Intermediate Algebra
Tutorial 8:
Introduction to Problem
Solving
 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. |
|
 Example
1: Twice the difference of a number and 1 is 4 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
|
 |
*Remove ( ) by using dist. prop.
*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.
|
| Example
2: One number is 3 less than another number. If
the sum of the two numbers is 177, 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 3 less than another number:
x - 3 = one number
|
 |
*Combine like terms
*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.
|
| Example
3: Find 45% of 125. |
| 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
|
 |
*Multiply |
56.25 is 45% of 125.
FINAL ANSWER:
The number is 56.25.
|
| Example
4: A math class has 30 students. Approximately
70%
passed their last math test. How many students passed the last
math
test? |
| 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
|
 |
*Multiply |
21 is 70% of 30.
FINAL ANSWER:
21 students passed the last math test.
|
| Example
5: I purchased a new tv at a local electronics store
for
$541.25, which included tax. If the tax rate is 8.25%, find the
price
of the tv before they added the tax. |
| 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.
|
 |
*Combine like terms
*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)
|
| Example
6: In a blueprint of a rectangular room, the length is
1 inch more than 3 times the width. Find the dimensions if the
perimeter
is to be 26 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 1 inch more than 3 times the width:
1 + 3w = length
|
 |
*Remove ( ) by using dist. prop.
*Combine like terms
*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.
|
|
Supplementary and Complementary
angles
|
| Supplementary angles sum up to be 180 degrees.
Complimentary angles sum up to be 90 degrees.
|
Example
7: Find the measure of each angle in the figure
below.
Note that since the angles make up a straight line, they are
supplementary
to each other.
|
| Make sure that you read the question carefully several
times.
We are already given in the figure that
x = 1 angle
5x = other angle
|
 |
*Combine like terms
*Inv. of mult. by 6 is div. by 6
|
| If x is 30, then 5x = 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.
|
Consecutive integers are integers that follow
one another in
order.
| 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.
|
|
| Example
8: The sum of 3 consecutive integers is 258.
Find
the integers. |
| 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
|
 |
*Combine like terms
*Inv. of add 3 is sub. 3
*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.
|
| Example
9: The ages of 3 sisters are 3 consecutive even
integers.
If the sum of twice the 1st even integer, 3 times the 2nd even integer,
and the 3rd even integer is 34, find each age. |
| 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
|
 |
*Remove ( ) by using dist. prop.
*Combine like terms
*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.
|
|
Business Problem: Breaking
Even
|
| 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.
|
| Example
10: The cost C to
produce
x
number of cd’s is C = 50 + 5x.
The cd’s are sold wholesale for $15 each, so revenue R
is given by R = 15x.
Find how many cd’s the manufacturer needs to produce and sell to break
even. |
| 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
x = the number of
cd’s
|
 |
*Get all x
terms
on one side
*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.
|
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 - 1g:
Solve the word problem.
|
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) |
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) |
| 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.
(answer/discussion
to 1e)
|
1g. The cost C to
produce x
numbers of VCR’s is C = 1000 + 100x.
The VCR’s are sold wholesale for $150 each, so the revenue is given by
R
= 150x. Find how many VCR’s the
manufacturer
needs to produce and sell to break even.
(answer/discussion
to 1g) |
Need Extra Help on These Topics?
|
All contents copyright (C) 2001 - 2008, WTAMU and Kim Seward. All rights reserved.
Last revised on June 22, 2003 by Kim Seward. |
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to the THEA Math Help Page)
