Monetary Conversions

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/