Sketch with values labeled:
90X = 120(7 - X) TIME OUT: X = 4
90X = 840 - 120X DISTANCE = RATE · TIME = 90(4) = 360 mi
210X = 840
MATH 27
WORD PROBLEM AEROBICS
DIRECTIONS: For each of the word problems below, answer the questions, draw the sketches, write out the equation in words, then write it algebraicially using ONLY ONR VARIABLE, and solve for the unknown(s).
1. The sum of three consecutive integers is 48. Find the integers. Let the first on be X.
How far apart from one another are consecutive integers? ______________
How do you represent the second in terms of X? ___________________
How do you represent the third in terms of X? ___________________
2. The sum of three consecutive EVEN integers is 66. Find the integers. Let the first one be X.
How far apart from one another are consecutive EVEN integers? ______________
How do you represent the second in terms of X? ___________________
How do you represent the third in terms of X? ___________________
3. Seven times the first of two consecutive ODD integers is five times the second. Find the integers. Let the first one be X.
How far apart from one another are consecutive ODD integers? ______________
How do you represent the second in terms of X? ___________________
4. Twice the smallest of three consecutive ODD integers is seven more than the largest. Find the integers. Let the first one be X.
How far apart from one another are consecutive ODD integers? ______________
How do you represent the second in terms of X? ___________________
How do you represent the third in terms of X? ___________________
5. A piggy bank contains 30 coins total in dimes and quarters. The coins have a total value of $5.40. Find the number of dimes and quarters in the bank. Let X represent the quantity of dimes.
How do you represent the quantity of quarters in terms of X? _______________________
All the money should be expressed in the same units, dollars or cents, it is easier to use cents. Thus $5.40 is ____________ cents.
Fill in the table:
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VALUE = |
FACE VALUE · |
QUANTITY |
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DIMES |
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QUARTERS |
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TOTAL |
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6. A drawer contains 15 cent stamps and 18 cent stamps. The number of 15 cent stamps is two less than the three times the number of 18 cent stamps. The total value of all the stamps is 96 cents. How many 15 cent stamps are in the drawer? Let X represent the quantity of 18 cent stamps.
How do you represent the quantity of 15 cent stamps in terms of X? _______________________
Fill in the table:
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VALUE = |
FACE VALUE · |
QUANTITY |
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15 cent stamps |
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18 cent stamps |
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TOTAL |
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7. Two small planes start from the same point, at the same time, and fly in opposite directions. The first plane is flying 25 mph SLOWER than the second. In two hours they are 430 mi apart. Find the rate of each plane. Let X represent the rate of the second plane. This is a distance-rate-time problem; D = r · t
How do you represent the rate of the first plane in terms of X? ______________________
What is the same or equivalent? Use this information to write the equation from table values.
YES NO
Their rates ____ ____
Distances they travel ____ ____
Time of travel ____ ____
Total distance = sum of distances ____ ____
Fill in the table:
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Distance = |
rate · |
time |
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1st plane |
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2nd plane |
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TOTAL |
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Sketch with values labeled:
8. As part of flight training, a student pilot is required to fly to an airport and then return. The average speed to the airport was 90 mph, and the average speed returning was 120 mph. Find the distance between the two airports, if the total flying time was 7 hours. Let X represent the time to the airport.
How do you represent the return time in terms of X? _______________
What is the same or equivalent? Use this information to write the equation from table values.
YES NO
The rates ____ ____
To & from distances ____ ____
To & from times ____ ____
Fill in the table:
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Distance = |
rate · |
time |
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To airport |
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From airport |
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TOTAL |
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Sketch with values labeled:
9. A car traveling at 48 mph overtakes a jogger who, running at 6 mph, has had a three hour head start. How far from the starting point does the car overtake the cyclist? Let X represent the time of travel of the car.
How do you represent the time of travel of the cyclist in terms of X? ________________
What is the same or equivalent? Use this information to write the equation from table values.
YES NO
The rates ____ ____
Distances traveled ____ ____
Times of travel ____ ____
Fill in the table:
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Distance = |
rate · |
time |
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Car |
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Jogger |
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TOTAL |
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Sketch with values labeled:
10. The perimeter of a triangle is 33 ft. The first side of the triangle is 1 ft. longer than the second side. The third side is 2 ft. longer than the second side. Find the length of each side.
Two of the sides are being referenced to one of them. Which is the one being referenced to __________________? Let it be represented by X.
What are the other two sides, and how are their lengths represented in terms of X?
______________________ _________________
______________________ _________________
What is the generic equation for the perimeter of a triangle? ______________________ Use this to write your final equation.
Sketch and label its values:
11. The perimeter of a rectangle is 42 m. The length of the rectangle is three meters less than twice the width. Find the length and width of the rectangle.
One of the sides is being referenced to the other. Which side is the point of reference? ______________. Let it be represented by X.
How do you represent the the other side in terms of X? _____________________________
What is the generic equation for the perimeter of a rectangle? ____________________________ Use this to write your final equation.
Sketch with values labeled:
12. The perimeter of a triangle is 110 cm. The first side is twice the second side. The third side is 30 cm more than the second side. Find the lengths of each side.
Two of the sides are being referenced to one of them. Which is the one being referenced to __________________? Let it be represented by X.
What are the other two sides, and how are their lengths represented in terms of X?
______________________ _________________
______________________ _________________
What is the generic equation for the perimeter of a triangle? ______________________ Use this to write your final equation.
Sketch and label its values:
SOLUTIONS
1. The sum of three consecutive integers is 48. Find the integers. Let the first on be X.
How far apart from one another are consecutive integers? ____ONE______
How do you represent the second in terms of X? ______X + 1_____________
How do you represent the third in terms of X? ________X + 2___________
FIRST + SECOND + THIRD = 48
X + (X + 1) + (X + 2) = 48 3X = 45
3X + 3 = 48 X = 15
2. The sum of three consecutive EVEN integers is 66. Find the integers. Let the first one be X.
How far apart from one another are consecutive EVEN integers? _______2_______
How do you represent the second in terms of X? ________X + 2___________
How do you represent the third in terms of X? __________X + 4_________
X + (X + 2) + (X + 4) = 66 FIRST: X = 20
3X + 6 = 66 SECOND: X + 2 = 22
3X = 60 THIRD: X + 4 = 24
3. Seven times the first of two consecutive ODD integers is five times the second. Find the integers. Let the first one be X.
How far apart from one another are consecutive ODD integers? ______2________
How do you represent the second in terms of X? _____________(X + 2)_____________
7X = 5(X + 2) FIRST: X = 5
7X = 5X + 10 SECOND: (X + 2) = 7
2X = 10
4. Twice the smallest of three consecutive ODD integers is seven more than the largest. Find the integers. Let the first one be X.
How far apart from one another are consecutive ODD integers? _______2_______
How do you represent the second in terms of X? ________(X + 2)___________
How do you represent the third in terms of X? ________(X + 4)___________
2X = (X + 4) + 7 FIRST: X = 11 SECOND: (X+ 2) = 13
2X = X + 11 THIRD: (X + 4) = 15
5. A piggy bank contains 30 coins total in dimes and quarters. The coins have a total value of $5.40. Find the number of dimes and quarters in the bank. Let X represent the quantity of dimes.
How do you represent the quantity of quarters in terms of X? __________(30 - X)________
All the money should be expressed in the same units, dollars or cents, it is easier to use cents. Thus $5.40 is ____540________ cents.
Fill in the table:
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VALUE = |
FACE VALUE · |
QUANTITY |
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DIMES |
10X
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10 |
X |
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QUARTERS |
25(30 - X)
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25 |
(30 - X) |
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TOTAL |
540
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VALUE OF DIMES + VALUE OF QUARTERS = TOTAL VALUE
10X + 25(30 - X) = 540 -15X = -210
10X + 750 - 25X = 540 DIMES: X = 14
-15X + 750 = 540 QUARTERS: 30 - 14 = 16
6. A drawer contains 15 cent stamps and 18 cent stamps. The number of 15 cent stamps is two less than the three times the number of 18 cent stamps. The total value of all the stamps is 96 cents. How many 15 cent stamps are in the drawer? Let X represent the quantity of 18 cent stamps.
How do you represent the quantity of 15 cent stamps in terms of X? _________(3X - 2)_______
Fill in the table:
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VALUE = |
FACE VALUE · |
QUANTITY |
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15 cent stamps |
15(3X - 2)
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15 |
(3X - 2) |
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18 cent stamps |
18X
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18 |
X |
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TOTAL |
96
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18X + 15(3X - 2) = 96 18 CENT STAMPS: X = 2
18X + 45X - 30 = 96 15 CENT STAMPS: 3(2) - 2 = 4
63X = 126
7. Two small planes start from the same point, at the same time, and fly in opposite directions. The first plane is flying 25 mph SLOWER than the second. In two hours they are 430 mi apart. Find the rate of each plane. Let X represent the rate of the second plane. This is a distance-rate-time problem; D = r · t
How do you represent the rate of the first plane in terms of X? _____(X - 25)______
What is the same or equivalent? Use this information to write the equation from table values.
YES NO
Their rates ____ _X__
Distances they travel ____ _X__
Time of travel __X_ ____
Total distance = sum of distances __X_ ____
Fill in the table:
|
Distance = |
rate · |
time |
|
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1st plane |
2(X - 25)
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(X - 25) |
2 |
|
2nd plane |
2X
|
X |
2 |
Sketch with values labeled:
2X + 2(X - 25) = 430 4X - 50 = 430 FASTER RATE: X = 120
2X + 2X - 50 = 430 4X = 480 SLOWER RATE: 120 - 25 = 95
8. As part of flight training, a student pilot is required to fly to an airport and then return. The average speed to the airport was 90 mph, and the average speed returning was 120 mph. Find the distance between the two airports, if the total flying time was 7 hours. Let X represent the time to the airport.
How do you represent the return time in terms of X? _______(7 - X)_____
What is the same or equivalent? Use this information to write the equation from table values.
YES NO
The rates ____ __X_
To & from distances __X_ ____
To & from times ____ __X_
Fill in the table:
|
Distance = |
rate · |
time |
|
|
To airport |
90X
|
90 |
X |
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From airport |
120(7 - X)
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120 |
(7 - X) |
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TOTAL |
|
7 |
Sketch with values labeled:
90X = 120(7 - X) TIME OUT: X = 4
90X = 840 - 120X DISTANCE = RATE · TIME = 90(4) = 360 mi
210X = 840
9. A car traveling at 48 mph overtakes a jogger who, running at 6 mph, has had a three hour head start. How far from the starting point does the car overtake the cyclist? Let X represent the time of travel of the car.
How do you represent the time of travel of the cyclist in terms of X? ____(X + 3)_____
What is the same or equivalent? Use this information to write the equation from table values.
YES NO
The rates ____ __X_
Distances traveled __X_ ____
Times of travel ____ __X_
Fill in the table:
|
Distance = |
rate · |
time |
|
|
Car |
48X
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48 |
X |
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Jogger |
6(X + 3)
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6 |
(X + 3) |
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TOTAL |
|
Sketch with values labeled:
Car Distance = Jogger distance 42X = 18
48X = 6(X + 3) Car time: X = 18/42 = 9/21 hr
48X = 6X + 18 Distance: 48X = 48(9/21) @ 20.6 mi
10. The perimeter of a triangle is 33 ft. The first side of the triangle is 1 ft. longer than the second side. The third side is 2 ft. longer than the second side. Find the length of each side.
Two of the sides are being referenced to one of them. Which is the one being referenced to _SECOND SIDE____? Let it be represented by X.
What are the other two sides, and how are their lengths represented in terms of X?
________FIRST______ ______X + 1_____
________THIRD________ ___X + 2_________
What is the generic equation for the perimeter of a triangle? ____P = S1 + S2 + S3____ Use this to write your final equation.
Sketch and label its values:
P = S1 + S2 + S3
33 = (X + 1) + X + (X + 2) Second side: X = 10
33 = 3X + 3 First side: X + 1 = 10 + 1 = 11
30 = 3X Third side: X + 2 = 10 + 2 = 12
11. The perimeter of a rectangle is 42 m. The length of the rectangle is three meters less than twice the width. Find the length and width of the rectangle.
One of the sides is being referenced to the other. Which side is the point of reference? ____width____. Let it be represented by X.
How do you represent the the other side in terms of X? ____________2X - 3__________
What is the generic equation for the perimeter of a rectangle? ________P = 2L + 2W________ Use this to write your final equation.
Sketch with values labeled:
P = 2W + 2L 36 = 6X
42 = 2X + 2(2X - 3) Width: X = 6
42 = 2X + 4X - 6 Length: 2X - 3 = 2(6) - 3 = 9
42 = 6X - 6
12. The perimeter of a triangle is 110 cm. The first side is twice the second side. The third side is 30 cm more than the second side. Find the lengths of each side.
Two of the sides are being referenced to one of them. Which is the one being referenced to ____Second_____? Let it be represented by X.
What are the other two sides, and how are their lengths represented in terms of X?
________First________ _________2X_____
________Third_______ ______X + 30_____
What is the generic equation for the perimeter of a triangle? ___P = S1 + S2 + S3______ Use this to write your final equation.
Sketch and label its values:
P = S1 + S2 + S3 2nd: X = 20
110 = 2X + X + (X + 30) 1st: 2X = 2(20) = 40
110 = 4X + 30 3rd: X + 30 = 20 + 30 = 50
80 = 4X