More Variables Than Equations in a System

Date: 01/21/2002 at 16:35:27
From: Penny
Subject: Equations with multiple variables

Zeke, Maisie and Ivan stopped at a fruit market for an afternoon
snack.  Zeke bought an apple and a mango, and he paid $1.40.  Maisie
bought a pear and an orange, and she paid 90 cents.  Ivan was really
hungry, so he bought a mango, an orange, a pear, and an apple.  How
much did Ivan pay?

I'm trying to use variables, but I have four variables and there
doesn't seem to be enough information to figure it all out.  Can you
help me?

Date: 01/21/2002 at 17:14:30
From: Doctor Riz
Subject: Re: Equations with multiple variables

Hi Penny -

While having more variables than equations is often a difficult
situation, in this particular problem you don't have to figure out how
much each piece of fruit costs in order to answer the question.  

I agree that we could use four variables.  Let's use these:

    a = cost of an apple
    m = cost of a mango
    o = cost of an orange
    p = cost of a pear

We know that Zeke paid $1.40 for an apple and a mango, and Maisie paid
90 cents for a pear and an orange, so:

    a + m = 1.40   and   p + o = 0.90

Ivan bought a mango, an orange, a pear, and an apple, so:

    m + o + p + a = Ivan's bill

As you said, we don't have enough information to figure out each
variable.  But let's rearrange Ivan's equation:

    (a + m) + (p + o) = Ivan's bill

Can you see that this is the same as his original equation?  We can
change the order in which we add things without changing the result. 

Now we can use some substitution.  Since we know from the earlier
equations that a + m = 1.40 and p + o = 0.90, we can substitute those
into Ivan's equation to get:

    1.40 + 0.90 = Ivan's bill
           2.30 = Ivan's bill

We can now see that Ivan would have paid $2.30 to buy one of each kind
of fruit.  

Again, the key idea in this problem is that we did not determine the
individual cost of any of the pieces of fruit, but we were still able
to find the cost of buying one of each.

Does that help?  Feel free to write back if you need more help.

- Doctor Riz, The Math Forum
  http://mathforum.org/dr.math/