Put on your thinking caps, katz’n’kittenz. Here comes a word problem!
[UPDATE: My solution appears in the comments below.]
Molly the Cat and Tigger live here in our house. They each came to live with us at different times, and of their own choosing. Those could be long stories, one of which I’ve already told here, so I’ll skip ahead to the math problem.
No sooner had they moved in than they started asking for food. Regular meals, and they were quite insistent about it. Being the good ex-hippies that we are, we took it upon ourselves to provide not just a tasty menu, but also excellent nutrition. It took a while, but we finally found a brand of canned food that they liked and that we thought was good for them, and no less than three brands of dry food (hereafter called “crunchies”).
Tigger is a boy, and a little bigger than Molly, and over time we figured out that he needed more food than Molly. No doubt he thought we were hopelessly stupid during the months it took us to come to this realization, but eventually we did, and here is how the daily diet eventually took shape: Breakfast is at 7:00 AM and dinner (“supper” to you Eastern seaboarders) is at 6:00 PM. At each seating, Molly gets one fifth of a can and Tigger gets one fourth of a can. Throughout the day and in the evenings both of them get all the crunchies they want. To make it easier to measure the fractions of cans, each critter eats only from his or her own can until it is empty, then moves on to the next can in the cupboard. So it takes four meals (or two days) for Tigger to empty his can, and five meals (or two and a half days) for Molly to do the same.
You wouldn’t think that both cans would be empty at the same time very often, would you? You’d be right. But for a long time I have had the feeling that that event (two cans empty at the same time — two fresh cans opened for the same meal) was happening a little too often. For about a year, I had that feeling. Somebody — either me or Mrs. Jones — was screwing up the measurements at feeding time. To be fair, it’s pretty hard to eyeball a fifth of a can, and both of us may have muffed it from time to time.
Last night we figured out exactly how often this should happen. I’m embarrassed to say that it took two college graduates a half hour to come up with the definitive answer, and even now we don’t understand it mathematically. How fast can you solve the problem?
Start with two full cans. Give Molly a fifth of her can at each meal, and Tigger a fourth of his can at each meal. Put plastic caps on them and refrigerate between meals. Whenever a can is empty, open a new one. How many days before you find yourself opening two new cans at the same time?
Go ahead and tell me the answer in the comments, if you can. We figured it out basically by running a model scenario all the way to the end, but there is also a mathematical formula that is much more elegant and sophisticated. Except I can’t figure it out and explain the “why” of it. So help me with that, too.
My answer will be posted soon.