**Learning Objectives**

After completing this tutorial, you should be able to:

- Use Polya's four step process to solve word problems involving numbers, percents, rectangles, supplementary angles, complementary angles, consecutive integers, and breaking even.

** Introduction**

Whether you like it or not, whether you are going to be
a mother, father,
teacher, computer programmer, scientist, researcher, business owner,
coach,
mathematician, manager, doctor, lawyer, banker (the list can go on and
on), ** **problem solving is everywhere. Some people
think
that you either can do it or you can't. Contrary to that belief,
it can be a learned trade. Even the best athletes and musicians
had
some coaching along the way and lots of practice. That's what it
also takes to be good at problem solving.

**George
Polya**,
known as the father of modern problem solving, did extensive studies
and
wrote numerous mathematical papers and three books about problem
solving.
I'm going to show you his method of problem solving to help step you
through
these problems.

** Tutorial**

As mentioned above, I use Polya’s four steps to problem
solving to
show students how to solve word problems. **Just
note
that your math teacher or math book may word it a little differently,
but
you will see it all basically means the same thing.**

If you follow these steps, it will help you become more
successful in
the world of problem solving.

Polya created his famous **four-step process for
problem solving,** which is used all over to aid people in problem solving:

**Step 1: Understand the problem.**

Sometimes the problem lies in **understanding
the problem**.
If you are unclear as to what needs to be solved, then you are probably
going to get the wrong results. In order to show an understanding
of the problem, you, of course, need to read the problem
carefully.
Sounds simple enough, but some people jump the gun and try to start
solving
the problem before they have read the whole problem. Once the
problem
is read, you need to list all the components and data that are
involved.
This is where you will be assigning your variable.

**Step 2: Devise a plan (translate).**

When you **devise a plan (translate)**, you
come up with a way to
solve the problem. Setting up an equation, drawing a diagram, and
making a chart are all ways that you can go about solving your
problem.
In this tutorial, we will be setting up equations for each
problem.
You will translate them just like we did in **Tutorial
2: Algebraic Expressions** and **Tutorial
5: Properties of Real Numbers.**

**Step 3: Carry out the plan (solve).**

The next step, **carry out the plan (solve)**,
is big. This is where
you solve the equation you came up with in your 'devise a plan'
step.
The equations in this tutorial will all be linear equations. If
you
need help solving them, by all means, go back to **Tutorial
7: Linear Equations in One Variable** and review that concept.

**Step 4: Look back (check
and interpret).**

You may be familiar with the expression 'don't
look back'. In
problem solving it is good to **look back (check and interpret).**.
Basically, check to see if you used all your information and that the
answer
makes sense. If your answer does check out, make sure that you
write
your final answer with the correct labeling.

A lot of numeric types of word problems revolve around
translating
English statements into mathematical ones. If you need a review
on
these translations, you can go back to **Tutorial
2: Algebraic Expressions** and/or **Tutorial
5: Properties of Real Numbers**.

**Just read and translate it left to right to set up
your equation**

Make sure that you read the question carefully several
times.

Since we are looking for a number, we will let

*x* = a number

***Get all the x terms on one side**

***Inv. of sub. 2 is add 2**

If you take twice the difference of 6 and 1, that is
the same as 4
more than 6, so this does check.

**FINAL ANSWER: ** **The number is 6.**

Make sure that you read the question carefully several
times.

We are looking for two numbers, and since we can write the one number in terms of another number, we will let

*x* = another number

ne number is 3 less than another number:

*x* - 3 = one number

***Inv. of sub 3 is add 3**

***Inv. of mult. 2 is div. 2**

If we add 90 and 87 (a number 3 less than 90) we do get
177.

**FINAL ANSWER: ** **One number is 90.**
**Another number is 87.**

Whenever you are working with a percent problem, you
need to make sure
you write your percent in decimal form. You do this by moving the
decimal place of the percent two to the left. For example, 32% in
decimal form is .32

When you are wanting to find the percentage of some
number, **remember
that ‘of ’ represents multiplication** - so you would multiply the
percent
(in decimal form) times the number you are taking the percent of.

Make sure that you read the question carefully several
times.

We are looking for a number that is 45% of 125, we will let

*x* = the value we are
looking for

56.25 is 45% of 125.

**FINAL ANSWER: ** **The number is 56.25.**

Make sure that you read the question carefully several
times.

We are looking for how many students passed the last math test, we will let

*x* = number of
students

21 is 70% of 30.

**FINAL ANSWER:** **21 students passed the last math test.**

Make sure that you read the question carefully several
times.

We are looking for the price of the tv before they added the tax, we will let

*x* = price of the tv
before tax was added.

***Inv of mult. 1.0825 is div. by
1.0825**

If you add on 8.25% tax to 500, you would get 541.25.

**FINAL ANSWER:** **The original price is $500.**

The following formula will come in handy for solving
example 6:

**Perimeter of a Rectangle = 2(length) + 2(width)**

Make sure that you read the question carefully several
times.

We are looking for the length and width of the rectangle. Since length can be written in terms of width, we will let

*w* = width

length is 1 inch more than 3 times the width:

**1 + 3 w = length**

***Inv. of add. 2 is sub. 2**

***Inv. of mult. by 8 is div. by 8**

If width is 3, then length, which is 1 inch more than 3
times the width
would have to be 10. The perimeter of a rectangle with width of 3
inches and length of 10 inches does come out to be 26.

**FINAL ANSWER:** **Width is 3 inches.** **Length is 10 inches.**

**Complimentary angles sum up to be 90 degrees.**

Make sure that you read the question carefully several
times.

We are already given in the figure that

*x* = one angle

**5 x = other angle**

***Inv. of mult. by 6 is div. by 6**

If *x* is 30, then 5*x* = 5(30) = 150. 150 and 30 do
add up to be
180, so they are supplementary angles.

**FINAL ANSWER:** **The two angles are 30 degrees and 150 degrees.**

For example, 5, 6, and 7 are three
consecutive integers.

If we let *x* represent the first integer,
how would we represent the second consecutive integer in terms of* x*? Well if we look at 5, 6, and 7 - note that 6 is one
more than 5, the first integer.

In general, **we could represent the second
consecutive integer by x + 1**. And what about the third consecutive integer.

Well, note how 7 is 2 more than 5. In
general, **we could represent
the third consecutive integer as x + 2.**

**Consecutive EVEN integers **are even integers that
follow one another
in order.

For example, 4, 6, and 8 are three consecutive
even integers.

If we let *x* represent the first EVEN integer,
how would we represent the second consecutive even integer in terms of *x*?
Note that 6 is two more than 4, the first even integer.

In general, **we could represent the second
consecutive EVEN integer
by x + 2**.

And what about the third consecutive even
integer? Well, note
how 8 is 4 more than 4. In general, **we could represent the
third
consecutive EVEN integer as x + 4.**

**Consecutive ODD integers **are odd integers that
follow one another
in order.

For example, 5, 7, and 9 are three consecutive
odd integers.

If we let *x* represent the first ODD integer,
how would we represent the second consecutive odd integer in terms of *x*?
Note that 7 is two more than 5, the first odd integer.

In general, **we could represent the second
consecutive ODD integer
by x + 2.**

And what about the third consecutive odd
integer? Well, note how
9 is 4 more than 5. In general, **we could represent the third
consecutive
ODD integer as x + 4.**

Note that a common misconception is that because
we want an odd number
that we should not be adding a 2 which is an even number. Keep in
mind that *x* is representing an ODD
number and
that the next odd number is 2 away, just like 7 is 2 away form 5, so we
need to add 2 to the first odd number to get to the second consecutive
odd number.

Make sure that you read the question carefully several
times.

We are looking for 3 consecutive integers, we will let

*x* = 1st consecutive integer

*x* + 1 = 2nd consecutive integer

*x* + 2 = 3rd consecutive integer

***Inv. of mult. by 3 is div. by 3**

The sum of 85, 86 and 87 does check to be 258.

**FINAL ANSWER:** **The three consecutive integers are 85, 86, and 87.**

Make sure that you read the question carefully several
times.

We are looking for 3 EVEN consecutive integers, we will let

*x* = 1st consecutive even integer

*x* + 2 = 2nd consecutive even integer

*x* + 4 = 3rd consecutive even integer

***Inv. of add. 10 is sub. 10**

***Inv. of mult. by 6 is div. by 6**

If we take the sum of two times 4, three times 6, and
8, we do get
34

**FINAL ANSWER: The ages of the three sisters are 4, 6, and 8.**

In a business related problem, the cost equation, *C* is the cost of manufacturing a product.

In the revenue equation, *R* is the amount
of money the manufacturer makes on a product.

**If a manufacturer wants to know how many items must
be sold to break
even, that can be found by setting the cost equal to the revenue.**

Make sure that you read the question carefully several
times.

We are looking for the number of cd’s needed to be sold to break even, we will let

***Inv. of mult. by 10 is div. by
10**

When *x* is 5 the cost and
the revenue both
equal 75.

**FINAL ANSWER:** **5 cd’s.**

These are practice problems to help bring you to the
next level.
It will allow you to check and see if you have an understanding of
these
types of problems. **Math works just like
anything
else, if you want to get good at it, then you need to practice
it.
Even the best athletes and musicians had help along the way and lots of
practice, practice, practice, to get good at their sport or instrument.**
In fact there is no such thing as too much practice.

To get the most out of these, **you should work the
problem out on
your own and then check your answer by clicking on the link for the
answer/discussion
for that problem**. At the link you will find the answer
as well as any steps that went into finding that answer.

Practice Problems 1a - 1g:Solve the word problem.

1a. The sum of a number and 2 is 6 less than
twice that number.

(answer/discussion to 1a)

(answer/discussion to 1a)

1b. Find 72% of 35.

(answer/discussion to 1b)

(answer/discussion to 1b)

1c. A local furniture store is having a terrific
sale.
They are marking down every price 45%. If the couch you have our
eye on is $440 after the markdown, what was the original
price?
How much would you save if you bought it at this sale?

(answer/discussion to 1c)

(answer/discussion to 1c)

1d. A rectangular garden has a width that is 8
feet less than
twice the length. Find the dimensions if the perimeter is 20
feet.

(answer/discussion to 1d)

(answer/discussion to 1d)

1e. Complimentary angles sum up to be 90
degrees. Find
the measure of each angle in the figure below. Note that since
the
angles make up a right angle, they are complementary to each other.

1f. The sum of 3 consecutive odd integers is
57. Find the
integers.

(answer/discussion to 1f)

(answer/discussion to 1f)

1g. The cost *C* to
produce *x* numbers of VCR’s is* C* = 1000 + 100*x*.
The VCR’s are sold wholesale for $150 each, so the revenue is given by *R* = 150*x*. Find how many VCR’s the
manufacturer
needs to produce and sell to break even.

(answer/discussion to 1g)

(answer/discussion to 1g)

** Need Extra Help on these Topics?**

**http://www.purplemath.com/modules/translat.htm**

This webpage gives you the basics of problem solving and helps you
with translating English into math.

**http://www.purplemath.com/modules/numbprob.htm**

This webpage helps you with numeric and consecutive integer problems.

**http://www.purplemath.com/modules/percntof.htm**

This webpage helps you with percent problems.

**http://www.math.com/school/subject2/lessons/S2U1L3DP.html**

This website helps you with the basics of writing equations.

**http://www.purplemath.com/modules/ageprobs.htm**

This webpage goes through examples of age problems, which are
like the numeric problems found on this page.

**Go to Get
Help Outside the
Classroom found in Tutorial 1: How to Succeed in a Math Class for
some
more suggestions.**

Last revised on July 1, 2011 by Kim Seward.

All contents copyright (C) 2002 - 2011, WTAMU and Kim Seward. All rights reserved.