Solving Algebra Word Problems

Date: 10/16/98 at 22:32:46
From: Kathy/middle school student's mom
Subject: Algebra word problems

Hello Dr. Math,

I am trying to help my daughter with her algebra. This one kind of 
problem stumps me. Here is an example:

Ohio has half the population of California, and twice the the 
population of Indiana. If the total population of the three states is 
38,500,000, find the population of each state.

It is this kind of problem that she is having most trouble with.

Thanks,
A grateful mom

Date: 10/17/98 at 07:06:01
From: Doctor Derrel
Subject: Re: Algebra word problems

Hi Kathy,

Word problems may be confusing when you first read them, but they are
usually more understandable if you read them in small chunks. 

First, a word about using letters for unknown values, which are 
commonly called variables. We need to make sure that the letters we use 
make sense. Using x, y, z, etc. does nothing to shed light on the 
problem, and they are just plain confusing besides. We want letters 
that will help us remember something about the problem. For example, 
if we are talking about chickens and pigs, c and p are good letters. 
Don't use the letter "O" because it can be confused for zero. In a 
case like that, I will choose something else that makes sense to me.

One strategy that I encourage my sixth and seventh graders to use is 
to rewrite a word problem in their own words before they start it. 
Rewriting helps them focus on all of the important information. In 
this case, we might rewrite the problem as, "Ohio, California, and 
Indiana have a combined population of 38,500,000. We know that Ohio 
has two times the population of Indiana. We also know that Ohio has 
one-half times the population of California. What is the population 
for each state?"

So, we then look at the problem in small chunks. I like to assign 
letters to the unknown values when I go through a problem to keep it 
from seeming so overwhelming. Let's look at the first sentence of the 
rewrite, "Ohio, California, and Indiana have a combined population of 
38,500,000." From this, I know that we are talking about population. I 
want to assign letters for everything I can in this sentence, even if 
I decide not to use some of them later. (This is important - if you 
pick and choose right at the beginning, you run the risk that you will 
overlook an important variable.) 

So, for the total population, I will use "P". For the populations of 
Ohio, California, and Indiana, I will use "H", "C", and "I" 
respectively. (Remember that I don't use the letter "O".) What else do 
I know from this sentence? I know that P = 38,500,000. Hmm, I also 
know that the sum of the population of the three states is equal to P, 
so I get my first equation:

   H + C + I = P  

If I were doing the problem, I would actually write it all like this:

   Total Population          =  P ; P = 38,500,000
   Population of Ohio        =  H
   Population of California  =  C
   Population of Indiana     =  I

   H + C + I = P                  (1)

The number in parenthesis is just an equation number so that I can 
refer back to the equation easily if I have to. Numbering or lettering 
equations is a good habit for your daughter to get into.

Now let's look at the next chunk, "We know that Ohio has two times the 
population of Indiana." Hmmm. Maybe I can write an equation from this. 
Hey, I can write:

   H = 2I                         (2)

Then look at the next chunk, "We also know that Ohio has one-half times 
the population of California." Hmm, looks like another equation that I 
can write. I can actually write it in several ways ("*" means multiply 
and "/" means divide.):

   H = 1/2 * C  or 
   H = C/2      or 
   H = 0.5 * C

(If you are not quite sure why these are all equivalent, write back.) 
It doesn't really matter which one of these equations you use, but I 
kind of like whole numbers, so I'm going to use:

   H = C/2                        (3)

What about the next chunk? Well, "What is the population for each 
state?" tells me that I have to figure out what H, C, and I are. I'm 
going to put everything in one place now so that it is easier to see. 
To review:

   Total Population          =  P ; P = 38,500,000
   Population of Ohio        =  H
   Population of California  =  C
   Population of Indiana     =  I

   H + C + I = P                  (1)
   H = 2I                         (2)
   H = C/2                        (3)

Now that we have all of our information, we can start solving the 
problem. If we solve for H, we should be able to solve for I and C. 
So, I'm going to rewrite equation (2) as:

   I = H/2                        (4)

I will also rewrite equation 3 as:

   C = 2H                         (5)

If you don't know how I was able to get equations (4) and (5) from (2) 
and (3), write back.

I now put (4) and (5) into equation (1) to get:

   H + H/2 + 2H = P               (6)

Adding all H together, we get:

       3H + H/2 = P  

(Rats! H/2 isn't neat. Maybe I should have used another form. What 
about 0.5H? Yeah, I think that will do, so combining H, we get:)

           3.5H = P               (7)

Notice that in equation (4) I had used H/2 to represent the fact that 
Indiana has half the population of Ohio. I could have used I=0.5H or 
I=(1/2)H, but decided to use H/2. However, when I started to do H terms  
in equation (6), I had to think about one of the other ways to write 
H/2 so that I could add the terms.

From equation (7), I can divide both sides of the equation by 3.5 to 
get:

              H = P/(3.5)         (8)

If you have questions about why we need to divide both sides of the 
equation by 3.5, write back.

Now that we know what H is, even if it is in terms of P, we can solve 
equations (4) and (5). Putting H from equation (8) into equation (4), 
we get:

   I = (P/(3.5))/2 = (P/(2 x 3.5)) ==>

   I = P/7                        (9)

The "==>" is just a way to show that the second equation follows from 
the first. Putting H from Equation (8)into equation (5) we get:

   C = 2 x (P/3.5) = 2P/3.5 ==>
 
   C = 4P/7                       (10)

You may be wondering how I went from 2P/3.5 to 4P/7. Well, I know that 
3.5 is the same value as the improper fraction 7/2. (Recall that 0.5 
is 1/2, 1 is 2/2, 1.5 is 3/2, 2 is 4/2, and so on.) Also P/(7/2) is 
the same as 2P/7. (If you don't know why this is, look in the Dr. Math 
Archives for information on dividing by fractions.) If I then multiply 
2P/7 by 2, I get 4P/7.

Now, we have all of the unknowns in terms of P. If put the value for P 
into equations (8), (9), and (10), we get:

   H = 11,000,000
   I =  5,500,000
   C = 22,000,000

We are not finished yet. We still have to check to make sure that we 
did not make a mistake in doing the arithmetic, so we put the values 
for H, I, and C into equation (1) and see if they sum to equal P: 

   11,000,000 + 5,500,000 + 22,000,000 = 38,500,000  (11)

The equation checks, so we know we didn't make an arithmetic mistake. 

We could solve this problem several different ways, but I've shown you 
one method using algebra. I would suggest that your daughter use a 
consistent method of working out these problems, and that she use 
plenty of paper. Many students for some reason feel compelled to take 
up as little space as possible when solving problems, and their 
writing ends up so cramped and compact that they can't follow their 
logic. They end up making many mistakes because they can't follow their 
own logic. One way to present the answer is to use the following format 
(I have changed the equation numbers):

====================================================

   POPULATION PROBLEM

   Total Population          =  P ; P = 38,500,000
   Population of Ohio        =  H
   Population of California  =  C
   Population of Indiana     =  I

   H + C + I = P                  (1)
   H = 2I  ==> I = H/2            (2)
   H = C/2 ==> C = 2H             (3)

Substitute (2) and (3) into (1):

   H + H/2 + 2H = P  ==>
   H = P/(3.5)                    (4)

Substitute H from (4) into (2):

   I = (P/(3.5))/2 = (P/(2 x 3.5)) ==>

   I = P/7                         (5)

Substitute H from (4) into (3):

   C = 2 x (P/3.5) = 2P/3.5 ==>

   C = 4P/7                         (6)

Substitue P into (4), (5), and (6):

   H = 11,000,000
   I =  5,500,000
   C = 22,000,000
    
CHECK

Substitute H, I, and C into (1):

   11,000,000 + 5,500,000 + 22,000,000 = 38,500,000  (7)

The answer for (7) equals P. The answers check.

======================================================

I know this looks like a lot of writing, but your daughter will find 
out that her problems are so much easier to follow, and she won't get 
lost trying to solve them.

I hope that this helped you and your daughter. If you have any more 
questions, write again.

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

Date: 10/17/98 at 08:30:32
From: Anonymous
Subject: Re: Algebra word problems

Dr. Math,

Thank you for you quick response. I deeply appreciate it.
 
Kathy