1. Slide 7- 1 Copyright © 2012 Pearson Education, Inc.

2. 6.5 Solving Applications Using Rational Equations ■ Problems Involving Work ■ Problems Involving Motion ■ Problems Involving Proportions

3. Slide 6- 3 Copyright © 2012 Pearson Education, Inc. Problems Involving Work Example: Juan and Rebecca work during the summer painting houses. Juan can paint an average size house in 12 days, while Rebecca requires 8 days to do the same job. How long would it take them, working together, to paint an average size house? 1. Familiarize. We familiarize ourselves with the problem by exploring two common, but incorrect, approaches. a) One common incorrect approach is to add the two times. 12 + 8 = 20 b) Another incorrect approach is to assume that Juan and Rebecca each do half the painting. Juan does ½ of 12 days = 6 days Rebecca does ½ of 8 days = 4 days 6 days + 4 days = 10 days.

4. Slide 6- 4 Copyright © 2012 Pearson Education, Inc. A correct approach is to consider how much of the painting job is finished in one day. It takes Juan 12 days to finish painting a house, so his rate is 1/12 of the job per day. It takes Rebecca 8 days to do the painting alone, so her rate is 1/8 of the job per day. Working together, they can complete 1/8 + 1/12, or 5/24 of the job in one day. We can form a table to help organize the information: continued t/8t1/8 Rebecca t/12t1/12 Juan Amount Completed TimeRate of Work

5. Slide 6- 5 Copyright © 2012 Pearson Education, Inc. continued 2. Translate. The time that we want is some number t for which or Portion of work done by Rebecca in t days Portion of work done by Juan in t days Portion of work done together in t days

6. Slide 6- 6 Copyright © 2012 Pearson Education, Inc. continued 3. Carry out. We can choose any one of the above equations to solve: 4. Check. 5. State. Together, it will take Juan and Rebecca 4 4/5 days to complete painting a house.

7. Slide 6- 7 Copyright © 2012 Pearson Education, Inc. Modeling Work Problems If a = the time needed for A to complete the work alone, b = the time needed for B to complete the work alone, and t = the time needed for A and B to complete the work together, then The following are equivalent equations that can also be used:

8. Slide 6- 8 Copyright © 2012 Pearson Education, Inc. Problems Involving Motion Example: Because of a tail wind, a jet is able to fly 20 mph faster than another jet that is flying into the wind. In the same time that it takes the first jet to travel 90 miles the second jet travels 80 miles. How fast is each jet traveling? 1. Familiarize. We try a guess. If the first jet is traveling 300 mph because of a tail wind the second jet would be traveling 300  20 or 280 mph. At 300 mph the first jet would travel 90/300, or 3/10 hr. At 280 mph, the other jet would travel 80/280 = 2/7 hr. Since both planes spend the same amount of time traveling, we see that our guess is incorrect. Let’s set up a table. r r + 20

9. Slide 6- 9 Copyright © 2012 Pearson Education, Inc. continued Distance = Rate  Time 2. Translate. Fill in the blank column in the table. time = distance/rate. Distance (in miles) Speed (in miles per hour) Time (in hours) Jet 180r Jet 290r + 20 Distance (in miles) Speed (in miles per hour) Time (in hours) Jet 180r80/r Jet 290r + 2090/(r + 20) The times must be the same

10. Slide 6- 10 Copyright © 2012 Pearson Education, Inc. continued Since the times must be the same for both planes, we have the equation 3. Carry out. To solve the equation, we first multiply both sides by the LCM of the denominators r(r + 20). Simplifying Using the distributive law Subtracting 80r from both sides Dividing both sides by 10

11. Slide 6- 11 Copyright © 2012 Pearson Education, Inc. continued We can also solve the problem graphically.

12. Slide 6- 12 Copyright © 2012 Pearson Education, Inc. continued Now we have a possible solution. The speed of one jet is 160 mph and the speed of the other jet is 180 mph. 4. Check. Reread the problem to confirm that we were to find the speeds. At 160 mph the jet would cover 80 miles in ½ hour and at 180 mph the other jet would cover 90 miles in ½ hour. Since the times are the same, the speeds check. 5. State. One jet is traveling at 160 mph and the second jet is traveling at 180 mph.

13. Slide 7- 13 Copyright © 2012 Pearson Education, Inc. Problems Involving Proportions A ratio of two quantities is their quotient. For example, 45% is the ratio of 45 to 100, or 45/100. A proportion is an equation stating that two ratios are equal. Proportion An equality of ratios, A/B = C/D, is called a proportion. The numbers within a proportion are said to be proportional to each other.

14. Slide 7- 14 Copyright © 2012 Pearson Education, Inc. Example Triangles ABC and XYZ are similar. Solve for b if x = 8, y = 12 and a = 7. Solution We make a drawing, write a proportion, and then solve. Note that side a is always opposite angle A, side x is always opposite angle X, and so on. A B C X Y Z a = 7 b x = 8 y = 12

15. Slide 7- 15 Copyright © 2012 Pearson Education, Inc. Solution continued We set up our proportion: A B C X Y Z a = 7 b x = 8 y = 12

16. Slide 7- 16 Copyright © 2012 Pearson Education, Inc. Example A sample of 186 hard drives contained 4 defective drives. How many defective hard drives would you expect in a sample of 1302? Solution Form a proportion in which the ratio of defective hard drives is expressed in two ways. You would expect to find 28 defective hard drives. defective drives total drives defective drives total drives

17. Slide 7- 17 Copyright © 2012 Pearson Education, Inc. Example To determine the number of humpback whales in a pod, a marine biologist, using tail markings, identifies 35 members of the pod. Several weeks later, 50 whales from the pod are randomly sighted. Of the 50 sighted, 18 are from the 35 originally identified. Estimate the number of whales in the pod. Solution 1. Familiarize. We need to reread the problem to look for numbers that could be used to approximate a percentage of the pod sighted.

18. Slide 7- 18 Copyright © 2012 Pearson Education, Inc. Solution continued 2. Translate. Since 18 of the 35 whales that were later sighted were among those originally identified, the ratio 18/35 estimates the percentage of the pod originally identified. We can then translate to a proportion: 3. Carry out. Original whales sighted later Whales sighted later Whales originally identified Entire pod

19. Slide 7- 19 Copyright © 2012 Pearson Education, Inc. Solution continued 4. Check. The check is left to the student. 5. State. There are about 97 whales in the pod.