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- 1. Rational Equations Word-Problems
- 2. Rational Equations Word-Problems Following are some basic applications of rational equations.
- 3. Rational Equations Word-Problems Problems from the Multiplication–Division Operations Following are some basic applications of rational equations.
- 4. Rational Equations Word-Problems Problems from the Multiplication–Division Operations Following are some basic applications of rational equations. All the multiplicative formulas of the form AB = C may be written as A = .C B
- 5. Rational Equations Word-Problems Problems from the Multiplication–Division Operations Following are some basic applications of rational equations. All the multiplicative formulas of the form AB = C may be written as A = . This divisional form leads to rational equations.C B
- 6. Rational Equations Word-Problems Problems from the Multiplication–Division Operations Following are some basic applications of rational equations. All the multiplicative formulas of the form AB = C may be written as A = . This divisional form leads to rational equations. The calculation of “per unit” is a good example: C B
- 7. Rational Equations Word-Problems Problems from the Multiplication–Division Operations Following are some basic applications of rational equations. All the multiplicative formulas of the form AB = C may be written as A = . This divisional form leads to rational equations. Total amount Number of units The calculation of “per unit” is a good example: Per unit amount = C B
- 8. Rational Equations Word-Problems Problems from the Multiplication–Division Operations Following are some basic applications of rational equations. All the multiplicative formulas of the form AB = C may be written as A = . This divisional form leads to rational equations. Total amount Number of units Cost per Unit The calculation of “per unit” is a good example: Per unit amount = C B
- 9. Rational Equations Word-Problems Problems from the Multiplication–Division Operations Following are some basic applications of rational equations. All the multiplicative formulas of the form AB = C may be written as A = . This divisional form leads to rational equations. Total amount Number of units Cost per Unit The calculation of “per unit” is a good example: Per unit amount = C B Total cost Number of units Per unit cost =
- 10. Rational Equations Word-Problems Problems from the Multiplication–Division Operations Following are some basic applications of rational equations. All the multiplicative formulas of the form AB = C may be written as A = . This divisional form leads to rational equations. Total amount Number of units For example, a group of 5 people rent a taxi that cost $20 so each person’s share is 20/5 or $4 per person. Cost per Unit The calculation of “per unit” is a good example: Per unit amount = C B Total cost Number of units Per unit cost =
- 11. Rational Equations Word-Problems Problems from the Multiplication–Division Operations Following are some basic applications of rational equations. All the multiplicative formulas of the form AB = C may be written as A = . This divisional form leads to rational equations. Total amount Number of units For example, a group of 5 people rent a taxi that cost $20 so each person’s share is 20/5 or $4 per person. However if one person backs out, then the cost goes up to s = 20/4 = $5 per person. Cost per Unit The calculation of “per unit” is a good example: Per unit amount = C B Total cost Number of units Per unit cost =
- 12. Rational Equations Word-Problems Problems from the Multiplication–Division Operations Following are some basic applications of rational equations. All the multiplicative formulas of the form AB = C may be written as A = . This divisional form leads to rational equations. Total amount Number of units For example, a group of 5 people rent a taxi that cost $20 so each person’s share is 20/5 or $4 per person. However if one person backs out, then the cost goes up to s = 20/4 = $5 per person. Each remaining person has to pay $1 more. Cost per Unit The calculation of “per unit” is a good example: Per unit amount = C B Total cost Number of units Per unit cost =
- 13. Rational Equations Word-Problems Problems from the Multiplication–Division Operations Following are some basic applications of rational equations. All the multiplicative formulas of the form AB = C may be written as A = . This divisional form leads to rational equations. Total amount Number of units For example, a group of 5 people rent a taxi that cost $20 so each person’s share is 20/5 or $4 per person. However if one person backs out, then the cost goes up to s = 20/4 = $5 per person. Each remaining person has to pay $1 more. Let’s formulate this as a word problem and solve it using rational equations. Cost per Unit The calculation of “per unit” is a good example: Per unit amount = C B Total cost Number of units Per unit cost =
- 14. Rational Equations Word-Problems A table is useful in organizing repeated calculations for different inputs.
- 15. Rational Equations Word-Problems Example A. A taxi is rented by a group of x people for $20 and the cost is shared equally. If there is one less person in the group then each of the remaining people has to pay $1 more. What is x? A table is useful in organizing repeated calculations for different inputs.
- 16. Rational Equations Word-Problems Example A. A taxi is rented by a group of x people for $20 and the cost is shared equally. If there is one less person in the group then each of the remaining people has to pay $1 more. What is x? A table is useful in organizing repeated calculations for different inputs. Total Cost No. of People Cost per Person = 20 20 Total Cost No. of people
- 17. Rational Equations Word-Problems Example A. A taxi is rented by a group of x people for $20 and the cost is shared equally. If there is one less person in the group then each of the remaining people has to pay $1 more. What is x? A table is useful in organizing repeated calculations for different inputs. Total Cost No. of People Cost per Person = 20 x 20 (x – 1) Total Cost No. of people
- 18. Rational Equations Word-Problems Example A. A taxi is rented by a group of x people for $20 and the cost is shared equally. If there is one less person in the group then each of the remaining people has to pay $1 more. What is x? A table is useful in organizing repeated calculations for different inputs. Total Cost No. of People Cost per Person = 20 x 20 (x – 1) Total Cost No. of people 20 x 20 (x – 1)
- 19. Rational Equations Word-Problems Example A. A taxi is rented by a group of x people for $20 and the cost is shared equally. If there is one less person in the group then each of the remaining people has to pay $1 more. What is x? A table is useful in organizing repeated calculations for different inputs. Total Cost No. of People Cost per Person = 20 x 20 (x – 1) Total Cost No. of people 20 x 20 (x – 1) Let’s compare the two different per/person costs. 20 (x – 1) 20 x
- 20. Rational Equations Word-Problems Example A. A taxi is rented by a group of x people for $20 and the cost is shared equally. If there is one less person in the group then each of the remaining people has to pay $1 more. What is x? A table is useful in organizing repeated calculations for different inputs. Total Cost No. of People Cost per Person = 20 x 20 (x – 1) Total Cost No. of people 20 x 20 (x – 1) Let’s compare the two different per/person costs. cost more cost less 20 (x – 1) 20 x
- 21. Rational Equations Word-Problems Example A. A taxi is rented by a group of x people for $20 and the cost is shared equally. If there is one less person in the group then each of the remaining people has to pay $1 more. What is x? A table is useful in organizing repeated calculations for different inputs. Total Cost No. of People Cost per Person = 20 x 20 (x – 1) Total Cost No. of people 20 x 20 (x – 1) Let’s compare the two different per/person costs. cost more cost less ($1 more) 20 (x – 1) 20 x
- 22. Rational Equations Word-Problems Example A. A taxi is rented by a group of x people for $20 and the cost is shared equally. If there is one less person in the group then each of the remaining people has to pay $1 more. What is x? Total Cost No. of People Cost per Person = 20 x 20 (x – 1) Total Cost No. of people 20 x 20 (x – 1) 20 (x – 1) 20 x Let’s compare the two different per/person costs. – = 1 cost more cost less ($1 more) A table is useful in organizing repeated calculations for different inputs.
- 23. Rational Equations Word-Problems Example A. A taxi is rented by a group of x people for $20 and the cost is shared equally. If there is one less person in the group then each of the remaining people has to pay $1 more. What is x? Total Cost No. of People Cost per Person = 20 x 20 (x – 1) Total Cost No. of people 20 x 20 (x – 1) 20 (x – 1) Let’s compare the two different per/person costs. – = 1[ ] x (x – 1) clear the denominators by LCM A table is useful in organizing repeated calculations for different inputs. 20 x
- 24. Rational Equations Word-Problems Example A. A taxi is rented by a group of x people for $20 and the cost is shared equally. If there is one less person in the group then each of the remaining people has to pay $1 more. What is x? Total Cost No. of People Cost per Person = 20 x 20 (x – 1) Total Cost No. of people 20 x 20 (x – 1) 20 (x – 1) 20 x Let’s compare the two different per/person costs. – = 1[ ] x (x – 1) clear the denominators by LCM x (x – 1) x (x – 1) A table is useful in organizing repeated calculations for different inputs.
- 25. Rational Equations Word-Problems Example A. A taxi is rented by a group of x people for $20 and the cost is shared equally. If there is one less person in the group then each of the remaining people has to pay $1 more. What is x? Total Cost No. of People Cost per Person = 20 x 20 (x – 1) Total Cost No. of people 20 x 20 (x – 1) 20 (x – 1) Let’s compare the two different per/person costs. = 1[ ] x (x – 1) clear the denominators by LCM x (x – 1) x (x – 1) 20x – 20(x – 1) = x(x – 1) A table is useful in organizing repeated calculations for different inputs. 20 x –
- 26. Rational Equations Word-Problems Example A. A taxi is rented by a group of x people for $20 and the cost is shared equally. If there is one less person in the group then each of the remaining people has to pay $1 more. What is x? Total Cost No. of People Cost per Person = 20 x 20 (x – 1) Total Cost No. of people 20 x 20 (x – 1) 20 (x – 1) Let’s compare the two different per/person costs. = 1[ ] x (x – 1) clear the denominators by LCM x (x – 1) x (x – 1) 20x – 20(x – 1) = x(x – 1) 20x – 20x + 20 = x2 – x A table is useful in organizing repeated calculations for different inputs. 20 x –
- 27. Rational Equations Word-Problems Example A. A taxi is rented by a group of x people for $20 and the cost is shared equally. If there is one less person in the group then each of the remaining people has to pay $1 more. What is x? Total Cost No. of People Cost per Person = 20 x 20 (x – 1) Total Cost No. of people 20 x 20 (x – 1) 20 (x – 1) Let’s compare the two different per/person costs. = 1[ ] x (x – 1) clear the denominators by LCM x (x – 1) x (x – 1) 20x – 20(x – 1) = x(x – 1) A table is useful in organizing repeated calculations for different inputs. 0 = x2 – x – 20 set one side as 0 20 x – 20x – 20x + 20 = x2 – x
- 28. Rational Equations Word-Problems 0 = x2 – x – 20
- 29. Rational Equations Word-Problems 0 = x2 – x – 20 0 = (x – 5)(x + 4)
- 30. Rational Equations Word-Problems 0 = x2 – x – 20 0 = (x – 5)(x + 4) x = 5 or x = –4.
- 31. Rational Equations Word-Problems 0 = x2 – x – 20 0 = (x – 5)(x + 4) x = 5 or x = –4. Hence there are 5 people.
- 32. Rational Equations Word-Problems Rate–Time–Distance 0 = x2 – x – 20 0 = (x – 5)(x + 4) x = 5 or x = –4. Hence there are 5 people.
- 33. Rational Equations Word-Problems Let R = rate (mph), T = time (hours) and D = Distance (m) then RT = D or that T = . .D R Rate–Time–Distance 0 = x2 – x – 20 0 = (x – 5)(x + 4) x = 5 or x = –4. Hence there are 5 people.
- 34. Rational Equations Word-Problems Let R = rate (mph), T = time (hours) and D = Distance (m) then RT = D or that T = . .D R We hiked a 6–mile trail from A to B at a rate of 3 mph so the trip took 6/3 = 2 hours. Rate–Time–Distance 0 = x2 – x – 20 0 = (x – 5)(x + 4) x = 5 or x = –4. Hence there are 5 people.
- 35. Rational Equations Word-Problems Let R = rate (mph), T = time (hours) and D = Distance (m) then RT = D or that T = . .D R We hiked a 6–mile trail from A to B at a rate of 3 mph so the trip took 6/3 = 2 hours. Going back from B to A our rate was 2 mph so the return took 6/2 = 3 hours. Rate–Time–Distance 0 = x2 – x – 20 0 = (x – 5)(x + 4) x = 5 or x = –4. Hence there are 5 people.
- 36. Rational Equations Word-Problems Let R = rate (mph), T = time (hours) and D = Distance (m) then RT = D or that T = . .D R We hiked a 6–mile trail from A to B at a rate of 3 mph so the trip took 6/3 = 2 hours. Going back from B to A our rate was 2 mph so the return took 6/2 = 3 hours. In a table, Distance (m) Rate (mph) Time = D/R (hr) Going Return Rate–Time–Distance 0 = x2 – x – 20 0 = (x – 5)(x + 4) x = 5 or x = –4. Hence there are 5 people.
- 37. Rational Equations Word-Problems Let R = rate (mph), T = time (hours) and D = Distance (m) then RT = D or that T = . .D R We hiked a 6–mile trail from A to B at a rate of 3 mph so the trip took 6/3 = 2 hours. Going back from B to A our rate was 2 mph so the return took 6/2 = 3 hours. In a table, Distance (m) Rate (mph) Time = D/R (hr) Going 6 3 2 Return Rate–Time–Distance 0 = x2 – x – 20 0 = (x – 5)(x + 4) x = 5 or x = –4. Hence there are 5 people.
- 38. Rational Equations Word-Problems Let R = rate (mph), T = time (hours) and D = Distance (m) then RT = D or that T = . .D R We hiked a 6–mile trail from A to B at a rate of 3 mph so the trip took 6/3 = 2 hours. Going back from B to A our rate was 2 mph so the return took 6/2 = 3 hours. In a table, Distance (m) Rate (mph) Time = D/R (hr) Going 6 3 2 Return 6 2 3 Rate–Time–Distance 0 = x2 – x – 20 0 = (x – 5)(x + 4) x = 5 or x = –4. Hence there are 5 people.
- 39. Rational Equations Word-Problems Let R = rate (mph), T = time (hours) and D = Distance (m) then RT = D or that T = . .D R We hiked a 6–mile trail from A to B at a rate of 3 mph so the trip took 6/3 = 2 hours. Going back from B to A our rate was 2 mph so the return took 6/2 = 3 hours. In a table, Distance (m) Rate (mph) Time = D/R (hr) Going 6 3 2 Return 6 2 3 Let’s turn this example into a rational–equation word problem. 0 = x2 – x – 20 0 = (x – 5)(x + 4) x = 5 or x = –4. Rate–Time–Distance Hence there are 5 people.
- 40. Rational Equations Word-Problems Example B. We hiked 6 miles from A to B at a rate of x mph. For the return trip our rate was 1 mph faster. It took us 5 hours for the entire round trip. What is x?
- 41. Rational Equations Word-Problems Example B. We hiked 6 miles from A to B at a rate of x mph. For the return trip our rate was 1 mph faster. It took us 5 hours for the entire round trip. What is x? The rate of the return is “1 mph faster then x” so it’s (x + 1).
- 42. Rational Equations Word-Problems Example B. We hiked 6 miles from A to B at a rate of x mph. For the return trip our rate was 1 mph faster. It took us 5 hours for the entire round trip. What is x? D = Distance (m) R = Rate (mph) Time = D/R (hr) Go Return The rate of the return is “1 mph faster then x” so it’s (x + 1). Put these into the table
- 43. Rational Equations Word-Problems Example B. We hiked 6 miles from A to B at a rate of x mph. For the return trip our rate was 1 mph faster. It took us 5 hours for the entire round trip. What is x? D = Distance (m) R = Rate (mph) Time = D/R (hr) Go 6 Return 6 The rate of the return is “1 mph faster then x” so it’s (x + 1). Put these into the table
- 44. Rational Equations Word-Problems Example B. We hiked 6 miles from A to B at a rate of x mph. For the return trip our rate was 1 mph faster. It took us 5 hours for the entire round trip. What is x? D = Distance (m) R = Rate (mph) Time = D/R (hr) Go 6 x Return 6 The rate of the return is “1 mph faster then x” so it’s (x + 1). Put these into the table
- 45. Rational Equations Word-Problems Example B. We hiked 6 miles from A to B at a rate of x mph. For the return trip our rate was 1 mph faster. It took us 5 hours for the entire round trip. What is x? D = Distance (m) R = Rate (mph) Time = D/R (hr) Go 6 x 6/x Return 6 The rate of the return is “1 mph faster then x” so it’s (x + 1). Put these into the table
- 46. Rational Equations Word-Problems Example B. We hiked 6 miles from A to B at a rate of x mph. For the return trip our rate was 1 mph faster. It took us 5 hours for the entire round trip. What is x? D = Distance (m) R = Rate (mph) Time = D/R (hr) Go 6 x 6/x Return 6 x + 1 The rate of the return is “1 mph faster then x” so it’s (x + 1). Put these into the table
- 47. Rational Equations Word-Problems Example B. We hiked 6 miles from A to B at a rate of x mph. For the return trip our rate was 1 mph faster. It took us 5 hours for the entire round trip. What is x? D = Distance (m) R = Rate (mph) Time = D/R (hr) Go 6 x 6/x Return 6 x + 1 6/(x + 1) The rate of the return is “1 mph faster then x” so it’s (x + 1). Put these into the table
- 48. Rational Equations Word-Problems Example B. We hiked 6 miles from A to B at a rate of x mph. For the return trip our rate was 1 mph faster. It took us 5 hours for the entire round trip. What is x? D = Distance (m) R = Rate (mph) Time = D/R (hr) Go 6 x 6/x Return 6 x + 1 6/(x + 1) The rate of the return is “1 mph faster then x” so it’s (x + 1). Put these into the table Now we use the information about the times.
- 49. Rational Equations Word-Problems Example B. We hiked 6 miles from A to B at a rate of x mph. For the return trip our rate was 1 mph faster. It took us 5 hours for the entire round trip. What is x? D = Distance (m) R = Rate (mph) Time = D/R (hr) Go 6 x 6/x Return 6 x + 1 6/(x + 1) The rate of the return is “1 mph faster then x” so it’s (x + 1). Put these into the table 6 (x + 1) 6 x number of hrs for going Now we use the information about the times. number of hrs for returning
- 50. Rational Equations Word-Problems Example B. We hiked 6 miles from A to B at a rate of x mph. For the return trip our rate was 1 mph faster. It took us 5 hours for the entire round trip. What is x? D = Distance (m) R = Rate (mph) Time = D/R (hr) Go 6 x 6/x Return 6 x + 1 6/(x + 1) The rate of the return is “1 mph faster then x” so it’s (x + 1). Put these into the table 6 (x + 1) 6 x Now we use the information about the times. (total trip 5 hrs)number of hrs for going number of hrs for returning
- 51. Rational Equations Word-Problems Example B. We hiked 6 miles from A to B at a rate of x mph. For the return trip our rate was 1 mph faster. It took us 5 hours for the entire round trip. What is x? D = Distance (m) R = Rate (mph) Time = D/R (hr) Go 6 x 6/x Return 6 x + 1 6/(x + 1) The rate of the return is “1 mph faster then x” so it’s (x + 1). Put these into the table 6 (x + 1) 6 x Now we use the information about the times. + = 5 (total trip 5 hrs)number of hrs for going number of hrs for returning
- 52. Rational Equations Word-Problems Example B. We hiked 6 miles from A to B at a rate of x mph. For the return trip our rate was 1 mph faster. It took us 5 hours for the entire round trip. What is x? D = Distance (m) R = Rate (mph) Time = D/R (hr) Go 6 x 6/x Return 6 x + 1 6/(x + 1) The rate of the return is “1 mph faster then x” so it’s (x + 1). Put these into the table 6 (x + 1) 6 x Now we use the information about the times. + = 5 use the cross multiplication method
- 53. Rational Equations Word-Problems Example B. We hiked 6 miles from A to B at a rate of x mph. For the return trip our rate was 1 mph faster. It took us 5 hours for the entire round trip. What is x? D = Distance (m) R = Rate (mph) Time = D/R (hr) Go 6 x 6/x Return 6 x + 1 6/(x + 1) The rate of the return is “1 mph faster then x” so it’s (x + 1). Put these into the table 6 (x + 1) 6 x Now we use the information about the times. + = 5 6(x + 1) + 6x x(x + 1) = 5 use the cross multiplication method
- 54. Rational Equations Word-Problems Example B. We hiked 6 miles from A to B at a rate of x mph. For the return trip our rate was 1 mph faster. It took us 5 hours for the entire round trip. What is x? D = Distance (m) R = Rate (mph) Time = D/R (hr) Go 6 x 6/x Return 6 x + 1 6/(x + 1) The rate of the return is “1 mph faster then x” so it’s (x + 1). Put these into the table 6 (x + 1) 6 x Now we use the information about the times. + = 5 6(x + 1) + 6x x(x + 1) = 5 5 1 =x(x + 1) 12x + 6 use the cross multiplication method
- 55. Rational Equations Word-Problems Example B. We hiked 6 miles from A to B at a rate of x mph. For the return trip our rate was 1 mph faster. It took us 5 hours for the entire round trip. What is x? D = Distance (m) R = Rate (mph) Time = D/R (hr) Go 6 x 6/x Return 6 x + 1 6/(x + 1) The rate of the return is “1 mph faster then x” so it’s (x + 1). Put these into the table 6 (x + 1) 6 x Now we use the information about the times. + = 5 6(x + 1) + 6x x(x + 1) = 5 5 1 =x(x + 1) 12x + 6 Cross again. You finish it. use the cross multiplication method
- 56. Rational Equations Word-Problems Example B. We hiked 6 miles from A to B at a rate of x mph. For the return trip our rate was 1 mph faster. It took us 5 hours for the entire round trip. What is x? D = Distance (m) R = Rate (mph) Time = D/R (hr) Go 6 x 6/x Return 6 x + 1 6/(x + 1) The rate of the return is “1 mph faster then x” so it’s (x + 1). Put these into the table 6 (x + 1) 6 x Now we use the information about the times. + = 5 6(x + 1) + 6x x(x + 1) = 5 5 1 =x(x + 1) 12x + 6 Cross again. You finish it. (Remember that x = 2 mph.) use the cross multiplication method
- 57. Rational Equations Word-Problems Job–Rates
- 58. Rational Equations Word-Problems Job–Rates If we can eat 6 pizzas in 2 hours then our pizza–eating rate =
- 59. Rational Equations Word-Problems Job–Rates If we can eat 6 pizzas in 2 hours then our 6 pizzas 2 hours pizza–eating rate =
- 60. Rational Equations Word-Problems Job–Rates If we can eat 6 pizzas in 2 hours then our 6 pizzas 2 hours pizza–eating rate = = 3 (piz/hr)
- 61. Rational Equations Word-Problems Job–Rates If we can eat 6 pizzas in 2 hours then our 6 pizzas 2 hours pizza–eating rate = = 3 (piz/hr) If we can eat 6 pizzas in 2 hours then our pizza–eating rate =
- 62. Rational Equations Word-Problems Job–Rates If we can eat 6 pizzas in 2 hours then our 6 pizzas 2 hours pizza–eating rate = = 3 (piz/hr) If we can eat 6 pizzas in 2 hours then our 2 pizzas 6 hours pizza–eating rate =
- 63. Rational Equations Word-Problems Job–Rates If we can eat 6 pizzas in 2 hours then our 6 pizzas 2 hours pizza–eating rate = = 3 (piz/hr) If we can eat 6 pizzas in 2 hours then our 2 pizzas 6 hours pizza–eating rate = = (piz/hr)1 3
- 64. Rational Equations Word-Problems Job–Rates If we can eat 6 pizzas in 2 hours then our 6 pizzas 2 hours pizza–eating rate = = 3 (piz/hr) If we can eat 6 pizzas in 2 hours then our 2 pizzas 6 hours pizza–eating rate = = (piz/hr)1 3 These types of rates (in per unit of time) are called job–rates.
- 65. Rational Equations Word-Problems Job–Rates If we can eat 6 pizzas in 2 hours then our 6 pizzas 2 hours pizza–eating rate = = 3 (piz/hr) If we can eat 6 pizzas in 2 hours then our 2 pizzas 6 hours pizza–eating rate = = (piz/hr)1 3 These types of rates (in per unit of time) are called job–rates. We note the following about job–rates.
- 66. Rational Equations Word-Problems Job–Rates If we can eat 6 pizzas in 2 hours then our 6 pizzas 2 hours pizza–eating rate = = 3 (piz/hr) If we can eat 6 pizzas in 2 hours then our 2 pizzas 6 hours pizza–eating rate = = (piz/hr)1 3 These types of rates (in per unit of time) are called job–rates. We note the following about job–rates. * In all such problems, a complete job to be done may be viewed as a pizza needing to be eaten and it’s set to be 1.
- 67. Rational Equations Word-Problems Job–Rates If we can eat 6 pizzas in 2 hours then our 6 pizzas 2 hours pizza–eating rate = = 3 (piz/hr) If we can eat 6 pizzas in 2 hours then our 2 pizzas 6 hours pizza–eating rate = = (piz/hr)1 3 These types of rates (in per unit of time) are called job–rates. We note the following about job–rates. * In all such problems, a complete job to be done may be viewed as a pizza needing to be eaten and it’s set to be 1. Hence if it takes 3 hrs to paint a room, to mow a lawn, or to fill a pool, then the job–rate for each is 1 job 3 hours
- 68. Rational Equations Word-Problems Job–Rates If we can eat 6 pizzas in 2 hours then our 6 pizzas 2 hours pizza–eating rate = = 3 (piz/hr) If we can eat 6 pizzas in 2 hours then our 2 pizzas 6 hours pizza–eating rate = = (piz/hr)1 3 These types of rates (in per unit of time) are called job–rates. We note the following about job–rates. * In all such problems, a complete job to be done may be viewed as a pizza needing to be eaten and it’s set to be 1. Hence if it takes 3 hrs to paint a room, to mow a lawn, or to fill a pool, then the job–rate for each is 1 job 3 hours = (room), (lawn), or (pool) / hr1 3
- 69. Rational Equations Word-Problems * The job–rate may be computed even if only a portion of the job is completed.
- 70. Rational Equations Word-Problems * The job–rate may be computed even if only a portion of the job is completed. For example, if it took us 6 hrs to paint 2/3 of the room then the job–rate is 2/3 room 6 hours
- 71. Rational Equations Word-Problems * The job–rate may be computed even if only a portion of the job is completed. For example, if it took us 6 hrs to paint 2/3 of the room then the job–rate is 2/3 room 6 hours = 2 3 1 6 *
- 72. Rational Equations Word-Problems * The job–rate may be computed even if only a portion of the job is completed. For example, if it took us 6 hrs to paint 2/3 of the room then the job–rate is 2/3 room 6 hours = (room/hr)1 9 2 3 1 6 * = 3
- 73. Rational Equations Word-Problems * The job–rate may be computed even if only a portion of the job is completed. For example, if it took us 6 hrs to paint 2/3 of the room then the job–rate is 2/3 room 6 hours = (room/hr)1 9 2 3 1 6 * = * The reciprocal of a job–rate is the amount of time it would take to complete one job. 3
- 74. Rational Equations Word-Problems * The job–rate may be computed even if only a portion of the job is completed. For example, if it took us 6 hrs to paint 2/3 of the room then the job–rate is 2/3 room 6 hours = (room/hr)1 9 2 3 1 6 * = * The reciprocal of a job–rate is the amount of time it would take to complete one job. (lawn/hr), 1 9 Hence if the job–rate is 9 1 then it would take = 9 hrs to complete the entire lawn. 3
- 75. Rational Equations Word-Problems * The job–rate may be computed even if only a portion of the job is completed. For example, if it took us 6 hrs to paint 2/3 of the room then the job–rate is 2/3 room 6 hours = (room/hr)1 9 2 3 1 6 * = * The reciprocal of a job–rate is the amount of time it would take to complete one job. (lawn/hr), 1 9 Hence if the job–rate is 9 1 then it would take = 9 hrs to complete the entire lawn. Example C. We can paint 3/4 of a wall in 4 ½ hrs, what is the job–rate? How long would it take to paint the entire wall? 3
- 76. Rational Equations Word-Problems * The job–rate may be computed even if only a portion of the job is completed. For example, if it took us 6 hrs to paint 2/3 of the room then the job–rate is 2/3 room 6 hours = (room/hr)1 9 2 3 1 6 * = * The reciprocal of a job–rate is the amount of time it would take to complete one job. (lawn/hr), 1 9 Hence if the job–rate is 9 1 then it would take = 9 hrs to complete the entire lawn. Example C. We can paint 3/4 of a wall in 4 ½ hrs, what is the job–rate? How long would it take to paint the entire wall? The job–rate is 3/4 4½ 3
- 77. Rational Equations Word-Problems * The job–rate may be computed even if only a portion of the job is completed. For example, if it took us 6 hrs to paint 2/3 of the room then the job–rate is 2/3 room 6 hours = (room/hr)1 9 2 3 1 6 * = * The reciprocal of a job–rate is the amount of time it would take to complete one job. (lawn/hr), 1 9 Hence if the job–rate is 9 1 then it would take = 9 hrs to complete the entire lawn. Example C. We can paint 3/4 of a wall in 4 ½ hrs, what is the job–rate? How long would it take to paint the entire wall? The job–rate is 3/4 4½ = 3/4 9/2 = 3 4 2 9 * 3
- 78. Rational Equations Word-Problems * The job–rate may be computed even if only a portion of the job is completed. For example, if it took us 6 hrs to paint 2/3 of the room then the job–rate is 2/3 room 6 hours = (room/hr)1 9 2 3 1 6 * = * The reciprocal of a job–rate is the amount of time it would take to complete one job. (lawn/hr), 1 9 Hence if the job–rate is 9 1 then it would take = 9 hrs to complete the entire lawn. Example C. We can paint 3/4 of a wall in 4 ½ hrs, what is the job–rate? How long would it take to paint the entire wall? The job–rate is 3/4 4½ (wall / hr)= 1 6= 3/4 9/2 = 3 4 2 9 * 2 3 3
- 79. Rational Equations Word-Problems * The job–rate may be computed even if only a portion of the job is completed. For example, if it took us 6 hrs to paint 2/3 of the room then the job–rate is 2/3 room 6 hours = (room/hr)1 9 2 3 1 6 * = * The reciprocal of a job–rate is the amount of time it would take to complete one job. (lawn/hr), 1 9 Hence if the job–rate is 9 1 then it would take = 9 hrs to complete the entire lawn. Example C. We can paint 3/4 of a wall in 4 ½ hrs, what is the job–rate? How long would it take to paint the entire wall? The job–rate is 3/4 4½ (wall / hr)= 1 6= 3/4 9/2 = 3 4 2 9 * 2 3 The reciprocal of this job–rate is 6 (hr / wall) or it would take 6 hours to paint the entire wall. 3
- 80. Rational Equations Word-Problems Example D. Pipe A can fill a full tank of water in 3 hours, pipe B can fill ¾ of the (same) tank of water in 1½ hr. a. What is unit of the fill–rate and what is the rate of each pipe?
- 81. Rational Equations Word-Problems Example D. Pipe A can fill a full tank of water in 3 hours, pipe B can fill ¾ of the (same) tank of water in 1½ hr. a. What is unit of the fill–rate and what is the rate of each pipe? Pipes A = Amount (tank) T = Time (hr) Rate = A/T (tank/hr) A B
- 82. Rational Equations Word-Problems Example D. Pipe A can fill a full tank of water in 3 hours, pipe B can fill ¾ of the (same) tank of water in 1½ hr. a. What is unit of the fill–rate and what is the rate of each pipe? The unit of the rate in question is tank/hr. Pipes A = Amount (tank) T = Time (hr) Rate = A/T (tank/hr) A B
- 83. Rational Equations Word-Problems Example D. Pipe A can fill a full tank of water in 3 hours, pipe B can fill ¾ of the (same) tank of water in 1½ hr. a. What is unit of the fill–rate and what is the rate of each pipe? The unit of the rate in question is tank/hr. Pipes A = Amount (tank) T = Time (hr) Rate = A/T (tank/hr) A 1 3 B
- 84. Rational Equations Word-Problems Example D. Pipe A can fill a full tank of water in 3 hours, pipe B can fill ¾ of the (same) tank of water in 1½ hr. a. What is unit of the fill–rate and what is the rate of each pipe? The unit of the rate in question is tank/hr. 1 3 The rate of A = tank/hr. Pipes A = Amount (tank) T = Time (hr) Rate = A/T (tank/hr) A 1 3 1/3 B
- 85. Rational Equations Word-Problems Example D. Pipe A can fill a full tank of water in 3 hours, pipe B can fill ¾ of the (same) tank of water in 1½ hr. a. What is unit of the fill–rate and what is the rate of each pipe? The unit of the rate in question is tank/hr. 1 3 The rate of A = tank/hr. Pipes A = Amount (tank) T = Time (hr) Rate = A/T (tank/hr) A 1 3 1/3 B 3/4 1 ½ = 3/2 1/2
- 86. Rational Equations Word-Problems Example D. Pipe A can fill a full tank of water in 3 hours, pipe B can fill ¾ of the (same) tank of water in 1½ hr. a. What is unit of the fill–rate and what is the rate of each pipe? The unit of the rate in question is tank/hr. 1 3 The rate of A = tank/hr. The rate of B = 3/4 1½ = Pipes A = Amount (tank) T = Time (hr) Rate = A/T (tank/hr) A 1 3 1/3 B 3/4 1 ½ = 3/2 1/2
- 87. Rational Equations Word-Problems Example D. Pipe A can fill a full tank of water in 3 hours, pipe B can fill ¾ of the (same) tank of water in 1½ hr. a. What is unit of the fill–rate and what is the rate of each pipe? The unit of the rate in question is tank/hr. 1 3 = The rate of A = tank/hr. The rate of B = 3/4 1½ = 3/4 3/2 Pipes A = Amount (tank) T = Time (hr) Rate = A/T (tank/hr) A 1 3 1/3 B 3/4 1 ½ = 3/2 1/2
- 88. Rational Equations Word-Problems Example D. Pipe A can fill a full tank of water in 3 hours, pipe B can fill ¾ of the (same) tank of water in 1½ hr. a. What is unit of the fill–rate and what is the rate of each pipe? The unit of the rate in question is tank/hr. 1 3 = 1 2 The rate of A = tank/hr. The rate of B = tank/hr. 3/4 1½ = 3/4 3/2 Pipes A = Amount (tank) T = Time (hr) Rate = A/T (tank/hr) A 1 3 1/3 B 3/4 1 ½ = 3/2 1/2
- 89. Rational Equations Word-Problems Example D. Pipe A can fill a full tank of water in 3 hours, pipe B can fill ¾ of the (same) tank of water in 1½ hr. a. What is unit of the fill–rate and what is the rate of each pipe? The unit of the rate in question is tank/hr. 1 3 = 1 2 The rate of A = tank/hr. The rate of B = tank/hr. 3/4 1½ = 3/4 3/2 b. How much time would it take for pipe B to fill the tank alone? Pipes A = Amount (tank) T = Time (hr) Rate = A/T (tank/hr) A 1 3 1/3 B 3/4 1 ½ = 3/2 1/2
- 90. Rational Equations Word-Problems Example D. Pipe A can fill a full tank of water in 3 hours, pipe B can fill ¾ of the (same) tank of water in 1½ hr. a. What is unit of the fill–rate and what is the rate of each pipe? The unit of the rate in question is tank/hr. 1 3 = 1 2 The rate of A = tank/hr. The rate of B = tank/hr. 3/4 1½ = 3/4 3/2 b. How much time would it take for pipe B to fill the full tank alone? We reciprocate the rate of B “½ (tank/hr)” and get 2 hr/tank or that it would take 2hrs for pipe B to fill a full tank. Pipes A = Amount (tank) T = Time (hr) Rate = A/T (tank/hr) A 1 3 1/3 B 3/4 1 ½ = 3/2 1/2
- 91. Rational Equations Word-Problems Pipes A = Amount (tank) T = Time (hr) Rate = A/T (tank/hr) A 1 3 1/3 B 3/4 1 ½ = 3/2 1/2 c. What is the combined rate if both pipes are used and how long would it take to fill the entire tank if both pipes are used?
- 92. Rational Equations Word-Problems The combined rate is the sum of the rates Pipes A = Amount (tank) T = Time (hr) Rate = A/T (tank/hr) A 1 3 1/3 B 3/4 1 ½ = 3/2 1/2 c. What is the combined rate if both pipes are used and how long would it take to fill the entire tank if both pipes are used? 1 3 + 1 2
- 93. Rational Equations Word-Problems 5 6= The combined rate is the sum of the rates tank/hr. Pipes A = Amount (tank) T = Time (hr) Rate = A/T (tank/hr) A 1 3 1/3 B 3/4 1 ½ = 3/2 1/2 c. What is the combined rate if both pipes are used and how long would it take to fill the entire tank if both pipes are used? 1 3 + 1 2
- 94. Rational Equations Word-Problems 5 6= The combined rate is the sum of the rates tank/hr. Pipes A = Amount (tank) T = Time (hr) Rate = A/T (tank/hr) A 1 3 1/3 B 3/4 1 ½ = 3/2 1/2 The reciprocal of the combined rate is 6/5 therefore it would take 1 1/5 hr if both pipes are used. c. What is the combined rate if both pipes are used and how long would it take to fill the entire tank if both pipes are used? 1 3 + 1 2
- 95. Rational Equations Word-Problems 5 6= The combined rate is the sum of the rates tank/hr. Pipes A = Amount (tank) T = Time (hr) Rate = A/T (tank/hr) A 1 3 1/3 B 3/4 1 ½ = 3/2 1/2 The reciprocal of the combined rate is 6/5 therefore it would take 1 1/5 hr if both pipes are used. c. What is the combined rate if both pipes are used and how long would it take to fill the entire tank if both pipes are used? 1 3 + 1 2 We note that in the above example, it takes pipe B one hour less to fill the tank then pipe A does and that together they got the job done in 6/5 hr.
- 96. Rational Equations Word-Problems 5 6= The combined rate is the sum of the rates tank/hr. Pipes A = Amount (tank) T = Time (hr) Rate = A/T (tank/hr) A 1 3 1/3 B 3/4 1 ½ = 3/2 1/2 The reciprocal of the combined rate is 6/5 therefore it would take 1 1/5 hr if both pipes are used. c. What is the combined rate if both pipes are used and how long would it take to fill the entire tank if both pipes are used? 1 3 + 1 2 We note that in the above example, it takes pipe B one hour less to fill the tank then pipe A does and that together they got the job done in 6/5 hr. Let’s treat this as a word problem.
- 97. Rational Equations Word-Problems Example E. Pipe B takes one hour less to fill a full tank of water then pipe A does and together they got the job done in 6/5 hr. How long will it take pipe A to fill the tank alone? (What should x be?)
- 98. Rational Equations Word-Problems Set x = number of hours for A to fill the tank Example E. Pipe B takes one hour less to fill a full tank of water then pipe A does and together they got the job done in 6/5 hr. How long will it take pipe A to fill the tank alone? (What should x be?)
- 99. Rational Equations Word-Problems Set x = number of hours for A to fill the tank Example E. Pipe B takes one hour less to fill a full tank of water then pipe A does and together they got the job done in 6/5 hr. How long will it take pipe A to fill the tank alone? (What should x be?) So the number of hours for B to fill the tank is (x – 1)
- 100. Rational Equations Word-Problems Set x = number of hours for A to fill the tank Example E. Pipe B takes one hour less to fill a full tank of water then pipe A does and together they got the job done in 6/5 hr. How long will it take pipe A to fill the tank alone? (What should x be?) So the number of hours for B to fill the tank is (x – 1) It takes 6/5 hr to use both. .
- 101. Rational Equations Word-Problems Set x = number of hours for A to fill the tank Pipes Time (hr) Rate (tank/hr) Example E. Pipe B takes one hour less to fill a full tank of water then pipe A does and together they got the job done in 6/5 hr. How long will it take pipe A to fill the tank alone? (What should x be?) So the number of hours for B to fill the tank is (x – 1) It takes 6/5 hr to use both. Put these into a table.
- 102. Rational Equations Word-Problems Set x = number of hours for A to fill the tank Pipes Time (hr) Rate (tank/hr) A B A & B Example E. Pipe B takes one hour less to fill a full tank of water then pipe A does and together they got the job done in 6/5 hr. How long will it take pipe A to fill the tank alone? (What should x be?) So the number of hours for B to fill the tank is (x – 1) It takes 6/5 hr to use both. Put these into a table.
- 103. Rational Equations Word-Problems Set x = number of hours for A to fill the tank Pipes Time (hr) Rate (tank/hr) A x B (x – 1) A & B 6/5 Example E. Pipe B takes one hour less to fill a full tank of water then pipe A does and together they got the job done in 6/5 hr. How long will it take pipe A to fill the tank alone? (What should x be?) So the number of hours for B to fill the tank is (x – 1) It takes 6/5 hr to use both. Put these into a table.
- 104. Rational Equations Word-Problems Set x = number of hours for A to fill the tank Pipes Time (hr) Rate (tank/hr) A x 1/x B (x – 1) 1/(x – 1) A & B 6/5 5/6 Example E. Pipe B takes one hour less to fill a full tank of water then pipe A does and together they got the job done in 6/5 hr. How long will it take pipe A to fill the tank alone? (What should x be?) So the number of hours for B to fill the tank is (x – 1) It takes 6/5 hr to use both. Put these into a table.
- 105. Rational Equations Word-Problems Set x = number of hours for A to fill the tank Pipes Time (hr) Rate (tank/hr) A x 1/x B (x – 1) 1/(x – 1) A & B 6/5 5/6 Example E. Pipe B takes one hour less to fill a full tank of water then pipe A does and together they got the job done in 6/5 hr. How long will it take pipe A to fill the tank alone? (What should x be?) So the number of hours for B to fill the tank is (x – 1) It takes 6/5 hr to use both. . Put these into a table. The rates for A and B add to the combined rate 5/6 so we get a rational equation in x.
- 106. Rational Equations Word-Problems Set x = number of hours for A to fill the tank Pipes Time (hr) Rate (tank/hr) A x 1/x B (x – 1) 1/(x – 1) A & B 6/5 5/6 Example E. Pipe B takes one hour less to fill a full tank of water then pipe A does and together they got the job done in 6/5 hr. How long will it take pipe A to fill the tank alone? (What should x be?) So the number of hours for B to fill the tank is (x – 1) It takes 6/5 hr to use both. . 5 6= 1 x + 1 (x – 1) Put these into a table. The rates for A and B add to the combined rate 5/6 so we get a rational equation in x.
- 107. Rational Equations Word-Problems 5 6= 1 x + 1 (x – 1) clear the denominatorsx (x – 1)][ Set x = number of hours for A to fill the tank Pipes Time (hr) Rate (tank/hr) A x 1/x B (x – 1) 1/(x – 1) A & B 6/5 5/6 Example E. Pipe B takes one hour less to fill a full tank of water then pipe A does and together they got the job done in 6/5 hr. How long will it take pipe A to fill the tank alone? (What should x be?) So the number of hours for B to fill the tank is (x – 1) It takes 6/5 hr to use both. . Put these into a table. The rates for A and B add to the combined rate 5/6 so we get a rational equation in x.
- 108. Rational Equations Word-Problems 5 6= 1 x + 1 (x – 1) clear the denominators6x (x – 1) 6x ][ Set x = number of hours for A to fill the tank Pipes Time (hr) Rate (tank/hr) A x 1/x B (x – 1) 1/(x – 1) A & B 6/5 5/6 Example E. Pipe B takes one hour less to fill a full tank of water then pipe A does and together they got the job done in 6/5 hr. How long will it take pipe A to fill the tank alone? (What should x be?) So the number of hours for B to fill the tank is (x – 1) Put these into a table. The rates for A and B add to the combined rate 5/6 so we get a rational equation in x. It takes 6/5 hr to use both. . 6(x – 1) x (x – 1)
- 109. Rational Equations Word-Problems 5 6= 1 x + 1 (x – 1) clear the denominators6x (x – 1) 6x ][ Set x = number of hours for A to fill the tank Pipes Time (hr) Rate (tank/hr) A x 1/x B (x – 1) 1/(x – 1) A & B 6/5 5/6 Example E. Pipe B takes one hour less to fill a full tank of water then pipe A does and together they got the job done in 6/5 hr. How long will it take pipe A to fill the tank alone? (What should x be?) So the number of hours for B to fill the tank is (x – 1) Put these into a table. The rates for A and B add to the combined rate 5/6 so we get a rational equation in x. It takes 6/5 hr to use both. . 6(x – 1) x (x – 1) 6x – 6 + 6x = 5x2 – 5x
- 110. Rational Equations Word-Problems 5 6= 1 x + 1 (x – 1) clear the denominators6x (x – 1) 6x ][ Set x = number of hours for A to fill the tank Pipes Time (hr) Rate (tank/hr) A x 1/x B (x – 1) 1/(x – 1) A & B 6/5 5/6 Example E. Pipe B takes one hour less to fill a full tank of water then pipe A does and together they got the job done in 6/5 hr. How long will it take pipe A to fill the tank alone? (What should x be?) So the number of hours for B to fill the tank is (x – 1) Put these into a table. The rates for A and B add to the combined rate 5/6 so we get a rational equation in x. It takes 6/5 hr to use both. . 6(x – 1) x (x – 1) 6x – 6 + 6x = 5x2 – 5x 0 = 5x2 – 17x + 6 (Question: Why is the other solution no good?) you finish this...
- 111. Rational Equations Word-Problems Ex. For problems 1 – 9, fill in the table, set up an equation and solve for x. (If you can guess the answer first, go ahead. But set up and solve the problem algebraically to confirm it.) 2. A group of x people are to share 6 slices of pizza equally. If there are three less people in the group then each person would get one more slice. What is x? 3. A group of x people are to share 6 slices of pizza equally. If one more person joins in to share the pizza then each person would get three less slices. What is x? 1. A group of x people are to share 6 slices of pizza equally. If there is one less person in the group then each person would get one more slice. What is x? Total Cost No. of People Cost per Person = Total Cost No. of people
- 112. Rational Equations Word-Problems 5. A group of x people are to share 12 slices of pizza equally. If two more people want to partake then each person would get one less slice. What is x? 6. A group of x people are to share 12 slices of pizza equally. If four people in the group failed to show up then each person would get four more slices. What is x? 7. A group of x people are to share the cost of $24 to rent a van equally. If three more people want to partake then each person would pay $4 less. What is x? 8. A group of x people are to share the cost of $24 to rent a van equally. If four more people join in the group then each person would pay $1 less. What is x? 9. A group of x people are to share the cost of $24 to rent a van equally. If there are two less people in the group then each person would pay $1 more. What is x?
- 113. Rational Equations Word-Problems Distance (m) Rate (mph) Time = D/R (hr) 1st trip 2nd trip 10. We hiked 6 miles from A to B at a rate of x mph. For the return trip our rate was 1 mph slower. It took us 5 hours for the entire round trip. What is x? HW C. For problems 10 – 16, fill in the table, set up an equation and solve for x. 11. We hiked 6 miles from A to B at a rate of x mph. For the return trip our rate was 3 mph faster. It took us 3 hours for the entire round trip. What is x? 12. We hiked 6 miles from A to B at a rate of x mph. For the return trip our rate was 1 mph faster. It took us 9 hours for the entire round trip. What is x?
- 114. Rational Equations Word-Problems 13. We hiked 3 miles from A to B at a rate of x mph. Then we hiked 2 miles from B to C at a rate that was 1 mph slower. It took us 2 hours for the entire round trip. What is x? 14. We hiked 6 miles from A to B at a rate of x mph. Then we hiked 3 miles from B to C at a rate that was 1 mph faster. It took us 3 hours for the entire round trip. What is x? 15. We ran 8 miles from A to B. Then we ran 5 miles from B to C at a rate that was 1 mph faster. It took one more hour to get from A to B than from B to C. What was the rate we ran from A to B? 16. We piloted a boat 12 miles upstream from A to B. Then we piloted it downstream from B back to A at a rate that was 4 mph faster. It took one more hour to go upstream from A to B than going downstream from B to C. What was the rate of the boat going up stream from A to B?
- 115. Rational Equations Word-Problems 17. Its takes 3 hrs to complete 2/3 of a job. What is the job–rate? How long would it take to complete the whole job? How long would it take to complete ½ of the job? For problems 17–22, don’t set up equations. Find the answers directly and leave the answer in fractional hours. 18. Its takes 4 hrs to complete 3/5 of a job. What is the job–rate? How long would it take to complete the whole job? How long would it take to complete 1/3 of the job? Time Rate A B A & B 19. Its takes 4 hrs for A to complete 3/5 of a job and 3 hr for B to do 2/3 of the job. Complete the table. What is their combined job–rate? How long would it take for A & B to complete the whole job together?
- 116. Rational Equations Word-Problems Time Rate A B A & B 20. Its takes 1/2 hr for A to complete 2/5 of a job and 3/4 hr for B to do 1/6 of the job. Complete the table. What is their combined job–rate? How long would it take for A & B to complete the whole job together? Time Rate A B A & B 21. It takes 3 hrs for pipe A to fill a pool and 5 hrs for pipe B to drain the pool. Complete the table. If we use both pipes to fill and drain simultaneously, what is their combined job–rate? Will the pool be filled eventually? If so, how long would it take for the pool to fill?
- 117. Rational Equations Word-Problems Time Rate leak pump both 22. A leak in our boat will fill the boat in 6 hrs at which time the boat sinks. We have a pump that can empty a filled boat in 9 hrs. How much time do we have before the boat sinks? For 23 –26, given the information, how many hours it would take for A or B to do the job alone. 23. It takes A one more hour than B to do the job alone. It takes them 2/3 hr to do the job together. 24. It takes A two more hours than B to do the job alone. It takes them 3/4 hr to do the job together. 25. It takes B two hour less than A to do the job alone. It takes them 4/3 hr to do the job together. 26. It takes A three more hours than B to do the job alone. It takes them 2 hr to do the job together.

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