# Would this make for a good high school algebra problem?

High school algebra problems tend not to be very interesting. I thought that this common recreational math problem given to children might provide for some interest.

A frog is at the bottom of a 30 foot well. The frog climbs 5 feet per day and then goes back down 4 feet every night. How long does it take the frog to escape the well.

The common response to this is 30 days, one foot each day. Algebraically, the distance d at the end of t days is d = (5–4)t, which gives t = d. For d = 30, t = 30. What is wrong with this reasoning and how can the algebra be corrected to answer this and similar problems?

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## 11 Answers

Well, a frog can jump almost 45cm high, and that is when the frog is jumping forward at an angle. If it diverts all of its power into a vertical jump, depending on the size of the frog’s foot, it can jump out of the well immediately.

I can see the trick but so far I don’t see the a neat equation, yet.

I will assume you are only interested in integer days and are not breaking it down to fractions of a day using the 5 ft/day climb rate.

Ok, I have to do it mentally and then do an iterative process. If the well were 4 ft the frog escapes the first day. I’ll call that day 1. It escapes at the end of the first day if the well is 5 ft. I’ll call that day 1 also. If the well is 6 feet it takes one more day. Call that day 2. 7 ft…day 3.. The system is linear so t (days) = d (depth) – 4 . We also add the condition if t is less than or equal to 0 then we call the result 1.

The frog gets out of the well in 29 days. Don’t expect me to explain this in algebraic terms though. I suck at math.

I am a little surprised you guys did not get this. The problem with the 30 day answer is that it measures after the frog pulls back. We want the maximum forward progress.

The frog travels 25 feet in 25 days. The next day, the frog travels 5 feet to reach 30 feet in 26 days.

Algebraically we have d = (5–4)(t-1) + 5. Solving for t gives t = d – 4. For f feet forward per day and b feet back each night we get d = (f-b)(t-1) + f. This gives the non-intuitive general formula of t = (d-b)/(f-b), which must be rounded up to the nearest whole number.

That seems like a good extra credit problem, unless the class has been doing several problems in their practice where the day the frog, person, or whatever is moving forward and losing gain gets out of their predicament, and so on that day they do not lose anything.

I figured I’d get it wrong or I just have a slow frog. lol

At least the problem got your attention, which makes for a good starting point

^^ I am interested in math I just have a hard time with it.

A frog? Climbs 5 feet per **day?** Then descends 4 feet? WTF?

No, that’s stupidly, insultingly wrong. Go take a remedial class on the behavior of frogs and other animals!

Try something more like a snail, and look up the speed of snails, and make the period more realistic (or at least round it to say an hour, not a day).

It’s a good problem, but it seems pretty straightforward to me. I would however welcome a whole battery of such problems IF they are really formulated well, and get students thinking about real situations and using math to figure them out. But I think you really need to make them GOOD examples of actual situations, and not surreal, or wrong in their models, or annoying things no one but a puzzle fan would ever really care what the answer was, etc.

The frog never gets out of the well. It climbs up so high but doesn’t have anything to sit on until the next day so it would fall back to the bottom of the well. (This is a logic problem, not a math problem.)

Yeah, no student should be required to do math for story problems where the story makes no sense.

* What would a frog land on 5 feet up a well?

* If there are places to jump and land every 5–6 feet, why doesn’t it just keep jumping up and up?

* What on earth could the 4 feet down part be about, that would make any sense for a frog in a well?

* And what frog would move on a scale of a few feet per day?

* What frog has such constant predictable movement?

The author and/or teacher should answer all those questions before presuming to educate anyone else by demanding answers to math/logic problems where the problems’ situations don’t make sense!

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