5.6 Applications of rational expressions

http://www.youtube.com/watch?v=VnOlvFqmWEY

1) Alex can paint his garage in 6 hours, Tim can do the same job in 4 hours. How long would it take them, working together, to finish painting the garage?

(t/6) + (t/4)t/4t/6t hr

=5/2=2.56/46/66 hr

=25/12=2.085/45/65 hr

=5/3=1.674/44/64 hr

=5/4=1.253/43/63 hr

=5/62/42/62 hr

Add = 5/121/41/61 hr

TogetherTimAlexTime

We need them to work together to finish 1 job (painting a garage)

Can you guess the answer?

• Let t be the number of hours Alex and Tim work together to finish painting the garage

• Equation: t/6 + t/4 = 1

• Solve on the board, t = 12/5 hours or 2.4 hours (or 2 hours and 24 minutes)

Practice

2) Geraldo and Luisa operate a small laundry, Luisa, working alone, can clean a day’s laundry in 9 hours. Geraldo can clean a day’s laundry in 8 hours. How long it take them if they work together?

3) It takes Peter and Mary 20 hours to paint the house if they work together. If each were working alone, it will take Peter 9 hours longer than Mary to complete the job. How long would it take each, working alone, to complete the job?

Let x be the number of hours that would take Mary working alone

Then x + 9 is the number of hours that would take Peter working alone

Go back to problem 1a) to look at the equation again.

In 1a) we don’t know the total number of hours for 2 people to work together, but know the number of hours for each person working alone.

In this problem 1b) we do know the total number of hours for 2 people to work together, but don’t know the number of hours for each person working alone.

Equation:

19

2020

xx

20(x+9) + 20x = 1x(x+9)

20x + 180 + 20x = x2 + 9x

X2 + 9x – 20x – 20x – 180 = 0

X2 – 31x -180 =0

(x-36)(x + 5) = 0

X = 36 or x = -5

Therefore, it takes Mary 36 hours and Peter 45 hours to complete the task alone.

Practice

4) An experienced employee can enter tax data into a computer twice as fast as a new employee. Working together, it takes the employees 2 hours. How long would it take the experienced employee working alone?

5) To determine the number of fish in a river, a fisherman catches 620 fish, tags them, and releases them. Later, 122 fish are caught, 31 of them are tagged. Estimate how many fish in the river.

Let x be the number of fish in the riverEquation: # of fish caught (1st time) = # of fish tagged

# of fish in the river # of fish caught (2nd)

620 = 31

x 122

So x = (620)(122) = 2440 fish in the river

31

6) The current in the Lazy River moves at a rate of 4mph. A boat can go 6mi upstream in the same time that it takes to go 12 mi downstream. What is the speed of the boat in still water?

12

X+4

X +412Down-

stream

6

X - 4

X - 46Up-

stream

T= RD

Let x is the speed of the boat in still water

• Rational equation:

6 = 12

X – 4 X + 4

Solve on the board, x = 12 mph is the speed of the boat in still water

7) A local bus travels 7mph slower than the express bus. The express bus travels 90 mi in the time it takes the local bus to travel 75mi. Find the speed of each bus.

90

x

X90Express

bus

75

X-7

X-775Local bus

T= RD

Let x be the speed of the express bus

• Rational equation:

75 = 90

x – 7 x

Solve on the board, x = 42 mph is the speed of the express bus and 35 mph is the speed of the local bus