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Applications

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Applications
2.5. APPLICATIONS
2.5
91
Applications
1. We follow the Requirements for Word Problem Solutions.
1. Set up a variable dictionary. Let θ represent the measure of the first
angle.
2. Set up an equation. A sketch will help summarize the information given
in the problem. First, we sketch two angles whose sum is 90 degrees.
The second angle is 6 degrees larger than 2 times the first angle, so the
second angle has measure 2θ + 6.
2θ + 6
θ
The angles are complementary, so their sum is 90 degrees. Thus the
equation is:
θ + (2θ + 6) = 90
3. Solve the equation. Simplify the left-hand side by combining like terms.
θ + (2θ + 6) = 90
3θ + 6 = 90
Subtract 6 from both sides of the equation, then divide both sides of the
resulting equation by 3.
3θ + 6 − 6 = 90 − 6
3θ = 84
3θ
84
=
3
3
θ = 28
4. Answer the question. To find the second angle, substitute 28 for θ in
2θ + 6 to get:
2θ + 6 = 2(28) + 6
= 62
Hence, the two angles are 28 and 62 degrees.
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5. Look back. Let’s label the angles with their numerical values.
62◦
28◦
Clearly, their sum is 90◦ , so we have the correct answer.
3. We follow the Requirements for Word Problem Solutions.
1. Set up a variable dictionary. Let θ represent the measure of the first
angle.
2. Set up an equation. A sketch will help summarize the information given
in the problem. First, we sketch two angles whose sum is 180 degrees.
The second angle is 10 degrees larger than 4 times the first angle, so the
second angle has measure 4θ + 10.
4θ + 10
θ
The angles are supplementary, so their sum is 180 degrees. Thus the
equation is:
θ + (4θ + 10) = 180
3. Solve the equation. Simplify the left-hand side by combining like terms.
θ + (4θ + 10) = 180
5θ + 10 = 180
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2.5. APPLICATIONS
93
Subtract 10 from both sides of the equation, then divide both sides of
the resulting equation by 5.
5θ + 10 − 10 = 180 − 10
5θ = 170
5θ
170
=
5
5
θ = 34
4. Answer the question. To find the second angle, substitute 34 for θ in
4θ + 10 to get:
4θ + 10 = 4(34) + 10
= 146
Hence, the two angles are 34 and 146 degrees.
5. Look back. Let’s label the angles with their numerical values.
146◦
34◦
Clearly, their sum is 180◦ , so we have the correct answer.
5. In the solution, we address each step of the Requirements for Word Problem
Solutions.
1. Set up a Variable Dictionary. An example of three consecutive integers
is 19, 20, and 21. These are not the integers we seek, but they do give us
some sense of the meaning of three consecutive integers. Note that each
consecutive integer is one larger than the preceding integer. Thus, if k
is the length of the first side of the triangle, then the next two sides are
k + 1 and k + 2. In this example, our variable dictionary will take the
form of a well-labeled figure.
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CHAPTER 2. SOLVING LINEAR EQUATIONS
94
k+2
k
k+1
2. Set up an Equation. The perimeter of the triangle is the sum of the three
sides. If the perimeter is 483 meters, then:
k + (k + 1) + (k + 2) = 483
3. Solve the Equation. To solve for k, first simplify the left-hand side of the
equation by combining like terms.
k + (k + 1) + (k + 2) = 483
3k + 3 = 483
3k + 3 − 3 = 483 − 3
3k = 480
480
3k
=
3
3
k = 160
Original equation.
Combine like terms.
Subtract 3 from both sides.
Simplify.
Divide both sides by 3.
Simplify.
4. Answer the Question. Thus, the first side has length 160 meters. Because
the next two consecutive integers are k + 1 = 161 and k + 2 = 162, the
three sides of the triangle measure 160, 161, and 162 meters, respectively.
5. Look Back. An image helps our understanding. The three sides are
consecutive integers.
162 meters
160 meters
161 meters
Note that the perimeter (sum of the three sides) is:
160 meters + 161 meters + 162 meters = 483 meters
(2.1)
Thus, the perimeter is 483 meters, as it should be. Our solution is correct.
Second Edition: 2012-2013
2.5. APPLICATIONS
95
7. We follow the Requirements for Word Problem Solutions.
1. Set up a variable dictionary. Let x represent the unknown number.
2. Set up an equation. The statement “four less than eight times a certain
number is −660” becomes the equation:
8x − 4 = −660
3. Solve the equation. Add 4 to both sides, then divide the resulting equation
by 8.
8x − 4 = −660
8x − 4 + 4 = −660 + 4
8x = −656
8x
−656
=
8
8
x = −82
4. Answer the question. The unknown number is −82.
5. Look back. “Four less than eight times −82” translates as 8(−82) − 4,
which equals −660. The solution make sense.
9. In the solution, we address each step of the Requirements for Word Problem
Solutions.
1. Set up a Variable Dictionary. Let d represent the distance left for Alan
to hike. Because Alan is four times further from the beginning of the
trail than the end, the distance Alan has already completed is 4d. Let’s
construct a little table to help summarize the information provided in
this problem.
Section of Trail
Distance (mi)
Distance to finish
Distance from start
d
4d
Total distance
70
2. Set up an Equation. As you can see in the table above, the second column
shows that the sum of the two distances is 70 miles. In symbols:
d + 4d = 70
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96
3. Solve the Equation. To solve for d, first simplify the left-hand side of the
equation by combining like terms.
d + 4d = 70
5d = 70
5d
70
=
5
5
d = 14
Original equation.
Combine like terms.
Divide both sides by 5.
Simplify.
4. Answer the Question. Alan still has 14 miles to hike.
5. Look Back. Because the amount left to hike is d = 14 miles, Alan’s
distance from the start of the trail is 4d = 4(14), or 56 miles. If we
arrange these results in tabular form, it is evident that not only is the
distance from the start of the trail four times that of the distance left to
the finish, but also the sum of their lengths is equal to the total length
of the trail.
Section of Trail
Distance (mi)
Distance (mi)
Distance to finish
Distance from start
d
4d
14
56
Total distance
70
70
Thus, we have the correct solution.
11. We follow the Requirements for Word Problem Solutions.
1. Set up a variable dictionary. Let p represent the percentage of Martha ’s
sixth grade class that is absent.
2. Set up the equation. The question is “what percent of the class size
equals the number of students absent?” The phrase “p percent of 36 is
2” becomes the equation:
p × 36 = 2
Or equivalently:
36p = 2
3. Solve the equation. Use a calculator to help divide both sides of the
equation by 36.
36p = 2
2
36p
=
36
36
p = 0.0555555556
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2.5. APPLICATIONS
97
4. Answer the question. We need to change our answer to a percent. First,
round the percentage answer p to the nearest hundredth.
Test digit
0.0 5 5 5555556
Rounding digit
Because the test digit is greater than or equal to 5, add 1 to the rounding
digit, then truncate. Hence, to the nearest hundredth, 0.0555555556 is
approximately 0.06. To change this answer to a percent, multiply by 100,
or equivalently, move the decimal two places to the right. Hence, 6% of
Martha’s sixth grade class is absent.
5. Look back. If we take 6% of Martha’s class size, we get:
6% × 36 = 0.06 × 36
= 2.16
Rounded to the nearest student, this means there are 2 students absent,
indicating we’ve done the problem correctly.
13. We follow the Requirements for Word Problem Solutions.
1. Set up a variable dictionary. Let x represent the length of the first piece.
2. Set up an equation. The second piece is 3 times as long as the first piece,
so the second piece has length 3x. The third piece is 6 centimeters longer
than the first piece, so the second piece has length x + 6. Let’s construct
a table to summarize the information provided in this problem.
Piece
Length (centimeters)
First
Second
Third
Total length
x
3x
x+6
211
As you can see in the table above, the second column shows that the sum
of the three pieces is 211 centimeters. Hence, the equation is:
x + 3x + (x + 6) = 211
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3. Solve the equation. First, simplify the left-hand side of the equation by
combining like terms.
x + 3x + (x + 6) = 211
5x + 6 = 211
Subtract 6 from both sides of the equation, then divide both sides of the
resulting equation by 5.
5x + 6 − 6 = 211 − 6
5x = 205
5x
205
=
5
5
x = 41
4. Answer the question. Let’s add a column to our table to list the length
of the three pieces. The lengths of the second and third pieces are found
by substituting 41 for x in 3x and x + 6.
Piece
First
Second
Third
Total length
Length (centimeters)
Length (centimeters)
x
3x
x+6
41
123
47
211
211
5. Look back. The third column of the table above shows that the lengths
sum to 211 centimeters, so we have the correct solution.
15. In the solution, we address each step of the Requirements for Word Problem
Solutions.
1. Set up a Variable Dictionary. An example of three consecutive even
integers is 18, 20, and 22. These are not the integers we seek, but they
do give us some sense of the meaning of three consecutive even integers.
Note that each consecutive even integer is two larger than the preceding
integer. Thus, if k is the length of the first side of the triangle, then the
next two sides are k+2 and k+4. In this example, our variable dictionary
will take the form of a well-labeled figure.
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2.5. APPLICATIONS
99
k+4
k
k+2
2. Set up an Equation. The perimeter of the triangle is the sum of the three
sides. If the perimeter is 450 yards, then:
k + (k + 2) + (k + 4) = 450
3. Solve the Equation. To solve for k, first simplify the left-hand side of the
equation by combining like terms.
k + (k + 2) + (k + 4) = 450
3k + 6 = 450
3k + 6 − 6 = 450 − 6
3k = 444
444
3k
=
3
3
k = 148
Original equation.
Combine like terms.
Subtract 6 from both sides.
Simplify.
Divide both sides by 3.
Simplify.
4. Answer the Question. Thus, the first side has length 148 yards. Because
the next two consecutive even integers are k+2 = 150 and k+4 = 152, the
three sides of the triangle measure 148, 150, and 152 yards, respectively.
5. Look Back. An image helps our understanding. The three sides are
consecutive even integers.
152 yards
148 yards
150 yards
Note that the perimeter (sum of the three sides) is:
148 yards + 150 yards + 152 yards = 450 yards
(2.2)
Thus, the perimeter is 450 yards, as it should be. Our solution is correct.
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100
17. We follow the Requirements for Word Problem Solutions.
1. Set up a variable dictionary. Let x represent the length of the first side
of the triangle.
2. Set up an equation. The second side is 7 times as long as the first side,
so the second side has length 7x. The third side is 9 yards longer than
the first side, so the second side has length x + 9. Let’s sketch a diagram
to summarize the information provided in this problem (the sketch is not
drawn to scale).
x+9
x
7x
The sum of the three sides of the triangle equals the perimeter. Hence,
the equation is:
x + 7x + (x + 9) = 414
3. Solve the equation. First, simplify the left-hand side of the equation by
combining like terms.
x + 7x + (x + 9) = 414
9x + 9 = 414
Subtract 9 from both sides of the equation, then divide both sides of the
resulting equation by 9.
9x + 9 − 9 = 414 − 9
9x = 405
9x
405
=
9
9
x = 45
4. Answer the question. Because the first side is x = 45 yards, the second
side is 7x = 315 yards, and the third side is x + 9 = 54 yards.
5. Look back. Let’s add the lengths of the three sides to our sketch.
54 yards
315 yards
Second Edition: 2012-2013
45 yards
2.5. APPLICATIONS
101
Our sketch clearly indicates that the perimeter of the triangle is Perimeter =
45 + 315 + 54, or 414 yards. Hence, our solution is correct.
19. We follow the Requirements for Word Problem Solutions.
1. Set up a variable dictionary. Let k represent the smallest of three consecutive odd integers.
2. Set up an equation. Because k is the smallest of three consecutive odd
integers, the next two consecutive odd integers are k + 2 and k + 4.
Therefore, the statement “the sum of three consecutive odd integers is
−543” becomes the equation:
k + (k + 2) + (k + 4) = −543
3. Solve the equation. First, combine like terms on the left-hand side of the
equaton.
k + (k + 2) + (k + 4) = −543
3k + 6 = −543
Subtract 6 from both sides, then divide both sides of the resulting equation by 3.
3k + 6 − 6 = −543 − 6
3k = −549
3k
−549
=
3
3
k = −183
4. Answer the question. The smallest of three consecutive odd integers is
−183.
5. Look back. Because the smallest of three consecutive odd integers is −183,
the next two consecutive odd integers are −181, and −179. If we sum
these integers, we get
−183 + (−181) + (−179) = −543,
so our solution is correct.
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102
21. We follow the Requirements for Word Problem Solutions.
1. Set up a variable dictionary. Let θ represent the measure of angle A.
2. Set up an equation. A sketch will help summarize the information given
in the problem. Because angle B is 4 times the size of angle A, the degree
measure of angle B is represented by 4θ. Because angle C is 30 degrees
larger than the degree measure of angle A, the degree measure of angle
C is represented by θ + 30.
C
θ + 30
θ
4θ
A
B
Because the sum of the three angles is 180◦, we have the following equation:
θ + 4θ + (θ + 30) = 180
3. Solve the equation. Start by combining like terms on the left-hand side
of the equation.
θ + 4θ + (θ + 30) = 180
6θ + 30 = 180
Subtract 30 from both sides of the equation and simplify.
6θ + 30 − 30 = 180 − 30
6θ = 150
Divide both sides by 6.
150
6θ
=
6
6
θ = 25
4. Answer the question. The degree measure of angle A is θ = 25◦ . The
degree measure of angle B is 4θ = 100◦. The degree measure of angle C
is θ + 30 = 55◦ .
5. Look back. Our figure now looks like the following.
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2.5. APPLICATIONS
103
C
55◦
25◦
100◦
A
B
Note that
25 + 100 + 55 = 180,
so our solution is correct.
23. We follow the Requirements for Word Problem Solutions.
1. Set up a variable dictionary. Let k represent the smallest of three consecutive integers.
2. Set up an equation. Because k is the smallest of three consecutive integers, the next two consecutive integers are k + 1 and k + 2. Therefore,
the statement “the sum of three consecutive integers is −384” becomes
the equation:
k + (k + 1) + (k + 2) = −384
3. Solve the equation. First, combine like terms on the left-hand side of the
equaton.
k + (k + 1) + (k + 2) = −384
3k + 3 = −384
Subtract 3 from both sides, then divide both sides of the resulting equation by 3.
3k + 3 − 3 = −384 − 3
3k = −387
−387
3k
=
3
3
k = −129
4. Answer the question. The smallest of three consecutive integers is k =
−129, so the next two consecutive integers are −128 and −127. Therefore,
the largest of the three consecutive integers is −127.
5. Look back. If we sum the integers −129, −128, and −127, we get
−129 + (−128) + (−127) = −384,
so our solution is correct.
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CHAPTER 2. SOLVING LINEAR EQUATIONS
25. We follow the Requirements for Word Problem Solutions.
1. Set up a variable dictionary. Let x represent the unknown number.
2. Set up an equation. The statement “seven more than two times a certain
number is 181” becomes the equation:
7 + 2x = 181
3. Solve the equation. Subtract 7 from both sides, then divide the resulting
equation by 2.
7 + 2x = 181
7 + 2x − 7 = 181 − 7
2x = 174
2x
174
=
2
2
x = 87
4. Answer the question. The unknown number is 87.
5. Look back. “Seven more than two times 87” translates as 7 + 2(87), which
equals 181. The solution make sense.
27. In the solution, we address each step of the Requirements for Word Problem
Solutions.
1. Set up a Variable Dictionary. An example of three consecutive odd integers is 19, 21, and 23. These are not the integers we seek, but they
do give us some sense of the meaning of three consecutive odd integers.
Note that each consecutive odd integer is two larger than the preceding
integer. Thus, if k is the length of the first side of the triangle, then the
next two sides are k+2 and k+4. In this example, our variable dictionary
will take the form of a well-labeled figure.
k+4
k
k+2
Second Edition: 2012-2013
2.5. APPLICATIONS
105
2. Set up an Equation. The perimeter of the triangle is the sum of the three
sides. If the perimeter is 537 feet, then:
k + (k + 2) + (k + 4) = 537
3. Solve the Equation. To solve for k, first simplify the left-hand side of the
equation by combining like terms.
k + (k + 2) + (k + 4) = 537
3k + 6 = 537
3k + 6 − 6 = 537 − 6
3k = 531
3k
531
=
3
3
k = 177
Original equation.
Combine like terms.
Subtract 6 from both sides.
Simplify.
Divide both sides by 3.
Simplify.
4. Answer the Question. Thus, the first side has length 177 feet. Because
the next two consecutive odd integers are k+2 = 179 and k+4 = 181, the
three sides of the triangle measure 177, 179, and 181 feet, respectively.
5. Look Back. An image helps our understanding. The three sides are
consecutive odd integers.
181 feet
177 feet
179 feet
Note that the perimeter (sum of the three sides) is:
177 feet + 179 feet + 181 feet = 537 feet
(2.3)
Thus, the perimeter is 537 feet, as it should be. Our solution is correct.
29. We follow the Requirements for Word Problem Solutions.
1. Set up a variable dictionary. Let M represent the marked price of the
article.
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CHAPTER 2. SOLVING LINEAR EQUATIONS
106
2. Solve the equation. Because the store offers a 14% discount, Yao pays
86% for the article. Thus, the question becomes “86% of the marked
price is $670.8.” This translates into the equation
86% × M = 670.8,
or equivalently,
0.86M = 670.8
Use a calculator to help divide both sides by 0.86.
0.86M
670.8
=
0.86
0.86
M = 780
3. Answer the question. Hence, the original marked price was $780.
4. Look back. Because the store offers a 14% discount, Yao has to pay 86%
for the article. Check what 86% of the marked price will be.
86% × 780 = 0.86 × 780
= 670.8
That’s the sales price that Yao paid. Hence, we’ve got the correct solution.
31. We follow the Requirements for Word Problem Solutions.
1. Set up a variable dictionary. Let k represent the smallest of three consecutive even integers.
2. Set up an equation. Because k is the smallest of three consecutive even
integers, the next two consecutive even integers are k + 2 and k + 4.
Therefore, the statement “the sum of three consecutive even integers is
−486” becomes the equation:
k + (k + 2) + (k + 4) = −486
3. Solve the equation. First, combine like terms on the left-hand side of the
equaton.
k + (k + 2) + (k + 4) = −486
3k + 6 = −486
Second Edition: 2012-2013
2.5. APPLICATIONS
107
Subtract 6 from both sides, then divide both sides of the resulting equation by 3.
3k + 6 − 6 = −486 − 6
3k = −492
3k
−492
=
3
3
k = −164
4. Answer the question. The smallest of three consecutive even integers is
−164.
5. Look back. Because the smallest of three consecutive even integers is
−164, the next two consecutive even integers are −162, and −160. If we
sum these integers, we get
−164 + (−162) + (−160) = −486,
so our solution is correct.
33. We follow the Requirements for Word Problem Solutions.
1. Set up a variable dictionary. Let M represent the amount invested in the
mutal fund.
2. Set up the equation. We’ll use a table to help summarize the information
in this problem. Because the amount invested in the certificate of deposit
is $3,500 more than 6 times the amount invested in the mutual fund,
we represent the amount invested in the certificate of deposit with the
expression 6M + 3500.
Investment
Mutual fund
Certificate of deposit
Totals
Amount invested
M
6M + 3500
45500
The second column of the table gives us the needed equation. The two
investment amounts must total $45,500.
M + (6M + 3500) = 45500
Second Edition: 2012-2013
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