Finding the Number of Fishbowls Sold
Date: 3/15/96 at 12:34:32 From: Ellen Borger Subject: fishbowls Dear Dr. Math, My son was given the following problem as homework. He is an 8th grade algebra student. Unfortunately, neither my husband or I were able to help him. Could you give us some help figuring this one out? Thanks for your assistance. Ellen Borger Problem: A young entrepreneur named Trivia took some fishbowls that she had bought cheaply to the flea market. In the first hour she sold one-third of them and a third more. In the second hour, she sold half of them, and a half more. In her third hour there, she sold one third of them and a third of one more. The next hour, she sold half of them and half of one more. Finally, she sold the last two and went home to Capitola. How many fishbowls did Trivia sell?
Date: 4/3/96 at 11:49:27 From: Doctor Ken Subject: Re: fishbowls Hello! I'm sorry we've taken a little while to get back to you. It took some time before the doctors could agree on what was meant by the wording of the problem (specifically, the parts like "one third of them and a third more"). But I think we've got it now. Here are a couple of hints about how to do this problem. I'll try to lead you through the first steps, and then see if you can finish it on your own. Let's say that Trivia starts out with a certain number of bowls, and call that number x. How many does she have after the first hour? Well, she sold a third of them plus a third of a bowl (this is what you meant, right?). So the way we can write that in algebraic notation is: Number of bowls left after 1 hour = x - x/3 - 1/3 = 2x/3 - 1/3 Does that make sense? Essentially, what you have to do in the rest of the problem is just keep doing that kind of step over and over. So let's figure out how many bowls she has left after the second hour: Number of bowls left after 2 hours = (2x/3 - 1/3) - (2x/3 - 1/3)/2 - 1/2 = 2x/6 - 1/6 - 1/2 = 2x/6 - 4/6 = x/3 - 2/3 Did that make sense to you? To get that equation on top, I just took the amount we had left (2x/3 - 1/3), subtracted half of it, and then subtracted 1/2. Then I just simplified to get x/3 - 2/3. So you'd keep doing this kind of thing, taking the result from the previous hour each time and figuring out how many you'd have the next hour. At the end of the last hour, you'll have 0 bowls, so you have an equation: set your last expression in x equal to 0, and then solve for x. Good luck! -Doctor Ken, The Math Forum
Date: 4/3/96 at 12:13:6 From: Ellen Borger Subject: Re: fishbowls Thanks for your response. I was able to get some assistance from one of our economics faculty here at UC Santa Cruz. We finally came up with the answer of 162 bowls. Thanks again for your help. Ellen Borger
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