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Virtual Math Lab

Intermediate Algebra
Tutorial 8: Introduction to Problem Solving
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deskLearning Objectives

 
After completing this tutorial, you should be able to:
  1. Use Polya's four step process to solve word problems involving numbers, percents, rectangles, supplementary angles, complementary angles, consecutive integers, and breaking even. 




desk 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.

 

 

desk 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.

 
 
Numeric Word Problems

 
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

 
notebook 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

 

 
example 1a

 

 
example 1b

*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.


 
 
 
notebook 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 

 

ne number is 3 less than another number:

x - 3 = one number

 

 
example 2a

 

 
example 2b

*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.

 
 
 
Percent Problems

 
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.

 
 
notebook 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

 

 
example 3a

 

 
example 3b

*Multiply

 

 
56.25 is 45% of 125.
 

FINAL ANSWER:  The number is 56.25.


 
 
notebook 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 

 

 
example 4a

 

 
example 4b

*Multiply

 

 
21 is 70% of 30.
 

FINAL ANSWER: 21 students passed the last math test.


 
 
 
notebook 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. 

 

 
example 5a

 

 
example 5b

*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.


 
 
 
 
Rectangle Problem

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

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

 
 
notebook 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

 

 
example 6a

 

 
example 6b

*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.

 
notebook 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.
example 7c

 

 
Make sure that you read the question carefully several times. 

We are already given in the figure that

x = one angle

5x = other angle

 

 
example 7a

 

 
example 7b

*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

 
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.

 
 
notebook 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

 

 
example 8a

 

 
example 8b

*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.

 
 
 
 
notebook 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

 

 
example 9a2

 

 
example 9b

*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.

 
 
notebook 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

 

 
example 10a

 

 
example 10b

*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.


 
 
desk 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.

 

pencil 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)

 
1b.  Find 72% of 35.
(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)

 
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.

problem 1

(answer/discussion to 1e)

 
1f.  The sum of 3 consecutive odd integers is 57.  Find the integers.
(answer/discussion to 1f)

 
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)

 


desk Need Extra Help on these Topics?


 
The following are webpages that can assist you in the topics that were covered on this page:
 

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.