Rational Functions: Applications

1) The difference of a whole number and its reciprocal is . 63 8 Number Problems 1) The difference of a whole number and its reciprocal is . 63 8 What is the number? a) Represent c) Solve Let x = the whole number x - = 1 x 63 8 8x 8x = reciprocal 1 x 8x2 – 8 = 63x 8x2 - 63x – 8 = 0 b) Equate (8x + 1) (x – 8) = 0 x - = 1 x 63 8 8x + 1 = 0 x – 8 = 0 8x = -1 x = 8 x = -1 8 Copyright © by Mr. Florben G. Mendoza

Number Problems 2) When five times the square of certain integer is divided by four more than three times the same number, the result is -10. What is the integer? a) Represent c) Solve Let x = integer 5x2 3x + 4 = -10 3x + 4 3x + 4 b) Equate 5x2 = -30x - 40 5x2 5x2 + 30x + 40 = 0 = -10 3x + 4 5 x2 + 6x + 8 = 0 x + 4 = 0 x + 2 = 0 x = -4 x = -2 Copyright © by Mr. Florben G. Mendoza

Number Problems: 3) The sum of the reciprocals of two consecutive even integers is equal to 10 times the reciprocal of the product of those integers. Find the two integers. Solution: a) Represent c) Solve Let x = 1st even integer + 1 x x + 2 = 10 x(x + 2) x(x+2) x(x+2) x + 2 = next even integer x + 2 + x = 10 b) Equate 2x = 10 - 2 1 x 1 x + 2 10 x(x + 2) 2x = 8 + = x = 4 Copyright © by Mr. Florben G. Mendoza

Work Problems 1) John can completely wash and dry the dishes in 20 minutes. His brother can do it in 30 minutes. How long will it take them working together? b) Equate/Solve a) Represent + t 20 30 = 1 60 John Brother Completed Job Together t 1 Alone 20 30 3t + 2t = 60 5t = 60 t = 12 Copyright © by Mr. Florben G. Mendoza

Work Problems 1) Two pipes are connected to the same tank. When working together, they can fill the tank in 2 hours. The larger pipe, working alone, can fill the tank in 3 hours less time than the smaller one. How long would the smaller one take, working alone, to fill the tank? b) Equate/Solve a) Represent + 2 t t - 3 = 1 t (t - 3) Smaller Pipe Larger Complete Job Together 2 1 Alone t t - 3 2t - 6 + 2t = t2 – 3t t2 + - 2t – 2t – 3t + 6 = 0 t2 - 7t + 6= 0 (t - 6) (t - 1) = 0 t - 6 = 0 t - 1= 0 t = 6 t = 1 Copyright © by Mr. Florben G. Mendoza

Work Problems: 1) Two pipes are connected to the same tank. When working together, they can fill the tank in 2 hours. The larger pipe, working alone, can fill the tank in 3 hours less time than the smaller one. How long would the smaller one take, working alone, to fill the tank? b) Equate/Solve a) Represent t = 6 t = 1 Smaller Pipe Larger Complete Job Together 2 1 Alone t t - 3 c) Substitute Smaller Pipe: 6 hours Larger Pipe: 3 hours Copyright © by Mr. Florben G. Mendoza

Work Problems: 2) An office contains two copy machines. Machine B is known to take 12 minutes longer than Machine A to copy the company's monthly report. Using both machines together, it takes 8 minutes to reproduce the report. How long would it take each machine alone to reproduce the report? b) Equate/Solve a) Represent + 8 t t + 12 = 1 t (t + 12) Machine A Machine B Complete Job Together 8 1 Alone t t + 12 8t + 96 + 8t = t2 + 12t t2 + 12t – 8t – 8t – 96 = 0 t2 - 4t– 96 = 0 (t - 12) (t + 8) = 0 t - 12 = 0 t + 8= 0 t = 12 t = -8 Copyright © by Mr. Florben G. Mendoza

Work Problems: 2) An office contains two copy machines. Machine B is known to take 12 minutes longer than Machine A to copy the company's monthly report. Using both machines together, it takes 8 minutes to reproduce the report. How long would it take each machine alone to reproduce the report? a) Represent b) Equate/Solve t = 12 t = -8 Machine A Machine B Complete Job Together 8 1 Alone t t + 12 c) Substitute Machine A: 12minutes Machine B: 24minutes Copyright © by Mr. Florben G. Mendoza

TRY THIS! Ella and Julia can paint a room in 6 hours. Working alone, Ella takes 16 hours longer than Julia would paint the same room. How long would it take Julia to paint the room? Copyright © by Mr. Florben G. Mendoza

Distance Problems: 1) A truck traveled the first 100 kilometers of a trip at one speed and the last 135 kilometers at an average speed of 5 kilometers per hour less. If the entire trip took 5 hours, what was the average speed for the first part of the trip? 135 = 100 t - 5 (5 – t) b) Equate/Solve Solution: a) Represent d = rt d r t 100 135 r - 5 5 - t 100 – 5t t 135 = (5 – t) t t 135t = (100 – 5t) (5 – t) t2 – 52t + 100 = 0 135t = 500 – 100t – 25t + 5t2 (t – 50) (t – 2) = 0 100 = r t r = 100/t 5t2 – 260t + 500 = 0 t – 50 = 0 t – 2 = 0 135 = (r – 5) (5 – t) t2 – 52t + 100 = 0 t = 50 t = 2 t2 – 52t + 100 = 0 Copyright © by Mr. Florben G. Mendoza

Distance Problems: 2) A truck traveled the first 100 kilometers of a trip at one speed and the last 135 kilometers at an average speed of 5 kilometers per hour less. If the entire trip took 5 hours, what was the average speed for the first part of the trip? t = 50 Solution: b) t = 2 a) Let d r t 100 r = d/t r = 100/2 r = 50 kph 2 hrs 135 r – 5 50 – 5 = r = 45 kph 5 – t 5 – 2 = 3hrs c) d r t 100 135 r - 5 5 - t d = rt 100 = r t r = 100/t 135 = (r – 5) (5 – t) Copyright © by Mr. Florben G. Mendoza

b) Equate/Solve ± Distance Problems 2) A car goes 12 kph faster than the other car and requires 1 hours less time to travel 400 kms. Find the rate of each. 1 9 b) Equate/Solve Solution: 400 = r (t - ) 10 9 a) Represent 400 r – 12 400 = r ( - ) 10 9 d = rt d r t 400 t – 10/9 r - 12 400 r – 12 400 = r ( - ) 10 9 9(r – 12) 9(r – 12) 400 = r (t - ) 10 9 3600r – 43200 = 3600r - 10r2 + 120r 10r2 -120r – 43200 = 0 r – 6 = ± 66 400 = (r – 12) t r2 -12r – 4320 = 0 r = ± 66 + 6 (r – 12) (r – 12) r2 -12r + 36 = 4320 + 36 r = 72 (r – 6)2 = 4356 ± r = -60 Copyright © by Mr. Florben G. Mendoza

b) c) Distance Problems: 1 9 2) A car goes 12 kph faster than the other car and requires 1 hours less time to travel 400 kms. Find the rate of each. Solution: b) r = 72 r = -60 a) Let d r t 400 72 kph t – 10/9 20/3 – 10/9 t = 50/9 hrs 72 – 12 60 kph t = d/r t = 400/60 t = 20/3 hrs c) d = rt d r t 400 t – 10/9 r - 12 400 = r (t - ) 10 9 400 = (r – 12) t Copyright © by Mr. Florben G. Mendoza

Distance Problems: Round Trip 3) Juancho rides his power boat up and down the Pasig river. The water in the river flows at 6 miles per hour. Juancho takes 5 hours longer to travel 360 miles against the current than he does to travel 360 miles with the current. What is the speed of Juancho's boat in still water? b) Equate/Solve 360 = ( + 5) (r – 6) 360 r + 6 Against: a) Represent d = rt t r d With r + 6 360 Against t + 5 r – 6 360 = ( + 5) (r – 6) 360 r + 6 (r + 6) 360r + 2160 = 360 + 5r +30 (r – 6) 360 r + 6 t = 360r + 2160 = (360 + 5r +30) (r – 6) With: 360r + 2160 = (360 + 5r +30) (r – 6) Against: 360 = (t + 5) (r – 6) 360r + 2160 = 360r – 2160 + 5r2 – 30r + 30r - 180 5r2 – 4500 = 0 r2 – 900 = 0 r = ± 900 r = ± 30 Copyright © by Mr. Florben G. Mendoza

Rate of the boat in still water is 30 mph Distance Problems: Round Trip 3) Juancho rides his power boat up and down the Pasig river. The water in the river flows at 6 miles per hour. Juancho takes 5 hours longer to travel 360 miles against the current than he does to travel 360 miles with the current. What is the speed of Juancho's boat in still water? b) Equate/Solve a) Represent r = 30 r = - 30 d = rt r t d With r + 6 360 Against t + 5 r – 6 c) Substitution d = rt t r d With t = d/r 10 hrs r + 6 36 mph 360 Against t + 5 15 hrs r – 6 24 mph Rate of the boat in still water is 30 mph 360 r + 6 t = With: Against: 360 = (t + 5) (r – 6) Copyright © by Mr. Florben G. Mendoza

Number Problems 1. Suppose that the sum of two whole numbers is 9 and the sum of their reciprocals is 1/2 . Find the numbers. 2. The difference between two whole numbers is 8 and the difference between their reciprocals is 1/6. Find the two numbers. 3. The sum of the reciprocals of two consecutive integers is equal to 11 times the reciprocal of the product of those integers. What are the two integers? 4. If the same number is added to the numerator and denominator of 2/5, the result is 4/5. What is that number? 5. One positive number is 2 more than another. If the sum of the reciprocals of the two numbers is 7/24, what are those numbers? Copyright © by Mr. Florben G. Mendoza

Distance Problems 1. A jet flew from Tokyo to Bangkok, a distance of 4800 km. On the return trip the speed was decreased by 200km/h. If the difference in the times of the flights was 2 hours, what was the speed from Bangkok to Tokyo?  2. On a 570-mile trip, Jubert averaged 5 miles per hour faster for the last 240 miles than he did for the first 330 miles. The entire trip took 10 hours. How fast did he travel for the first 330 miles?  3. Ann can row her boat at a speed of 5 km/hr in still water. If it takes her one hour more to row her boat 5.25 km upstream than to return downstream, find the speed of the stream.  4. A head wind reduced a cyclist’s speed by 3km/hr, as a result he took 30 minutes more to travel a distance of 30 km. What would have been his speed had there been no head wind.   5. It takes one hour more to complete a journey of 180 km if that journey is made during the rush hours. If the average speed during the rush hour is 30 kph slower than the average speed during an off-hour, how long does it take to make the trip during the rush hour? Copyright © by Mr. Florben G. Mendoza

Work Problems 1. Khen and Kyle can eat 1260 hamburgers in 12 hours. Eating by himself, it would take Khen 7 hours longer to eat 1260 hamburgers than it would take Kyle to eat 1260 hamburgers. How long would it take Kyle to eat 1260 hamburgers by himself?   2. AC and Edward can paint a room of house in 6 hours. Working alone, Edward takes 16 hours longer than AC would paint the room. How long does it take Edward to paint the room? 3. Two taps A and B fill a swimming pool together in 1 hour and 20 minutes. Alone, it takes tap A two hours longer than B to fill the same pool. How many hours does it take each tap to fill the pool separately? 4. Ben and Eduardo, working together, can cover the roof of a house in 6 days, Ben working alone; can complete this job in 5 days less than Eduardo. How long will it take Ben to make this job? 5. Two teams, working together can lay all of the bricks at Angelicum in 12 hours, Team A, if works separately, can lay all of the bricks in 18 hours less than Team B. How long will it take for each team to complete the job alone? Copyright © by Mr. Florben G. Mendoza