Transcript 6.7 Applications of Rational Expressions

6.7 Applications of
Rational
Expressions
Objective 1
Solve problems about numbers.
Slide 6.7-3
CLASSROOM
EXAMPLE 1
Solving a Problem about an Unknown Number
A certain number is added to the numerator and subtracted from the
5
5
.
denominator of The new number equals the reciprocal of . Find the
8
8
number.
Solution:
5 x 8

8 x 5
LCD : 5(8  x)
5 x 8
5 8  x  
  5 8  x 
8 x 5
5(5  x)  8(8  x)
25  5x  64  8x
13x  39
x 3
It is important to check your solution from the words of the problem because the
equation may be solved correctly, but set up incorrectly.
Slide 6.7-4
Objective 2
Solve problems about distance, rate,
and time.
Slide 6.7-5
Solve problems about distance, rate, and time.
Recall the following formulas that relate distance, rate, and time.
Slide 6.7-6
CLASSROOM
EXAMPLE 2
Solving a Problem about Distance, Rate, and Time
A boat can go 10 mi against a current in the same time it can go 30 mi
with the current. The current flows at 4mph. Find the speed of the boat
with no current.
Solution:
d
t
r
30
10

 x  4  x  4
30  x  4  10  x  4
d
r
t
Downstream
30
x+4
30
x4
Upstream
10
x−4
10
x4
tdown 
30
 x  4
tup 
10
 x  4
30x 120  10x  40
20x  160
x 8
The speed of the boat with no current equals 8 miles per hour.
Slide 6.7-7
Objective 3
Solve problems about work.
Slide 6.7-8
Solve problems about work.
Rate of Work
If a job can be completed in t units of time, then the rate of work is
1
job per unit of time.
t
PROBLEM-SOLVING HINT
Recall that the formula d = rt says that distance traveled is equal to
rate of travel multiplied by time traveled. Similarly, fractional part of a
job accomplished is equal to the rate of work multiplied by the time
worked.
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CLASSROOM
EXAMPLE 3
Solving a Problem about Work Rates
Al and Mario operate a small roofing company. Mario can roof an average
house alone in 9 hr. Al can roof a house alone in 8 hr. How long will if take
them to do the job if they work together?
Solution:
1
1
x  x 1
9
8
LCD = 72
1 
1
72  x  x   1 72 
8 
9
It will take Mario and Al
4
4
17
8x  9x  72
17 x  72
x
72
4
or 4
hr
17
17
hours if they work together.
Slide 6.7-10
Solve problems about work.
PROBLEM-SOLVING HINT
A common error students make when solving a work problem, like the
one in Example 3, is to simply add the two times. In this case, adding
Al’s time of 8 hours and Mario’s time of 9 hours together for a total
time of 17 hours. This obviously is not a plausible answer since either
of the workers alone could complete the project in much less time.
Another common error is to try to split the job in half between the two
workers so that All would work half of his normal 8 hours and Mario
would work half of his normal 9 hours. By doing this, Al would work 4
hours and Mario would work one half hour longer that Al, which would
mean that they aren’t actually working together for the last half hour.
Slide 6.7-11