Using Variables to Solve Word Problems

Date: 05/10/98 at 16:23:47
From: Christina
Subject: using algebra( with "let" statements)

We started this "let x be the first number" stuff last week and I 
still have no clue ... at all! Can you help me with this?!

For example:

Susan puts only dimes and quarters in her coin bank. She has 50 coins 
with a value of $8.30 in her bank. How many coins of each kind are 
there?

Help!

Date: 05/12/98 at 13:29:45
From: Doctor Loni
Subject: Re: using algebra( with 

Yes, it can be confusing at first, but pretty soon you'll be a pro!

First, look at the problem to see what you are looking for. The 
problem asks how many dimes and how many quarters there are in Susan's 
piggy bank, and you are given clues to help you figure it out.

There are 50 coins in her bank, and because she only has dimes and 
quarters, the total number of dimes and quarters added together is 50. 
We can write this as:

   Number of dimes + Number of quarters = 50

At this point, we don't know what either one of those numbers is. This 
is where that "let x ..." part comes in. All the x does is stand for 
something. We could actually name it anything we wanted. Say, for 
instance, we let D stand for the number of dimes -- that would work 
just as well. In fact, let's use D for the number of dimes. Now 
instead of writing "number of dimes," we will write D:

   D + Number of quarters = 50

Now, if you knew the number of dimes, how would you find the number of 
quarters? You know that the total number is 50, so if you subtracted 
the number of dimes from 50, you would know the number of quarters:

   Number of quarters = 50 - D

So even though we don't know the number of coins yet, we have labels 
for them. Just this amount of information isn't enough, so you were 
given how much value the coins have. The total value of the coins is 
$8.30. If you knew the number of dimes and number of quarters you had, 
could you figure out how much they were worth? You would simply count 
the money -- 25 cents for each quarter and 10 cents for each dime. In 
this case, you know how much they add up to, you just don't know how 
many there are. But you do know that the number of quarters times 25 
cents plus the number of dimes times 10 cents will give you $8.30. We 
could write it like this:

   .25(Number of quarters) + .10(Number of dimes) = 8.30

We have already written expressions for the number of quarters and 
number of dimes, so we can just substitute those in:

   .25(50 - D) + .10(D) = 8.30

Now you have an equation with just one variable and you can solve for 
D, where D is the number of dimes. To get the number of quarters, look 
back to the expression for quarters, plug the number in for D, and you 
will have the number of quarters.

The problem would be the same if you had x stand for the number of 
dimes instead of D. The important thing is to have a symbol that 
stands for what you are looking for.

Hope that helps. Let me know if you have more questions.

-Doctor Loni, The Math Forum
Check out our web site! http://mathforum.org/dr.math/