1.10
Introduction
to Word Problems
Part I
from Basic Algebra: One Step at
a Time © 2002
P.
63-83
Dr. Robert J. Rapalje
Seminole Community College
ANSWERS TO ALL EXERCISES ARE INCLUDED AT THE
END OF THIS PAGE
|
To see
selected solutions in
Living
C
O
L
O
R
click here! |
|
In
this section, the following "categories" of word problems will
be considered:
I. Number
problems
II.
Consecutive number problems
III.
Perimeter problems
IV. Coin
problems
V. Mixture
problems (Optional)
Before
describing these categories of word problems, it will be helpful to
identify five steps in setting up and solving word problems:
|
GUIDELINES
TO SOLVING WORD PROBLEMS
STEP 1:
IDENTIFY THE VARIABLE. State exactly what it is
that the variable represents.
For example, "Let x = the number” or "Let x =
the smaller of two numbers” or “Let x = the width of a
rectangle” or “Let x = the number of dimes”
Then express all other quantities to be used in the
problem in terms of x. This is the most important, often the most difficult, and
usually the most overlooked step of the problem.
STEP
2: WRITE
THE EQUATION. Having completed Step 1, use this step in writing the
equation. This is
often no more than translating a sentence of the problem into an
equation. Read the
problem carefully, paraphrasing as necessary.
STEP 3:
SOLVE THE EQUATION. This is usually the easy
part!
STEP 4:
ANSWER THE QUESTION. After solving for x,
there may be other quantities to be determined.
Be sure you have answered the question before going on to
the next exercise.
STEP 5:
CHECK.
Check the answers in
the worded problem itself and make sure the solution actually
works. Reject any extraneous (i.e., inappropriate) answers.
|
I.
NUMBER PROBLEMS.
Probably
the simplest type of application is the number problem.
If only one number involved, then begin by writing, “Let x = the
number.” If more than one
number is involved, then begin “Let x = one of the numbers,” “Let x
= first number,” “Let x = the smaller number,”
“Let x = the larger number,” or something like this.
How to decide what to let x equal will be explained later.
EXAMPLE
1.
Four
times a number plus 12 is equal to 40.
Find the number.
Solution:
Step
1 Let x= the number (since
there is only one number)
Step
2 Write the equation,
(translate the sentence into math!)
4x + 12 = 40
Step
3 Solve the equation, (as in
the previous section!).
4x + 12 = 40
- 12
-12
4x
= 28
x = 7
Step
4 Answer the question: The
number is 7.
Step
5 Check:
Substitute
7 back into the original wording of the problem to see if it works: 4(7) +
12 = 40.
EXAMPLE
2.
Six less than twice a number is ten more than the number.
Find the number.
Solution:
Step
1 “Let x = the number”
Step
2 Write the equation:
2x - 6 = x + 10
Step
3 Solve the equation: 2x - 6 = x
+ 10
-x
+ 6 -x
+ 6
x
=
16
Step
4 Answer the question: The
number is 16.
Step
5 Check
2(16) - 6 = 16 + 10
32
- 6 = 26 (This
checks!)
EXERCISES.
Write equations for and solve each of the following word problems.
1. Five times a number is 20
more than the number. Find
the number.
Step 1 “Let x = __________”
Step 2 Write the equation:
Step 3 Solve the equation:
Step 4 Answer the question:
Step 5 Check
2. Twice a number minus ten
is equal to 22 more than the number.
Step 1 “Let x = __________”
Step 2 Write the equation:
Step 3 Solve the equation:
Step 4 Answer the question:
Step 5
Check
3. Three times a number plus
8 is equal to five times the number.
Find the number.
Step 1 “Let x = ___________
Step 2 Write the equation:
Step 3 Solve the equation:
Step 4 Answer the question:
Step 5 Check
4. Six more than four
times a number is equal to 18 less than the number.
Find the number.
Step 1
Step 2
Step 3
Step 4
Step 5
5. Three less than four times a
number is equal to 27 more than the number.
Find the number.
6. Five more than four times a number is
equal to 35 less than twice the number.
Find the number.
7.
Five more than four times a number is equal to 35 less twice the
number. Find the number.
(NOTE: this looks like #6, but read the problem carefully!!)
8.
Seven times a number, minus 5, is equal to three times the number
plus 19. Find the number.
9.
Four less than three times a number is equal to 28 less than the
negative of the number. Find
the number.
10.
Twice a number less 10 is equal to the negative of the number plus
14. Find the number.
EXAMPLE
3.
The larger of two numbers is twice the smaller number.
Their sum is 243. Find
the numbers. [Hint: The first sentence tells about the larger number, but
it assumes you already know the smaller.
Therefore, let x = the smaller number.]
Solution:
Step 1
“Let x = the smaller number
2x = the larger
number
Step 2
Write the equation: x
+ 2x = 243
Step 3
Solve the equation: x
+ 2x = 243
3x
= 243
x =
81
Step 4
Answer the question: The smaller number is x = 81.
The
larger number is 2x = 162.
Step 5
Check (optional!) The
sum of the numbers (81 + 162) is 243.
(This checks!)
EXERCISES.
11.
Two numbers are such that the larger number is three times the
smaller number. The sum of
the numbers is 60. Find the
numbers.
Step 1
“Let x = the _______
number
_____= the
_______ number
Step 2
Write the equation:
Step 3
Solve the equation:
Step 4
Answer the question:
Step 5
Check (optional!)
12.
Two numbers are such that the larger is 6 more than twice the
smaller number. The sum of
the numbers is 60. Find the numbers.
13.
Two numbers are such that the larger is 6 less than twice the
smaller number. The sum of
the numbers is 60. Find the numbers.
14.
The smaller of two numbers is 12 less than the larger.
The sum of the numbers is 60. Find the numbers. [Hint: Let x = the larger number (see end of first sentence!)]
15.
The smaller of two numbers is 60 less than twice the larger.
The sum of the numbers is 60. Find the numbers.
16.
The smaller of two numbers is 30 less than three times the larger.
The sum of the numbers is 22. Find the numbers.
EXAMPLE
4.
Three numbers are such that the second is five more than the first,
and the third is four less than three times the first.
The sum of the numbers is 36.
Find the numbers.
Solution: Let
x = first number
x + 5 = second number
3x - 4 = third
number
Equation:
x + (x + 5) + (3x - 4) = 36
5x +
1 =
36
5x =
35
x
= 7
Answer
the question:
x =
7 first
number
x + 5
= 12 second
number
3x
- 4 = 17
third number
Check:
7 + 12 + 17 = 36
EXERCISES.
17.
Three numbers are such that the second number is 5 more than the
first, and the third number is 4 more than three times the second.
The sum of the three numbers is 134.
Find the numbers.
Solution:
Let
x = first number
_________ =
second number
_________ =
third number
Equation:
Answer
the question: x
= _____ first number
_____ = _____
second number
_____ = _____
third number
Check:
18.
Three numbers are such that the second number is 3 less than
the first, and the third is twice the second number.
The sum of the numbers is 91.
Find the numbers.
19.
Three numbers are such that the second number is 4 more than the
first, and the third number is equal to the sum of the first two numbers.
The sum of the three numbers is 256.
Find the numbers.
20.
Three numbers are such that the second number is 6 less than twice
the first, and the third number is five more than the sum of the first
two. The sum of the numbers
is 293. Find the numbers.
II.
CONSECUTIVE INTEGER PROBLEMS
Word
problems frequently refer to consecutive
numbers or consecutive integers.
As examples, 5, 6, 7, and 23, 24, 25 are two sets of
consecutive integers. Whatever
the first integer may be, you must add 1 to the first integer to get the
second integer, and you must add 2 to the first integer to get the third
integer. In general, if x
represents the first integer, then the second integer is x
+ 1, and the third integer is x
+ 2.
Examples
of consecutive odd integers are
7, 9, 11 or 29, 31, 33. Notice
that beginning with the first integer, you must add 2 to obtain the second
integer, and you must add 4 to the first integer to obtain the third
integer. In general, if x represents the first integer, then the second integer is x
+ 2, and the third integer is x
+ 4.
Examples
of consecutive even integers
are 6, 8, 10 or 28, 30, 32. Notice
that, like with the consecutive odd integers, beginning with the first
integer, you must add 2 to obtain the second integer, and you must add 4
to the first integer to obtain the third integer.
In general, if x represents
the first integer, then the second integer is x
+ 2, and the third integer is x
+ 4.