups and plates (we count those trays from both the cups and the plates side). Since there are no empty trays, cups only + plates only + cups/plates counted once gives us the total number of trays, which is 25. What is left (the difference between 15+21 and 25) is that extra counted number of cups/plates trays. So the solution is (15+21)-25=11.

The solution with percentages uses the same approach but adds an extra level of complexity to calculations. It transforms numbers to percentages at the beginning, and at the end converts percentages back to numbers: (60%+84%)-100%=44%.

For some very mathy people, functions, formulas, and processes aren't the right way to understand the thinking. You have to understand the mental (likely visual) model that leads to that approach. That's one of the flaws I see with the inflexibility of classic math instruction particularly for kids who are further out there.

In this problem, if you see each quantity as a stick. You start with three sticks of the same length and clip off parts of two of them to represent the subsets (cups and plates), then you stack those sticks together and compare to the whole (trays). Clip off the part that expands past the base value and that is your overlap.

That model favors his approach. And percentages are a great generalization that allow answering many more questions about the problem set. If he answers it that quickly, then he probably has a solid mental model and the percent calculation is likely close cognitively free for him.

This is a classic use of the Inclusion–Exclusion principle
http://en.wikipedia.org/wiki/Inclusion-exclusion_principle

The diversion into percentages is unnecessary.

It's sensible, but pointlessly hard :-)

Expanding the expression of his approach: 60% of the trays contain cups, 84% of the trays contain plates. 100% of the trays (no more, no less!) contain a cup, a plate or both. There are 100% - 60% = 40% of the trays that do NOT contain a cup, so these must be among the 84% of the trays that contain a plate. The remaining 84% - 40% = 44% of the trays that contain a plate must be the ones that contain both. 44% of the 25 trays is 11 trays.

The reason why it's pointlessly hard is that this is strictly easier (generated by copying, pasting and simplifying):

15 of the trays contain cups, 21 of the trays contain plates. 25 of the trays (no more, no less!) contain a cup, a plate or both. There are 25 - 15 = 10 of the trays that do NOT contain a cup, so these must be among the 21 of the trays that contain a plate. The remaining 21 - 10 = 11 of the trays that contain a plate must be the ones that contain both.

If I were you, I'd make sure he understands exactly why his method was correct, but unnecessarily hard (e.g., show him this post), just to make sure he wouldn't get stuck on a problem that didn't allow for being turned easily into percentages!

Using different methods to solve the same problem is fun and can give you better understanding of some concepts. Let's say, you are calculating 50 - 5. 50 is 200% of 25, 5 is 20% of 25, 200% - 20% = 180%, 25 * 180% = 45. Now, understanding why using percentages gives the same result as simple subtraction is not obvious at all and requires some heavy abstract thinking.

It's Distributivity.
http://en.wikipedia.org/wiki/Distributive_property

Clever, messy, adding an extra level of complexity, unnecessary, and pointlessly hard. Yes, that describes perfectly my son's approach to many things in math!

He will sit there and cover two sheets of paper in unnecessary calculations to do something the hard way, to show how he came up with the answer that he had instantly in his head.

It's utterly fascinating, and supremely frustrating, to watch! I think at least some of it comes from having the intuitive grasp of the abstract but no practical knowledge of the methods most people use to get there. If you don't know how to do something, you have to invent a new way.

Of course, I should have realized I could plug another set of numbers in to check it -- I've only told him that a million times. And I like the stick thing!

I, on the other hand, looked at the question and insisted for a long time that there was no single answer. I thought it depended on which ones you assigned to the second slot -- if you designated 15 with cups, then it depended on which 21 you assigned to have plates. But I finally figured out that the phrasing of the question does not allow for any trays that have neither, which limits it to 11 with both. And it never once crossed my mind to convert anything to percentages! :p