Learning Objectives
Introduction
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
In that situation, you want to
Example
1: Write the phrase as an algebraic
expression:
The sum of -5 and a number.
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:
Example
2: Write the phrase as an algebraic
expression:
The product of 20 and a number.
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:
Example
3: Write the phrase as an algebraic
expression:
7 less than twice a number.
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:
Example
4: Write the phrase as an algebraic expression:
The quotient of 9 and the difference of 8 and a number.
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:
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).
Numeric Word Problems
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.
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
FINAL ANSWER:
Example
6: One number is 5 less than another number. If
the sum of the two numbers is 205, 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
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
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.
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
FINAL ANSWER:
The number of touchdowns the running back made was 21
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 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.
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
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.
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
FINAL ANSWER:
The ages of the three sisters are 9, 11 and 13.
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.
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
FINAL ANSWER:
Length is 18 inches.
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.
We are looking for a number that is 73% of 225, we will let
x = the value we are looking for
FINAL ANSWER:
Example
12: A math class has 45 students. Approximately
60% are females. How many students are females?
We are looking for how many students are females, we will let
x = number of female students
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.
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
FINAL ANSWER:
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.
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 = 2L + 2W for W.
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
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.
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)
Need Extra Help on these Topics?
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.
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