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
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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:
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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.
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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.
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Consecutive integers: x,
x+1, x+2
Consecutive odd integers:
x, x+2, x+4
Consecutive even
integers: x, x+2, x+4
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EXAMPLE
5. Find
two consecutive integers whose sum is 75.
Solution:
Let
x = the first integer
x + 1 = the
second integer
Equation:
x + x + 1 = 75
Solve:
2x + 1 =
75
2x
= 74
x
= 37
x + 1
= 38
Check:
37 + 38 = 75
EXAMPLE
6.
Find
three consecutive even integers whose sum is 78.
Solution:
Let
x = the first
integer
x + 2 = the
second integer
x + 4 = the
third integer
Equation:
x + x + 2 + x + 4 = 78
Solve:
3x
+ 6 = 78
3x
= 72
x
= 24
x + 2
= 26
x
+ 4 = 28
Check:
24 + 26+ 28 = 78
EXAMPLE
7. Find three consecutive odd integers whose sum is 123.
Solution:
Let x
= the first integer
x + 2 =
the second integer
x + 4 =
the third integer
Equation:
x + x + 2 + x + 4 = 123
Solve:
3x +
6 =
123
3x
= 117
x
= 39
x + 2 =
41
x + 4 =
43
Check:
39 + 41+ 43 = 123
EXERCISES.
21.
Find two consecutive integers whose sum is 71.
22. Find two
consecutive even integers whose sum is 86.
23. Find three
consecutive even integers whose sum is 396.
24. Find two consecutive odd
integers whose sum is 100.
25. Find three
consecutive odd integers whose sum is 399.
26. Find three
consecutive integers whose sum is 369.
EXAMPLE
8.
Three
consecutive even integers are such that twice the first, plus four times
the second, plus the third is equal to 96. Find the numbers.
Solution:
Let
x = the first integer
x + 2 = the
second integer
x + 4 = the
third integer
Equation:
2(x) + 4(x+2) + (x+4) = 96
Solve:
2x + 4x + 8 + x + 4 =
96
7x
+ 12 = 96
7x
= 84
x
= 12
x + 2
= 14
x + 4
= 16
Check:
2(12)+ 4(14) + 16 = 96
24 +
56 + 16 =
96
EXERCISES.
27. Two consecutive integers are such that
twice the first, plus three times the second, is equal to 78. Find the
numbers.
Let
x = first
x
+ 1 = second
Equation: 2(
) + 3(
) = 78
Solve:
28. Two consecutive odd integers are such
that twice the second, plus three times the first, is equal to 29. Find
the numbers.
29. Two consecutive even integers are such that
twice the first, plus three times the second, is equal to 156. Find the
numbers.
30. Three consecutive even integers are such
that the first, plus twice the second, plus three times the third, is
equal to 100. Find the numbers.
31. Three consecutive odd integers are such
that the sum of the integers is 7 less than four times the smallest
number. Find the numbers.
32. Three consecutive integers are such that
the first, plus twice the second, plus three times the third is equal to
200. Find the numbers.
III. PERIMETER PROBLEMS
The
perimeter of a geometric figure (shape) is the total distance around the
outside of that figure. For a
rectangle, the perimeter consists of two widths and two lengths.
For a triangle, the perimeter is just the sum of the three sides.
For a circle, the perimeter (for circles it is called the “circumference”)
is π times
the diameter of the circle, where π
is
approximately 3.14.
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PERIMETER
Triangle: P =
a + b + c
Rectangle:
P = 2W + 2L
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EXAMPLE
9.
The
second side of a triangle is twice the first side, and the third side is 6
less than the second side. The
perimeter of the triangle is 94 feet.
Find the sides of the triangle
Solution:
Let
x = first side
2x = second side
2x - 6 = third
side
Equation:
x + 2x + 2x - 6 = 94
5x - 6 =
94
5x
= 100
x
= 20 feet (first
side)
2x
= 40 feet
(second side)
2x - 6 =
34 feet (third side)
Check:
20 + 40 + 34 = 94
EXERCISES.
33.
The second side of a triangle is 6 more than the first side, and
the third side is 10 more than the first side.
The perimeter of the triangle is 70 meters.
Find the sides of the triangle.
Let
x
= first side
__________ =
second side
__________ =
third side
Equation:
__________ + __________ + __________ = 70
Solve:
34.
The second side of a triangle is 6 more than the first side, and
the third side is 10 more than the second side. The perimeter of the triangle is 70 meters.
Find the sides of the triangle.
35.
The second side of a triangle is 4 more than the first side, and
the third side is 10 less than the second side. The perimeter of the triangle is 70 meters.
Find the sides of the triangle.
36. The second side of a triangle
is twice the first side, and the third side is 10 more than the second
side. The perimeter of the
triangle is 70 feet. Find the sides of the triangle.
37. The third side of a triangle is
6 less than the first side, and the second side is twice the third side.
The perimeter of the triangle is 82 feet. Find the sides of the triangle.
38. The third side of a triangle is twice
the first side, and the second side is 10 less than the third side.
The perimeter of the triangle is 70 feet. Find the sides of the triangle.
EXAMPLE 10.
The
length of a rectangle is 5 more than the width. The perimeter of the rectangle is 50 meters.
Find the dimensions of the rectangle.
Solution:
Let x
= width of rectangle
x
+ 5 =
length of rectangle
Equation:
2W + 2L =
Perimeter
2(x) + 2(x+5) =
50
2x
+ 2x + 10 = 50
4x + 10
= 50
4x =
40
x = 10 meters
Answer
question:
x = 10 m. width
of rectangle
x + 5 = 15 m.
length of rectangle
Ch
2W +
2L =
P
2(10) + 2(15)
= 50
20
+ 30 =
50
EXAMPLE
11.
The
length of a rectangle is 8 less than twice the width. The perimeter of the rectangle is 104 centimeters.
Find the dimensions of the rectangle.
Solution:
Let
x = width of rectangle
2x-8
= length of rectangle
Equation:
2W
+ 2L = Perimeter
2(x) + 2(2x-8)
= 104
2x + 4x - 16
= 104
6x
- 16
= 104
6x
= 120
x
= 20
centimeters
Answer
question:
x = 20 cm width of rectangle
2x - 8 = 32 cm
length of rectangle
Check:
2W +
2L = P
2(20)
+ 2(32) = 104
40 +
64 = 10
EXERCISES.
39.
The length of a rectangle is 5 meters longer than the width.
The perimeter is 50 meters.
Find the dimensions of the rectangle
Solution:
Let
x = width of the rectangle
_____ = length
of the rectangle
Equation:
2W +
2L
= Perimeter
2( ) + 2(
) = ______
Check:
40.
The length of a rectangle is twice
the width. The perimeter is 24 meters.
Find the dimensions of the rectangle.
Solution:
Let
x = width of the
rectangle
______= length
of the rectangle
Equation:
2W +
2L
= Perimeter
2( ) + 2(
) = ______
Check:
41. The length of a rectangle is 5 meters
more than twice the width. The
perimeter is 130 meters. Find
the dimensions of the rectangle.
Solution:
Let x = width of the
rectangle
_____ = length
of the rectangle
Equation:
2W +
2L =
Perimeter
2( ) + 2(
) = ______
Check:
42.
The length of a rectangle is 5 meters less than twice the width.
The perimeter is 50 meters. Find
the dimensions of the rectangle.
Solution:
Let
x = width of the rectangle
_____
= length of the rectangle (Not 5 - 2x)
43.
The length of a rectangle is 50 feet less than three times the
width. The perimeter is 500
feet. Find the dimensions of
the rectangle.
44.
The length of a rectangle is 50 meters less than twice the width.
The perimeter is 1100 meters.
Find the dimensions.
45.
The length of a rectangle is 75 meters less than twice the width.
The perimeter is 600 meters. Find
the length and width of the rectangle.
46.
The width of a rectangle is 50 feet less than the length.
If the perimeter is 400 feet, find the length and width of the
rectangle.
47.
The length of a rectangle is 3 more than twice the width.
The perimeter is 56 meters. Find
the dimensions of the rectangle. (Note: the length and/or width do not
have to come out even! Express
the answer as in fractional form or as a repeating decimal.)
48.
The length of a rectangle is three less than five times the width.
The perimeter is ten times the width.
Find the dimensions and perimeter of the rectangle.
49. EXTRA CHALLENGE. The perimeter of a
rectangle is 46 meters. Twice
the length is 4 more than five times the width.
Find the dimensions of the rectangle.
Solution:
Let x = width
__________ =
two lengths
2 widths + 2
lengths = Perimeter
ANSWERS
1.10
p. 65 -
100:
1. 5;
2. 32;
3. 4;
4. -8;
5. 10;
6. -20;
7. 5;
8. 6;
9. -6; 10. 8; 11. 15, 45; 12.
18, 42; 13. 22, 38; 14.
36, 24; 15. 40, 20; 16. 13, 9;
17.
22, 27, 85;
18. 25, 22, 44;
19. 62, 66, 128;
20. 50, 94, 149;
21. 35, 36;
22. 42, 44; 23. 130,
132, 134; 24. 49, 51;
25.
131, 133, 135;
26. 122, 123, 124; 27.
15, 16; 28. 5, 7;
29. 30, 32; 30.
14, 16, 18; 31. 13, 15, 17;
32. 32, 33, 34;
33.
18m, 24m, 28m;
34. 16m, 22m, 32m;
35. 24m, 28m,
18m;
36. 12ft, 24ft, 34ft;
37. 25ft, 38ft, 19ft;
38. 16ft, 22ft,
32ft;
39.
W=10m, L=15m; 40. W=4m, L=8m; 41.
W=20m,L=45m; 42. W=10m,
L=15m; 43. W=75ft, L=175ft;
44. W=200m, L=350m; 45. W=125m, L=175m;
46. W=75ft,L=125ft; 47. W=8 1/3m,
L=19 2/3m; 48. W=3, L=12
P=30; 49. W=6, L=17;
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