2. Instructions
Each problem in this slide show is worked
out step-by-step.
For each problem, try to work it out by
yourself first. If you get stuck read
through my solution until you get unstuck
then work it from there.
Then check your work against mine.
Remember, you need to use an equation
with a variable to solve each problem.
3. Some Advice
• Always define your variable to represent
one of the quantities you are being asked
for.
• This section is based on rational
problems, your equation should be a
rational equation.
• Your equation should be meaningful. If
you can’t express it in a sentence, it isn’t
correct.
5. Problem #1
Find two consecutive even integers such
7
that the sum of their reciprocals is .
24
6. Problem #1 (step 1)
Find two consecutive even integers such
7
that the sum of their reciprocals is .
24
Define your variable.
Express all unknown quantities in terms of
that variable.
7. Problem #1 (step 1)
Find two consecutive even integers such
7
that the sum of their reciprocals is .
24
Define your variable.
Let x be the first even integer.
Express all unknown quantities in terms of
that variable.
x + 2 is the second even integer.
8. Problem #1 (step 2)
Find two consecutive even integers such
7
that the sum of their reciprocals is .
24
Write an equation using your variable and
the information in the problem as it is
written.
9. Problem #1 (step 2)
Find two consecutive even integers such
7
that the sum of their reciprocals is .
24
Write an equation using your variable and
the information in the problem as it is
written.
1 1 7
x x 2 24
14. Problem #1 (Step 4)
6 and 8 are the two consecutive even integers.
15. Problem #2
The sum of the reciprocal of a number and the
reciprocal of four less than the number is six
times the reciprocal of the original number. Find
the original number.
16. Problem #2
The sum of the reciprocal of a number and the
reciprocal of four less than the number is six
times the reciprocal of the original number. Find
the original number.
Define your variable.
Express all unknown quantities in terms of that
variable.
17. Problem #2
The sum of the reciprocal of a number and the
reciprocal of four less than the number is six
times the reciprocal of the original number. Find
the original number.
Define your variable.
Let n be the number.
Express all unknown quantities in terms of that
variable.
n – 4 is four less than the number.
23. Problem #2
The sum of the reciprocal of a number and the
reciprocal of four less than the number is six
times the reciprocal of the original number. Find
the original number.
Answer the question asked.
24. Problem #2
The sum of the reciprocal of a number and the
reciprocal of four less than the number is six
times the reciprocal of the original number. Find
the original number.
Answer the question asked.
5 is the original number.
25. Motion Problems
Remember when you are setting up the
problem that you can define a variable to
represent the quantity you are looking for.
Also, your equation can now be about time,
rate, or distance.
26. Problem #3 (Motion)
Garth likes to kayak in the river. One day he went
kayaking. He was told that the current of the river was 2
miles per hour. If it took Garth the same amount of time to
travel 10 miles downstream as 2 miles
upstream, determine the speed of his kayak in still water.
27. Problem #3 (Motion)
Garth likes to kayak in the river. One day he went
kayaking. He was told that the current of the river was 2
miles per hour. If it took Garth the same amount of time to
travel 10 miles downstream as 2 miles
upstream, determine the speed of his kayak in still water.
Identify and organize the information we have already and
define a variable to represent what we’re looking for.
28. Problem #3 (Motion)
Garth likes to kayak in the river. One day he went
kayaking. He was told that the current of the river was 2
miles per hour. If it took Garth the same amount of time to
travel 10 miles downstream as 2 miles
upstream, determine the speed of his kayak in still water.
Identify and organize the information we have already and
define a variable to represent what we’re looking for.
Rate (mph) Time (hours) Distance (miles)
Upstream 2
Downstream 10
29. Problem #3 (Motion)
Rate (mph) Time (hours) Distance (miles)
Upstream 2
Downstream 10
30. Problem #3 (Motion)
Rate (mph) Time (hours) Distance (miles)
Upstream 2
Downstream 10
31. Problem #3 (Motion)
Garth likes to kayak in the river. One day he went
kayaking. He was told that the current of the river was 2
miles per hour. If it took Garth the same amount of time to
travel 10 miles downstream as 2 miles
upstream, determine the speed of his kayak in still water.
Rate (mph) Time (hours) Distance (miles)
Upstream 2
Downstream 10
Because the time column is where all the information
comes together, create an equation about time.
32. Problem #3 (Motion)
Rate (mph) Time (hours) Distance (miles)
Upstream 2
Downstream 10
36. Problem #3 (Motion)
Garth likes to kayak in the river. One day he went
kayaking. He was told that the current of the river was 2
miles per hour. If it took Garth the same amount of time to
travel 10 miles downstream as 2 miles upstream,
determine the speed of his kayak in still water.
Answer the question asked.
37. Problem #3 (Motion)
Garth likes to kayak in the river. One day he went
kayaking. He was told that the current of the river was 2
miles per hour. If it took Garth the same amount of time to
travel 10 miles downstream as 2 miles upstream,
determine the speed of his kayak in still water.
Answer the question asked.
Garth kayaks at a rate of
3 miles per hour in still water.
38. Problem #4 (Motion)
A Subaru Outback travels 1 mile per hour faster
than a Ford Explorer. In the time it takes the
Explorer to travel 368 miles, the Outback travels
376 miles. Find the speed of each vehicle.
39. Problem #4 (Motion)
A Subaru Outback travels 1 mile per hour faster
than a Ford Explorer. In the time it takes the
Explorer to travel 368 miles, the Outback travels
376 miles. Find the speed of each vehicle.
Identify and organize the information we have already and
define a variable to represent what we’re looking for.
40. Problem #4 (Motion)
A Subaru Outback travels 1 mile per hour faster
than a Ford Explorer. In the time it takes the
Explorer to travel 368 miles, the Outback travels
376 miles. Find the speed of each vehicle.
Identify and organize the information we have already and
define a variable to represent what we’re looking for.
Rate Time Distance
Subaru 376
Explorer 368
43. Problem #4 (Motion)
A Subaru Outback travels 1 mile per hour faster
than a Ford Explorer. In the time it takes the
Explorer to travel 368 miles, the Outback travels
376 miles. Find the speed of each vehicle.
Rate Time Distance
Subaru 376
Explorer 368
Because the time column is where all the information
comes together, create an equation about time.
48. Problem #4 (Motion)
A Subaru Outback travels 1 mile per hour faster
than a Ford Explorer. In the time it takes the
Explorer to travel 368 miles, the Outback travels
376 miles. Find the speed of each vehicle.
Answer the question asked.
50. Problem #5 (Motion)
Jamie ran up to the top of the hill at 5 miles per hour. She
then turned around and ran back down the hill at 8 miles
per hour. If her total running time was 26 minutes, how far
is it to the top of the hill?
51. Problem #5 (Motion)
Jamie ran up to the top of the hill at 5 miles per hour. She
then turned around and ran back down the hill at 8 miles
per hour. If her total running time was 26 minutes, how far
is it to the top of the hill?
Identify and organize the information we have already and
define a variable to represent what we’re looking for.
52. Problem #5 (Motion)
Jamie ran up to the top of the hill at 5 miles per hour. She
then turned around and ran back down the hill at 8 miles
per hour. If her total running time was 26 minutes, how far
is it to the top of the hill?
Identify and organize the information we have already and
define a variable to represent what we’re looking for.
Rate (mph) Time (hours) Distance (miles)
Uphill 5
Downhill 8
53. Problem #5 (Motion)
Rate (mph) Time (hours) Distance (miles)
Uphill 5
Downhill 8
54. Problem #5 (Motion)
Rate (mph) Time (hours) Distance (miles)
Uphill 5
Downhill 8
55. Problem #5 (Motion)
Jamie ran up to the top of the hill at 5 miles per hour. She
then turned around and ran back down the hill at 8 miles
per hour. If her total running time was 26 minutes, how far
is it to the top of the hill?
Rate (mph) Time (hours) Distance (miles)
Uphill 5
Downhill 8
Because the time column is where all the information
comes together, create an equation about time.
56. Problem #5 (Motion)
Jamie ran up to the top of the hill at 5 miles per hour. She
then turned around and ran back down the hill at 8 miles
per hour. If her total running time was 26 minutes, how far
is it to the top of the hill?
Rate (mph) Time (hours) Distance (miles)
Uphill 5
Downhill 8
Because the time column is where all the information
comes together, create an equation about time.
Be careful with units on this one!
57. Problem #5 (Motion)
Rate (mph) Time (hours) Distance (miles)
Uphill 5
Downhill 8
61. Problem #5 (Motion)
Jamie ran up to the top of the hill at 5 miles per hour. She
then turned around and ran back down the hill at 8 miles
per hour. If her total running time was 26 minutes, how far
is it to the top of the hill?
Answer the question asked.
62. Problem #5 (Motion)
Jamie ran up to the top of the hill at 5 miles per hour. She
then turned around and ran back down the hill at 8 miles
per hour. If her total running time was 26 minutes, how far
is it to the top of the hill?
Answer the question asked.
The distance to the top of the hill is 1 and 1/3
miles.
64. Problem #6 (Work/Drain)
Adriel can paint the living room in 4 hours, but it
takes Max 6 hours. How long would it take them
to paint the living room if they worked together?
65. Problem #6 (Work/Drain)
Adriel can paint the living room in 4 hours, but it
takes Max 6 hours. How long would it take them
to paint the living room if they worked together?
Identify the information you know already and
assign a variable to the quantity for which you are
being asked.
66. Problem #6 (Work/Drain)
Adriel can paint the living room in 4 hours, but it
takes Max 6 hours. How long would it take them
to paint the living room if they worked together?
Identify the information you know already and
assign a variable to the quantity for which you are
being asked.
Working Rate Time Together Work Done
Alone (hrs) (rooms/hr) (hrs) (rooms)
Adriel 4
Max 6
67. Problem #6 (Work/Drain)
Adriel can paint the living room in 4 hours, but it
takes Max 6 hours. How long would it take them
to paint the living room if they worked together?
Identify the information you know already and
assign a variable to the quantity for which you are
being asked.
Working Rate Time Together Work Done
Alone (hrs) (rooms/hr) (hrs) (rooms)
Adriel 4
Max 6
68. Problem #6 (Work/Drain)
Adriel can paint the living room in 4 hours, but it
takes Max 6 hours. How long would it take them
to paint the living room if they worked together?
Identify the information you know already and
assign a variable to the quantity for which you are
being asked.
Working Rate Time Together Work Done
Alone (hrs) (rooms/hr) (hrs) (rooms)
Adriel 4
Max 6
69. Problem #6 (Work/Drain)
Adriel can paint the living room in 4 hours, but it
takes Max 6 hours. How long would it take them
to paint the living room if they worked together?
Identify the information you know already and
assign a variable to the quantity for which you are
being asked. Work = (rate)(time)
Working Rate Time Together Work Done
Alone (hrs) (rooms/hr) (hrs) (rooms)
Adriel 4
Max 6
70. Problem #6 (Work/Drain)
Adriel can paint the living room in 4 hours, but it
takes Max 6 hours. How long would it take them
to paint the living room if they worked together?
Identify the information you know already and
assign a variable to the quantity for which you are
being asked. Work = (rate)(time)
Working Rate Time Together Work Done
Alone (hrs) (rooms/hr) (hrs) (rooms)
Adriel 4
Max 6
71. Problem #6 (Work/Drain)
Adriel can paint the living room in 4 hours, but it
takes Max 6 hours. How long would it take them to
paint the living room if they worked together?
Working Rate Time Together Work Done
Alone (hrs) (rooms/hr) (hrs) (rooms)
Adriel 4
Max 6
Use this information to create an equation.
72. Problem #6 (Work/Drain)
Working Rate Time Together Work Done
Alone (hrs) (rooms/hr) (hrs) (rooms)
Adriel 4
Max 6
73. Problem #6 (Work/Drain)
Working Rate Time Together Work Done
Alone (hrs) (rooms/hr) (hrs) (rooms)
Adriel 4
Max 6
79. Problem #7 (Work/Drain)
Antonio can paint a fence by himself in 12 hours. With
Carlotta’s help it only takes 5 hours. How long would it
take Carlotta to paint the fence by herself?
80. Problem #7 (Work/Drain)
Antonio can paint a fence by himself in 12 hours. With
Carlotta’s help it only takes 5 hours. How long would it
take Carlotta to paint the fence by herself?
Identify the information you know already and assign a
variable to the quantity for which you are being asked.
81. Problem #7 (Work/Drain)
Antonio can paint a fence by himself in 12 hours. With
Carlotta’s help it only takes 5 hours. How long would it
take Carlotta to paint the fence by herself?
Identify the information you know already and assign a
variable to the quantity for which you are being asked.
Working Rate Time Together Work Done
Alone (hrs) (fences/hr) (hrs) (fences)
Antonio 12
Carlotta
82. Problem #7 (Work/Drain)
Antonio can paint a fence by himself in 12 hours. With
Carlotta’s help it only takes 5 hours. How long would it
take Carlotta to paint the fence by herself?
Identify the information you know already and assign a
variable to the quantity for which you are being asked.
Working Rate Time Together Work Done
Alone (hrs) (fences/hr) (hrs) (fences)
Antonio 12
Carlotta
83. Problem #7 (Work/Drain)
Antonio can paint a fence by himself in 12 hours. With
Carlotta’s help it only takes 5 hours. How long would it
take Carlotta to paint the fence by herself?
Identify the information you know already and assign a
variable to the quantity for which you are being asked.
Working Rate Time Together Work Done
Alone (hrs) (fences/hr) (hrs) (fences)
Antonio 12
Carlotta
84. Problem #7 (Work/Drain)
Antonio can paint a fence by himself in 12 hours. With
Carlotta’s help it only takes 5 hours. How long would it
take Carlotta to paint the fence by herself?
Identify the information you know already and assign a
variable to the quantity for which you are being asked.
Work = (rate)(time)
Working Rate Time Together Work Done
Alone (hrs) (fences/hr) (hrs) (fences)
Antonio 12
Carlotta
85. Problem #7 (Work/Drain)
Antonio can paint a fence by himself in 12 hours. With
Carlotta’s help it only takes 5 hours. How long would it
take Carlotta to paint the fence by herself?
Identify the information you know already and assign a
variable to the quantity for which you are being asked.
Work = (rate)(time)
Working Rate Time Together Work Done
Alone (hrs) (fences/hr) (hrs) (fences)
Antonio 12
Carlotta
86. Problem #7 (Work/Drain)
Antonio can paint a fence by himself in 12 hours. With
Carlotta’s help it only takes 5 hours. How long would it
take Carlotta to paint the fence by herself?
Working Rate Time Together Work Done
Alone (hrs) (fences/hr) (hrs) (fences)
Antonio 12
Carlotta
Use this information to create an equation.
87. Problem #7 (Work/Drain)
Working Rate Time Together Work Done
Alone (hrs) (fences/hr) (hrs) (fences)
Antonio 12
Carlotta
88. Problem #7 (Work/Drain)
Working Rate Time Together Work Done
Alone (hrs) (fences/hr) (hrs) (fences)
Antonio 12
Carlotta
92. Problem #7 (Work/Drain)
Antonio can paint a fence by himself in 12 hours. With
Carlotta’s help it only takes 5 hours. How long would it
take Carlotta to paint the fence by herself?
Answer the question asked.
95. Problem #8 (Work/Drain)
A large, multi-page document can be scanned by one
scanner in 3 hours. After the first scanner has worked for 1
hour, a second scanner is added to the job. It takes an
additional 1.5 hours to scan the rest of the document. How
long would it take to scan the whole document on just the
second scanner?
96. Problem #8 (Work/Drain)
A large, multi-page document can be scanned by one
scanner in 3 hours. After the first scanner has worked for 1
hour, a second scanner is added to the job. It takes an
additional 1.5 hours to scan the rest of the document. How
long would it take to scan the whole document on just the
second scanner?
Identify the information you know already and assign a
variable to the quantity for which you are being asked.
97. Problem #8 (Work/Drain)
A large, multi-page document can be scanned by one
scanner in 3 hours. After the first scanner has worked for 1
hour, a second scanner is added to the job. It takes an
additional 1.5 hours to scan the rest of the document. How
long would it take to scan the whole document on just the
second scanner?
Identify the information you know already and assign a
variable to the quantity for which you are being asked.
Working Rate Time Working Work Done
Alone (hrs) (documents/hr) (hrs) (documents)
First 3
Scanner
Second
Scanner
98. Problem #8 (Work/Drain)
A large, multi-page document can be scanned by one
scanner in 3 hours. After the first scanner has worked for 1
hour, a second scanner is added to the job. It takes an
additional 1.5 hours to scan the rest of the document. How
long would it take to scan the whole document on just the
second scanner?
Identify the information you know already and assign a
variable to the quantity for which you are being asked.
Working Rate Time Working Work Done
Alone (hrs) (documents/hr) (hrs) (documents)
First 3
Scanner
Second
Scanner
99. Problem #8 (Work/Drain)
A large, multi-page document can be scanned by one
scanner in 3 hours. After the first scanner has worked for 1
hour, a second scanner is added to the job. It takes an
additional 1.5 hours to scan the rest of the document. How
long would it take to scan the whole document on just the
second scanner?
Identify the information you know already and assign a
variable to the quantity for which you are being asked.
Working Rate Time Working Work Done
Alone (hrs) (documents/hr) (hrs) (documents)
First 3
Scanner
Second
Scanner
100. Problem #8 (Work/Drain)
A large, multi-page document can be scanned by one
scanner in 3 hours. After the first scanner has worked for 1
hour, a second scanner is added to the job. It takes an
additional 1.5 hours to scan the rest of the document. How
long would it take to scan the whole document on just the
second scanner?
Identify the information you know already and assign a
variable to the quantity for which you are being asked.
Working Rate Time Working Work Done
Alone (hrs) (documents/hr) (hrs) (documents)
First 3
Scanner
Second
Scanner
Work = (rate)(time)
101. Problem #8 (Work/Drain)
A large, multi-page document can be scanned by one
scanner in 3 hours. After the first scanner has worked for 1
hour, a second scanner is added to the job. It takes an
additional 1.5 hours to scan the rest of the document. How
long would it take to scan the whole document on just the
second scanner?
Identify the information you know already and assign a
variable to the quantity for which you are being asked.
Working Rate Time Working Work Done
Alone (hrs) (documents/hr) (hrs) (documents)
First 3
Scanner
Second
Scanner
Work = (rate)(time)
102. Problem #8 (Work/Drain)
A large, multi-page document can be scanned by one
scanner in 3 hours. After the first scanner has worked for 1
hour, a second scanner is added to the job. It takes an
additional 1.5 hours to scan the rest of the document. How
long would it take to scan the whole document on just the
second scanner?
Working Rate Time Working Work Done
Alone (hrs) (documents/hr) (hrs) (documents)
First 3
Scanner
Second
Scanner
Use the information to create an equation.
103. Problem #8 (Work/Drain)
Working Rate Time Working Work Done
Alone (hrs) (documents/hr) (hrs) (documents)
First 3
Scanner
Second
Scanner
107. Problem #8 (Work/Drain)
A large, multi-page document can be scanned by one
scanner in 3 hours. After the first scanner has worked for 1
hour, a second scanner is added to the job. It takes an
additional 1.5 hours to scan the rest of the document. How
long would it take to scan the whole document on just the
second scanner?
Answer the question asked.
108. Problem #8 (Work/Drain)
A large, multi-page document can be scanned by one
scanner in 3 hours. After the first scanner has worked for 1
hour, a second scanner is added to the job. It takes an
additional 1.5 hours to scan the rest of the document. How
long would it take to scan the whole document on just the
second scanner?
Answer the question asked.
9 hours
109. Problem #9 (Work/Drain)
A certain bathtub can be filled in 10 minutes and drained in
6 minutes. If the water is running and the drain is open,
how long will it take for the tub to empty? Assume the tub
starts out full.
110. Problem #9 (Work/Drain)
A certain bathtub can be filled in 10 minutes and drained in
6 minutes. If the water is running and the drain is open,
how long will it take for the tub to empty? Assume the tub
starts out full.
Identify the information you know already and assign a
variable to the quantity for which you are being asked.
111. Problem #9 (Work/Drain)
A certain bathtub can be filled in 10 minutes and drained in
6 minutes. If the water is running and the drain is open,
how long will it take for the tub to empty? Assume the tub
starts out full.
Identify the information you know already and assign a
variable to the quantity for which you are being asked.
Working Rate Time Together Work Done
Alone (mins) (tubs/min) (min) (tubs)
Fill 10
Drain
112. Problem #9 (Work/Drain)
A certain bathtub can be filled in 10 minutes and drained in
6 minutes. If the water is running and the drain is open,
how long will it take for the tub to empty? Assume the tub
starts out full.
Identify the information you know already and assign a
variable to the quantity for which you are being asked.
Working Rate Time Together Work Done
Alone (mins) (tubs/min) (min) (tubs)
Fill 10
Drain
113. Problem #9 (Work/Drain)
A certain bathtub can be filled in 10 minutes and drained in
6 minutes. If the water is running and the drain is open,
how long will it take for the tub to empty? Assume the tub
starts out full.
Identify the information you know already and assign a
variable to the quantity for which you are being asked.
Working Rate Time Together Work Done
Alone (mins) (tubs/min) (min) (tubs)
Fill 10
Drain
114. Problem #9 (Work/Drain)
A certain bathtub can be filled in 10 minutes and drained in
6 minutes. If the water is running and the drain is open,
how long will it take for the tub to empty? Assume the tub
starts out full.
Identify the information you know already and assign a
variable to the quantity for which you are being asked.
Work = (rate)(time)
Working Rate Time Together Work Done
Alone (mins) (tubs/min) (min) (tubs)
Fill 10
Drain
115. Problem #9 (Work/Drain)
A certain bathtub can be filled in 10 minutes and drained in
6 minutes. If the water is running and the drain is open,
how long will it take for the tub to empty? Assume the tub
starts out full.
Identify the information you know already and assign a
variable to the quantity for which you are being asked.
Work = (rate)(time)
Working Rate Time Together Work Done
Alone (mins) (tubs/min) (min) (tubs)
Fill 10
Drain
116. Problem #9 (Work/Drain)
A certain bathtub can be filled in 10 minutes and drained in
6 minutes. If the water is running and the drain is open,
how long will it take for the tub to empty? Assume the tub
starts out full.
Working Alone Rate Time Together Work Done
(mins) (tubs/min) (min) (tubs)
Fill 10
Drain
Use the information to create an equation.
117. Problem #9 (Work/Drain)
Working Alone Rate Time Together Work Done
(mins) (tubs/min) (min) (tubs)
Fill 10
Drain
118. Problem #9 (Work/Drain)
Working Alone Rate Time Together Work Done
(mins) (tubs/min) (min) (tubs)
Fill 10
Drain
