Date: 01/27/2001 at 16:27:24 From: al Subject: Math You are a researcher of primitive cultures. You visit the island of Ooga and learn of these monetary exchanges: 2 coconuts = 1 banana 3 bananas = 2 mangos 4 papayas = 1 coconut What is the exchange for banana to papaya and for papaya to mango? I tried but I just don't get it.
Date: 01/29/2001 at 12:25:18 From: Doctor Greenie Subject: Re: Math Hi, Al - This kind of problem can be very confusing. I have seen many different methods tried for teaching how to solve this type of problem, and none of the methods I have seen seems to be very effective for a large percentage of students. I will show you a couple of ways I solve these problems. The first method is easier to understand for most; the second method I find to be much faster, but also much harder to understand. I will find the exchange rate for banana to papaya by each of the two methods and let you try the same methods on the papaya-to-mango problem. First Method: I want to find the conversion rate for bananas to papayas. I am given the conversion rates for bananas to either coconuts or mangos. I am not given the conversion rate for mangos to anything else, so I can't use the bananas-to-mangos conversion to solve my problem. But I am given the conversion from coconuts to papayas, so I should be able to convert bananas to papayas by converting first to coconuts. So I know 1 banana is equal to 2 coconuts, and 1 coconut is equal to 4 papayas. If I think of exchanging my 1 banana first for 2 coconuts, then I need to figure out how many papayas those 2 coconuts are worth. Each of them is worth 4 papayas, so 2 of them are worth 8 papayas. So my original 1 banana is worth 8 papayas. I have solved this problem using words, but if I want to use mathematical symbols to see how the method works, I can use B, C, and P to represent the numbers of bananas, coconuts, and papayas; then I can write 1B = 2C (1 banana = 2 coconuts) and 1C = 4P (1 coconut = 4 papayas) Then I can "double" the second equation to get 2C = 8P The reason I do this is because I have "2C" in my first equation. Now I have "2C" in both equations, and I can therefore "telescope" the two equations and write 1B = 2C = 8P so 1 banana is equal to 8 papayas. Second Method: I can think of each given exchange rate as a fraction with a value equal to 1. For example, since 1 banana is equal to 2 coconuts, then the fraction 1B ---- 2C has the value 1, as do the fractions 3B ---- 2M and 1C ---- 4P The reciprocals of all these fractions also have the value 1; for example, 4P ---- 1C Now in each of these fractions I can think of the units (bananas, coconuts, and so on) as parts of the fraction, so that when I multiply these fractions I can try to cancel the units. In the problem where I want to find the exchange rate for bananas to papayas, I want to find a numerical value for the fraction ?B ---- ?P From the given information I know the fractions for ?B ---- ?C and ?C ---- ?P and I know that ?B ?C ---- * ---- ?C ?P will give me ?B ---- ?P because the "C" (coconuts) units will cancel out. So I have 1B 1C 1B ---- * ---- = ---- 2C 4P 8P This tells me that the fraction "1 banana over 8 papayas" has the value 1. In other words, that "1 banana" and "8 papayas" have the same value, so 1 banana is equal to 8 papayas. Here's how to start using this second method for the papayas-to-mangos problem: You need to find the fraction for ?P ---- ?M With the exchange rates which are given, you know the fractions for ?P ?C ?B ---- , ---- , and ---- ?C ?B ?M If you multiply these three "fractions" together, the "C" (coconut) and "B" (banana) units will cancel out, leaving you with the desired fraction ?P ---- ?M Good luck with these. - Doctor Greenie, The Math Forum http://mathforum.org/dr.math/
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