Understanding word problems
is probably the most important part of mathematics.
Mathematics without applications (‘pure’ mathematics)
is a beautiful work of art, part of our global culture,
and good practice for precise thinking.
But only applied mathematics is directly useful.
Almost everything in Algebra can be taught to a computer,
and while it's good to know how computer algebra systems work,
in the end it's more reliable to use them to solve problems
than to solve them yourself,
just as your calculator is more likely to do 276,435/6,250 correctly
than you are,
no matter how good you are at arithmetic.
But word problems cannot be taught to a computer,
and without word problems, Algebra is ultimately useless.
The reason that we can't teach word problems to a computer
is that there is no way to make the principles behind them fully precise.
This means that I also have no way to teach them to you perfectly.
However, I can give you some advice:
- First make sure that you understand what's going on in the problem
without worrying about the mathematical parts.
- Next make a list of
all of the numerical quantities, known and unknown, in the problem.
You don't necessarily have to write this down,
but at least think it through in your head.
(Instead, you might want to organize these with a diagram or a table.)
- Now think about
how you could calculate the unknown quantities
given the information in the problem.
Your goal is to identify a single quantity x
such that, if you knew what x is,
then you would know how to calculate everything else.
- When you identify this quantity, make up a name for it.
(You can call it x like I did a moment ago,
or you can call it something that reminds you more of what it means.)
You should write this down;
don't let yourself forget what this variable stands for!
Write ‘Let x be […].’
and fill this in with a precise description of a number.
(For example, ‘Let a be Alice's annual income in dollars.’,
not ‘a = Alice’.)
- Go through every other quantity in your list (or diagram or table)
and write down an algebraic expression for it involving your variable.
Since you would know how to calculate these quantities
if you knew the value of your variable,
your job here is simply
to write down expressions that tell how to do these calculations.
- If there's enough information in the problem to solve it,
then there will be some quantity
that you can write two different expressions for.
(The specific type of problem that you're solving
might have a general formula that you can use here.)
Write down these expressions with an equals sign between them.
There's your equation!
- Solve this equation,
and use the solution to calculate the value of all of the quantities.
- Read through the problem again,
filling in the values that you calculated for all of the quantities,
and make sure that the story makes sense.
- Be sure to answer the questions that the problem actually asked,
with appropriate units, in words.
If you just cannot find a single quantity x
that every other quantity can be written with,
then you may need to use more than one variable,
in which case you also need to find more than one equation.
This is the topic of Chapter 4, which we'll get to later on.
In general, if you can find as many equations as variables,
then you can make it work.
If you just cannot find a quantity with two different expressions,
or more generally if you can't find as many equations as you have variables,
then there might not be enough information to solve the problem.
If you can find at least two different ways
to simultaneously assign values to all of the quantities
that makes the entire story make sense,
then you know that this is the case!
But there shouldn't be any trick questions like that
in the assignments in this class.
Go back to the the course homepage.
This web page was written between 2007 and 2016 by Toby Bartels,
last edited on 2016 August 15.
Toby reserves no legal rights to it.
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