Sharing the Cost of the Weekend Trip
Date: 07/24/2003 at 21:58:21 From: Jill Subject: Algebra word problem A group of people planned to rent a large beach house for a weekend trip. They were to share the $800 cost equally. However, two people were unable to go and this increased the cost for each person by $20. How many persons were in the original group? I know the answer is 10, but how do you figure it out algebraically and not just guess and check? I tried a system of two equations with two unknowns, but that wouldn't do it. I'm actually a math teacher (fairly new), but I think maybe I'm making it too complicated. Please help. Thanks. x is number in original group y is cost per person x-2 is original group less the two that dropped out 800/x = y 800/(x-2) = y+20 Use elimination of y, but this is not a linear system, so it can't be solved that way.
Date: 07/24/2003 at 23:27:36 From: Doctor Ian Subject: Re: Algebra word problem Hi Jill, Don't knock guess-and-check. If you do it the right way, it's often faster than using algebra. Also, in all seriousness, guess-and-check is often a good first step toward figuring out what equations you need. But if it's algebra you want, it's algebra you'll get. You don't necessarily need two variables. If all N people go, the cost per person is $800/N, right? And if two fewer people go, then the cost per person is $800/(N-2). And we know that the latter is $20 higher than the former, so $800/(N-2) = $800/N + 20 So now we have one equation with one variable. It's ugly, but solvable. Now, how did I know to do this? What the problem is really telling me is the price per person with N-2 people = $20 + the price per person with N people This is a clue that I should look for a way to express the 'price per person' in both cases. Does that make sense? Or we could try to be clever by distorting the problem a little bit - enough to let us see it in a different way, but not enough to change the meaning of the problem. Suppose that instead of just skipping out on their part of the bill, the two people who can't come distribute their share among the remaining people. Their share of the total is (2/N) * $800 and since they're handing $20 to everyone else, their share must be (N-2) times $20. So their share = their share (2/N) * 800 = (N - 2) * 20 This is a little less messy, and if we multiply both sides by N, we'll have a well-behaved (i.e., factorable) quadratic equation. Does this help? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
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