Learning Objectives
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
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.
Step 2: Devise a plan (translate).
Step 3: Carry out the plan (solve).
Step 4: Look back (check
and interpret).
Just read and translate it left to right to set up your equation
Example
1: Twice the difference of a number and 1 is 4 more
than
that number. Find the number.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
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.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
FINAL ANSWER: One number is 90. Another number is 87.
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. We are looking for a number that is 45% of 125, we will let
x = the value we are looking for
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?We are looking for how many students passed the last math test, we will let
x = number of students
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.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
FINAL ANSWER: The original price is $500.
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.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
*Inv. of add. 2 is sub. 2
*Inv. of mult. by 8 is div. by 8
FINAL ANSWER: Width is 3 inches. Length is 10 inches.
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. 
We are already given in the figure that
x = one angle
5x = other angle
*Inv. of mult. by 6 is div. by 6
FINAL ANSWER: The two angles are 30 degrees and 150 degrees.
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.
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.
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.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
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.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
FINAL ANSWER: The ages of the three sisters are 4, 6, and 8.
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.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
FINAL ANSWER: 5 cd’s.
Practice Problems
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.
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.