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Section 6.7 Practice Problems

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  • 1. 6.7 Practice Problems Math 98
  • 2. InstructionsEach 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.
  • 4. Number PuzzlesThe first two problems are aboutsolving for numbers.
  • 5. Problem #1Find 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 . 24Define 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 . 24Define 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 . 24Write 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 . 24Write an equation using your variable and the information in the problem as it is written. 1 1 7 x x 2 24
  • 10. Problem #1 (step 3)Solve the equation. 1 1 7 x x 2 24
  • 11. Problem #1 (step 3)Solve the equation. 1 1 7LCD is 24x(x + 2) x x 2 24
  • 12. Problem #1 (step 3)Solve the equation. 1 1 7LCD is 24x(x + 2) x x 2 24
  • 13. Problem #1 (Step 4)
  • 14. Problem #1 (Step 4)6 and 8 are the two consecutive even integers.
  • 15. Problem #2The 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 #2The 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 #2The 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.
  • 18. Problem #2
  • 19. Problem #2
  • 20. Problem #2
  • 21. Problem #2
  • 22. Problem #2
  • 23. Problem #2The 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 #2The 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 ProblemsRemember when you are setting up theproblem that you can define a variable torepresent 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 wentkayaking. He was told that the current of the river was 2miles per hour. If it took Garth the same amount of time totravel 10 miles downstream as 2 milesupstream, determine the speed of his kayak in still water.
  • 27. Problem #3 (Motion)Garth likes to kayak in the river. One day he wentkayaking. He was told that the current of the river was 2miles per hour. If it took Garth the same amount of time totravel 10 miles downstream as 2 milesupstream, determine the speed of his kayak in still water.Identify and organize the information we have already anddefine a variable to represent what we’re looking for.
  • 28. Problem #3 (Motion)Garth likes to kayak in the river. One day he wentkayaking. He was told that the current of the river was 2miles per hour. If it took Garth the same amount of time totravel 10 miles downstream as 2 milesupstream, determine the speed of his kayak in still water.Identify and organize the information we have already anddefine a variable to represent what we’re looking for. Rate (mph) Time (hours) Distance (miles)Upstream 2Downstream 10
  • 29. Problem #3 (Motion) Rate (mph) Time (hours) Distance (miles)Upstream 2Downstream 10
  • 30. Problem #3 (Motion) Rate (mph) Time (hours) Distance (miles)Upstream 2Downstream 10
  • 31. Problem #3 (Motion)Garth likes to kayak in the river. One day he wentkayaking. He was told that the current of the river was 2miles per hour. If it took Garth the same amount of time totravel 10 miles downstream as 2 milesupstream, determine the speed of his kayak in still water. Rate (mph) Time (hours) Distance (miles)Upstream 2Downstream 10Because the time column is where all the informationcomes together, create an equation about time.
  • 32. Problem #3 (Motion) Rate (mph) Time (hours) Distance (miles)Upstream 2Downstream 10
  • 33. Problem #3 (Motion)
  • 34. Problem #3 (Motion)
  • 35. Problem #3 (Motion)
  • 36. Problem #3 (Motion)Garth likes to kayak in the river. One day he wentkayaking. He was told that the current of the river was 2miles per hour. If it took Garth the same amount of time totravel 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 wentkayaking. He was told that the current of the river was 2miles per hour. If it took Garth the same amount of time totravel 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 fasterthan a Ford Explorer. In the time it takes theExplorer to travel 368 miles, the Outback travels376 miles. Find the speed of each vehicle.
  • 39. Problem #4 (Motion)A Subaru Outback travels 1 mile per hour fasterthan a Ford Explorer. In the time it takes theExplorer to travel 368 miles, the Outback travels376 miles. Find the speed of each vehicle.Identify and organize the information we have already anddefine a variable to represent what we’re looking for.
  • 40. Problem #4 (Motion)A Subaru Outback travels 1 mile per hour fasterthan a Ford Explorer. In the time it takes theExplorer to travel 368 miles, the Outback travels376 miles. Find the speed of each vehicle.Identify and organize the information we have already anddefine a variable to represent what we’re looking for. Rate Time Distance Subaru 376 Explorer 368
  • 41. Problem #4 (Motion) Rate Time DistanceSubaru 376Explorer 368
  • 42. Problem #4 (Motion) Rate Time DistanceSubaru 376Explorer 368
  • 43. Problem #4 (Motion)A Subaru Outback travels 1 mile per hour fasterthan a Ford Explorer. In the time it takes theExplorer to travel 368 miles, the Outback travels376 miles. Find the speed of each vehicle. Rate Time Distance Subaru 376 Explorer 368Because the time column is where all the informationcomes together, create an equation about time.
  • 44. Problem #4 (Motion) Rate Time DistanceSubaru 376Explorer 368
  • 45. Problem #4 (Motion)
  • 46. Problem #4 (Motion)
  • 47. Problem #4 (Motion)
  • 48. Problem #4 (Motion)A Subaru Outback travels 1 mile per hour fasterthan a Ford Explorer. In the time it takes theExplorer to travel 368 miles, the Outback travels376 miles. Find the speed of each vehicle.Answer the question asked.
  • 49. Problem #4 (Motion) Rate Subaru Explorer
  • 50. Problem #5 (Motion)Jamie ran up to the top of the hill at 5 miles per hour. Shethen turned around and ran back down the hill at 8 milesper hour. If her total running time was 26 minutes, how faris 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. Shethen turned around and ran back down the hill at 8 milesper hour. If her total running time was 26 minutes, how faris it to the top of the hill?Identify and organize the information we have already anddefine 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. Shethen turned around and ran back down the hill at 8 milesper hour. If her total running time was 26 minutes, how faris it to the top of the hill?Identify and organize the information we have already anddefine a variable to represent what we’re looking for. Rate (mph) Time (hours) Distance (miles)Uphill 5Downhill 8
  • 53. Problem #5 (Motion) Rate (mph) Time (hours) Distance (miles)Uphill 5Downhill 8
  • 54. Problem #5 (Motion) Rate (mph) Time (hours) Distance (miles)Uphill 5Downhill 8
  • 55. Problem #5 (Motion)Jamie ran up to the top of the hill at 5 miles per hour. Shethen turned around and ran back down the hill at 8 milesper hour. If her total running time was 26 minutes, how faris it to the top of the hill? Rate (mph) Time (hours) Distance (miles)Uphill 5Downhill 8Because the time column is where all the informationcomes together, create an equation about time.
  • 56. Problem #5 (Motion)Jamie ran up to the top of the hill at 5 miles per hour. Shethen turned around and ran back down the hill at 8 milesper hour. If her total running time was 26 minutes, how faris it to the top of the hill? Rate (mph) Time (hours) Distance (miles)Uphill 5Downhill 8Because the time column is where all the informationcomes together, create an equation about time. Be careful with units on this one!
  • 57. Problem #5 (Motion) Rate (mph) Time (hours) Distance (miles)Uphill 5Downhill 8
  • 58. Problem #6 (Motion)
  • 59. Problem #6 (Motion)
  • 60. Problem #6 (Motion)
  • 61. Problem #5 (Motion)Jamie ran up to the top of the hill at 5 miles per hour. Shethen turned around and ran back down the hill at 8 milesper hour. If her total running time was 26 minutes, how faris 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. Shethen turned around and ran back down the hill at 8 milesper hour. If her total running time was 26 minutes, how faris it to the top of the hill?Answer the question asked.The distance to the top of the hill is 1 and 1/3miles.
  • 63. Work and Drain Problems
  • 64. Problem #6 (Work/Drain)Adriel can paint the living room in 4 hours, but ittakes Max 6 hours. How long would it take themto paint the living room if they worked together?
  • 65. Problem #6 (Work/Drain)Adriel can paint the living room in 4 hours, but ittakes Max 6 hours. How long would it take themto paint the living room if they worked together?Identify the information you know already andassign a variable to the quantity for which you arebeing asked.
  • 66. Problem #6 (Work/Drain)Adriel can paint the living room in 4 hours, but ittakes Max 6 hours. How long would it take themto paint the living room if they worked together?Identify the information you know already andassign a variable to the quantity for which you arebeing 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 ittakes Max 6 hours. How long would it take themto paint the living room if they worked together?Identify the information you know already andassign a variable to the quantity for which you arebeing 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 ittakes Max 6 hours. How long would it take themto paint the living room if they worked together?Identify the information you know already andassign a variable to the quantity for which you arebeing 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 ittakes Max 6 hours. How long would it take themto paint the living room if they worked together?Identify the information you know already andassign a variable to the quantity for which you arebeing 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 ittakes Max 6 hours. How long would it take themto paint the living room if they worked together?Identify the information you know already andassign a variable to the quantity for which you arebeing 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 ittakes Max 6 hours. How long would it take them topaint the living room if they worked together? Working Rate Time Together Work Done Alone (hrs) (rooms/hr) (hrs) (rooms) Adriel 4 Max 6Use this information to create an equation.
  • 72. Problem #6 (Work/Drain) Working Rate Time Together Work Done Alone (hrs) (rooms/hr) (hrs) (rooms)Adriel 4Max 6
  • 73. Problem #6 (Work/Drain) Working Rate Time Together Work Done Alone (hrs) (rooms/hr) (hrs) (rooms)Adriel 4Max 6
  • 74. Problem #6 (Work/Drain)
  • 75. Problem #6 (Work/Drain)
  • 76. Problem #6 (Work/Drain)
  • 77. Problem #6 (Work/Drain)
  • 78. Problem #6 (Work/Drain)
  • 79. Problem #7 (Work/Drain)Antonio can paint a fence by himself in 12 hours. WithCarlotta’s help it only takes 5 hours. How long would ittake Carlotta to paint the fence by herself?
  • 80. Problem #7 (Work/Drain)Antonio can paint a fence by himself in 12 hours. WithCarlotta’s help it only takes 5 hours. How long would ittake Carlotta to paint the fence by herself?Identify the information you know already and assign avariable to the quantity for which you are being asked.
  • 81. Problem #7 (Work/Drain)Antonio can paint a fence by himself in 12 hours. WithCarlotta’s help it only takes 5 hours. How long would ittake Carlotta to paint the fence by herself?Identify the information you know already and assign avariable 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. WithCarlotta’s help it only takes 5 hours. How long would ittake Carlotta to paint the fence by herself?Identify the information you know already and assign avariable 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. WithCarlotta’s help it only takes 5 hours. How long would ittake Carlotta to paint the fence by herself?Identify the information you know already and assign avariable 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. WithCarlotta’s help it only takes 5 hours. How long would ittake Carlotta to paint the fence by herself?Identify the information you know already and assign avariable 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. WithCarlotta’s help it only takes 5 hours. How long would ittake Carlotta to paint the fence by herself?Identify the information you know already and assign avariable 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. WithCarlotta’s help it only takes 5 hours. How long would ittake Carlotta to paint the fence by herself? Working Rate Time Together Work Done Alone (hrs) (fences/hr) (hrs) (fences)Antonio 12CarlottaUse this information to create an equation.
  • 87. Problem #7 (Work/Drain) Working Rate Time Together Work Done Alone (hrs) (fences/hr) (hrs) (fences)Antonio 12Carlotta
  • 88. Problem #7 (Work/Drain) Working Rate Time Together Work Done Alone (hrs) (fences/hr) (hrs) (fences)Antonio 12Carlotta
  • 89. Problem #7 (Work/Drain)
  • 90. Problem #7 (Work/Drain)
  • 91. Problem #7 (Work/Drain)
  • 92. Problem #7 (Work/Drain)Antonio can paint a fence by himself in 12 hours. WithCarlotta’s help it only takes 5 hours. How long would ittake Carlotta to paint the fence by herself? Answer the question asked.
  • 93. Problem #7 (Work/Drain)
  • 94. Problem #7 (Work/Drain)
  • 95. Problem #8 (Work/Drain)A large, multi-page document can be scanned by onescanner in 3 hours. After the first scanner has worked for 1hour, a second scanner is added to the job. It takes anadditional 1.5 hours to scan the rest of the document. Howlong would it take to scan the whole document on just thesecond scanner?
  • 96. Problem #8 (Work/Drain)A large, multi-page document can be scanned by onescanner in 3 hours. After the first scanner has worked for 1hour, a second scanner is added to the job. It takes anadditional 1.5 hours to scan the rest of the document. Howlong would it take to scan the whole document on just thesecond scanner?Identify the information you know already and assign avariable 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 3ScannerSecondScanner
  • 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 3ScannerSecondScanner
  • 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 3ScannerSecondScanner
  • 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 3ScannerSecondScanner 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 3ScannerSecondScanner 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 3ScannerSecondScanner 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 3ScannerSecondScanner
  • 104. Problem #8 (Work/Drain)
  • 105. Problem #8 (Work/Drain)
  • 106. Problem #8 (Work/Drain)
  • 107. Problem #8 (Work/Drain)A large, multi-page document can be scanned by onescanner in 3 hours. After the first scanner has worked for 1hour, a second scanner is added to the job. It takes anadditional 1.5 hours to scan the rest of the document. Howlong would it take to scan the whole document on just thesecond scanner?Answer the question asked.
  • 108. Problem #8 (Work/Drain)A large, multi-page document can be scanned by onescanner in 3 hours. After the first scanner has worked for 1hour, a second scanner is added to the job. It takes anadditional 1.5 hours to scan the rest of the document. Howlong would it take to scan the whole document on just thesecond scanner?Answer the question asked. 9 hours
  • 109. Problem #9 (Work/Drain)A certain bathtub can be filled in 10 minutes and drained in6 minutes. If the water is running and the drain is open,how long will it take for the tub to empty? Assume the tubstarts out full.
  • 110. Problem #9 (Work/Drain)A certain bathtub can be filled in 10 minutes and drained in6 minutes. If the water is running and the drain is open,how long will it take for the tub to empty? Assume the tubstarts out full.Identify the information you know already and assign avariable 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 in6 minutes. If the water is running and the drain is open,how long will it take for the tub to empty? Assume the tubstarts out full.Identify the information you know already and assign avariable 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 in6 minutes. If the water is running and the drain is open,how long will it take for the tub to empty? Assume the tubstarts out full.Identify the information you know already and assign avariable 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 in6 minutes. If the water is running and the drain is open,how long will it take for the tub to empty? Assume the tubstarts out full.Identify the information you know already and assign avariable 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 in6 minutes. If the water is running and the drain is open,how long will it take for the tub to empty? Assume the tubstarts out full.Identify the information you know already and assign avariable 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 in6 minutes. If the water is running and the drain is open,how long will it take for the tub to empty? Assume the tubstarts out full.Identify the information you know already and assign avariable 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 in6 minutes. If the water is running and the drain is open,how long will it take for the tub to empty? Assume the tubstarts out full. Working Alone Rate Time Together Work Done (mins) (tubs/min) (min) (tubs) Fill 10 DrainUse the information to create an equation.
  • 117. Problem #9 (Work/Drain) Working Alone Rate Time Together Work Done (mins) (tubs/min) (min) (tubs)Fill 10Drain
  • 118. Problem #9 (Work/Drain) Working Alone Rate Time Together Work Done (mins) (tubs/min) (min) (tubs)Fill 10Drain
  • 119. Problem #9 (Work/Drain)
  • 120. Problem #9 (Work/Drain)
  • 121. Problem #9 (Work/Drain)
  • 122. Problem #9 (Work/Drain)
  • 123. Problem #9 (Work/Drain)

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