**Learning Objectives**

After completing this tutorial, you should be able to:

- Translate an English phrase into an algebraic expression.
- Use Polya's four step process to solve word problems.
- Solve a formula for a given variable.

** 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 also
what
it 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**

Sometimes, you find yourself having to write out your
own algebraic
expression based on the wording of a problem.

In that situation, you want to

- read the problem carefully,
- pick out key words and phrases and determine their equivalent mathematical meaning,
- replace any unknowns with a variable, and
- put it all together in an algebraic expression.

** Example
1: ** Write the phrase as an algebraic
expression:
The sum of -5 and a number.

In this example, we are not evaluating an expression,
but we are wanting
to rewrite it as an algebraic expression.

It looks like **the only reference to a mathematical
operation is the
word sum,** so what operation will we have in this
expression?
If you said **addition,** you are correct!!!

The phrase 'a number' indicates that it is an unknown
number - there
was no specific value given to it. So we will replace the
phrase
'a number' with the variable *x*. We
want
to let our variable represent any number that is unknown

**Putting everything together, we can translate the
given English phrase
with the following algebraic expression:**

The sum of -5 and a number

Let’s try another!

** Example
2:** Write the phrase as an algebraic
expression:
The product of 20 and a number.

Again, we are wanting to rewrite this as an algebraic
expression, not
evaluate it.

This time, **the phrase that correlates with our
operation is 'product'** - so what operation will we be doing this time? If you said **multiplication**,
you are right on.

Again, we have the phrase 'a number', which, again, is
going to be replaced
with a variable since we do not know what the number is.

**Let’s see what we get for this answer:**

The product of 20 and a number

** Example
3:** Write the phrase as an algebraic
expression:
7 less than twice a number.

First of all, we have** the phrase 'less than' which
mathematically
translates as subtraction.** You need to be careful with this
phrase.
It is very tempting to start off with 7 and put your subtraction sign
after
the 7. However, think about it, if you want 7 less than
something,
you are 7 below it. In order to be 7 below something you would have to
subtract the 7!!! So you would not have 7 minus but minus 7 as
PART
of your expression.

The other part of the expression involves the phrase
'twice a number'.
'Twice' translates as two times a number and, as above, we will replace
the phrase 'a number' with our variable *x.*

**Putting this together we get:**

7 less than twice a number

** Example
4:** Write the phrase as an algebraic expression:
The quotient of 9 and the difference of 8 and a number.

First of all, the** term 'quotient'** is going to be
replaced with
what mathematical operation? If you said **division,** you
are
right on the mark!!

Note how 9 immediately follows the phrase 'the quotient of', this means that 9 is going to be in the numerator. The phrase that immediately follows the word 'quotient' is going to be in the numerator of the fraction.

After the word ‘and', you have the phrase 'the
difference of 8 and a
number'. That is the second part of your quotient which means it
will go in the denominator. And what operation will we have when
we do write that **difference **down below? I hope you said **subtraction**.

**Let’s see what we get when we put all of this
together:**

The quotient of 9 and the difference of 8 and a number.

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 above.

**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
14: 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.

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

** Example
5**: Three times the difference of a number and 4 is 8
more
than that number. Find the number.

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. 12 is add. 12**

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

If you take three times the difference of 10 and 4,
that is the same
as 8 more than 10, so this does check.

**FINAL ANSWER: **

** Example
6**: One number is 5 less than another number. If
the sum of the two numbers is 205, find each number.

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

one number is 5 less than another number:

*x* - 5 = one number

***Inv. of sub. 5 is add. 5**

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

If we add 105 and 100 (a number 5 less than 105) we do
get 205.

**FINAL ANSWER: **

**Another number is 105.**

** Example
7**: Last year, the star running back of the local
football
team made 6 more touchdowns than the star receiver. Together,
they
had 36 touchdowns. Determine the number of touchdowns for each
player
last year.

Make sure that you read the question carefully several
times.

We are looking for two numbers, and since we can write the number of touchdowns the running back made in terms of the number of touchdowns the receiver made we will let

*x* = the number of
touchdowns the receiver
made

The running back made 6 more touchdowns than the receiver:

*x* + 6 = number of
touchdowns the running
back made

***Inv. of add. 6 is sub. 6**

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

If we add 15 and 21 (a number 6 more than 15) we do get
36.

**FINAL ANSWER:**

**The number of touchdowns the running back made was 21**

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 from 5, so we
need to add 2 to the first odd number to get to the second consecutive
odd number.

** Example
8:** The sum of 4 consecutive integers is 406.
Find
the integers.

Make sure that you read the question carefully several
times.

We are looking for 4 **consecutive
integers**,
we will let

*x* = 1st consecutive
integer

*x *+ 1 = 2nd
consecutive integer

*x* + 2 = 3rd
consecutive integer

*x *+ 3 = 4th
consecutive integer

***Inv. of mult. by 4 is div. by 4**

The sum of 100, 101, 102, and 103 does check to be 406.

**FINAL ANSWER:**

**The three consecutive integers are 100, 101, 102, and
103.**

** Example
9:** The ages of 3 sisters are 3 consecutive odd
integers.
If the sum of the 1st odd integer, 3 times the 2nd odd integer, and
twice
the 3rd odd integer is 68, find each age.

Make sure that you read the question carefully several
times.

We are looking for 3 **ODD consecutive
integers**,
we will let

*x* = 1st consecutive odd
integer

*x* + 2 = 2nd consecutive
odd integer

*x* + 4 = 3rd
consecutive odd
integer

***Inv. of add. 14 is sub. 14**

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

If we take the sum of 9, three times 11, and twice 13,
we do get 68.

**FINAL ANSWER:**

**The ages of the three sisters are 9, 11 and 13.**

The following formula will come in handy for solving
example 10:

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

** Example
10**: In a blueprint of a rectangular room, the length
is
2 inches less than 4 times the width. Find the dimensions if the
perimeter is 46 inches.

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 2 inches less than 4 times the width:

*4w* - 2* = *length

***Inv. of sub. 4 is add. 4**

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

If the width is 5, then the length, which is 2 inches
less than 4 times
the width, would have to be 18. The perimeter of a rectangle with
width of 5 inches and length of 18 inches does come out to be 46
inches.

**FINAL ANSWER:**

**Length is 18 inches.**

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.

** Example
11**: Find 73% of 225.

Make sure that you read the question carefully several
times.

We are looking for a number that is 73% of 225, we will let

*x* = the value we are
looking for

164.25 is 73% of 225.

**FINAL ANSWER: **

** Example
12**: A math class has 45 students. Approximately
60% are females. How many students are females?

Make sure that you read the question carefully several
times.

We are looking for how many students are females, we will let

*x* = number of female
students

27 is 60% of 45.

**FINAL ANSWER:**

** Example
13**: You purchased a new computer monitor at a local electronics
store
for $216.50, which included tax. If the tax rate is 8.25%, find
the
price of the monitor before they added the tax.

Make sure that you read the question carefully several
times.

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

*x* = price of the computer monitor
before tax was added.

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

If you add on 8.25% tax to 200, you would get 216.50.

**FINAL ANSWER:**

A formula is an equation that involves two or more
variables that have
a specific relationship with each other.

Even though there is more than one variable in a
formula, you solve
for a specific variable using **the
exact same steps that you do with an equation in one variable, as shown
in Tutorial 14: Linear Equations in One Variable.**

It is really easy to get overwhelmed when there is more
than one variable
involved. Sometimes your head feels like it is spinning when you
see all of those variables. Isn’t math suppose to be about
numbers?
Well, **just remember that a variable
represents
a number. So if you need to move it to the other side of the
equation
you use inverse operations, just like you would do with a number.**

** Example
14**: Solve the equation for* l*.

Do you recognize this formula?

This happens to be the formula for the volume of a rectangular solid, where*V* = volume, *l *=
length, *w* = width, and *h* = height.

This happens to be the formula for the volume of a rectangular solid, where

**In this problem, we need to solve for l.
This means we need to get l on one side
and
EVERYTHING ELSE on the other side using inverse operations.**

Let’s solve this formula for *l*:

***Formula solved for l**

** Example
15**: Solve the equation *P* = 2*L* + 2*W* for *W*.

Do you recognize this formula?

This happens to be the formula for the perimeter of a rectangle, where*P* = perimeter, *L* = length, and *W* = width.

This happens to be the formula for the perimeter of a rectangle, where

**In this problem, we need to solve for W.
This means we need to get W on one side
and
EVERYTHING ELSE on the other side using inverse operations.**

Let’s solve this formula for *W*:

***Inverse of mult. by 2 is div.
by 2**

***Formula solved for W**

** Practice Problems**

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 - 1d:Solve the given word problem.

1a. In last night’s basketball game, Sally scored
9 less than
twice what Lucy scored. The sum of their scores is 27. How
many points did Sally and Lucy make individually?

(answer/discussion to 1a)

(answer/discussion to 1a)

1b. The heights in inches of three basketball
players are 3 consecutive
integers. If the sum of twice the 1st, 3 times the 2nd, and the
3rd
is 437, what are the three heights.

(answer/discussion
to 1b)

1c. A rectangular floor has a perimeter of 54
feet. If the
length is 3 more than the width, what are the dimensions of the floor?

(answer/discussion
to 1c)

1d. The original price of a CD player was marked
down 15%
and is now $127.50, how much was the original price?

(answer/discussion
to 1d)

Practice Problems 2a - 2b:Solve for the given variable.

** Need Extra Help on these Topics?**

This website helps you with basic problem solving.

**http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/****int_alg_tut9_form.htm**

This webpage helps you with solving a formula for a given variable.

**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.purplemath.com/modules/ageprobs.htm**

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

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

This webpage helps you with solving formulas for a specified variable.

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

Videos at this site were created and produced by Kim Seward and Virginia Williams Trice.

Last revised on Dec. 15, 2009 by Kim Seward.

All contents copyright (C) 2002 - 2010, WTAMU and Kim Seward. All rights reserved.