33, Lesson Section 6.5
Application Problems Using Rational Equations
1.
2.
3.
4.
1)
Define a Variable
Develop A Plan
Write an Equation
Solve and Answer the Question
The reciprocal of 5, plus the reciprocal of 7, is the reciprocal of what number?
Let x = the number
Reciprocal of 5 plus reciprocal of 7 = reciprocal of x
2)
The sum of a number and 21 times its reciprocal is 10 . Find the number.
Let n = the number
The number plus 21(reciprocal of number) = 10
1
Job or Work Problems:
Suppose Joe can paint a house in 9 hours and Clyde can paint the same house in 12 hours.
Examine the table below that shows the fractional part of the job that each man does plus
the part of the job that is completed together at that time.
Fraction
of the house
Painted
TIME
Joe
Clyde
Together
1 hour
1
1
11
7
+
or
9
12
9 12
36
2 hours
2
2
1
21
7
or
+ or
9
12
6
96
18
3 hours
3
1
3
1
11
7
or
or
+ or
9
3
12
4
34
12
t hours
t
t
tt
+
9
12
9 12
If the job is to be completed by both together, then
tt
+ = 1 , where t is time both
9 12
worked together.
This leads to the following summary to model uniform job or work problems.
Let a = time for A to complete the work alone; rate is
Let b = time for B to complete the work alone; rate is
1
/ unit of time.
a
1
/ unit of time.
b
Let t = time both A and B work together.
Plan: rate A(time) + rate B(time) = 1 job
1 1
t + t = 1
a
b
Then, the following equation results:
tt
+ =1
ab
3)
Stan needs 45 minutes to do the dishes, while Bobby can do them in 30 minutes.
How long will it take them, if they work together?
2
4)
A community water tank can be filled in 18 hours by the town office well alone
and in 22 hours by the high school well alone. How long will it take to fill the
tank if both wells are working?
5)
Ron can paint his den in 6 hours working alone. If Dee helps him, the job takes 4
hours. Estimate how long it would take Dee alone to paint the den.
6)
The HP Laser Jet 9000 works twice as fast as the Laser Jet 2300. (Source:
www.hewlettpackard.com ) If the machines work together, a university can
produce all its staff manuals in 15 hours. Find time the it would take each
machine, working alone, to complete the same job.
Let x = time for the 9000 model, 2 x = time for the 2300 model
3
7)
Jake can cut and split a cord of firewood in 6 fewer hours than Skyler can. When
they work together, it takes them 4 hours. How long would it take Jake alone to
cut and split a cord of firewood?
Uniform Motion Problems:
8)
A canal has a current of 2 miles per hour. Find the speed of Caseys boat in still
water, if the boat travels 11 miles down the canal in the same time that it goes 8
miles up the canal.
Distance
Rate
Time
Down the Canal
Up the Canal
4
9)
A local bus travels 7 mph slower than the Express bus. The express travels 45
miles in the same time it takes the local to travel 38 miles. Find the speed of each
bus.
Distance
Rate
Time
Local
Express
10)
There is a moving sidewalk at the local airport that moves at 1.5 feet per second.
Lisa can walk forward 570 feet on the moving sidewalk in the same time she can
walk 420 feet on a regular floor of the airport (without the moving sidewalk).
How fast does Lisa ordinarily walk without the moving sidewalk?
Distance
Rate
Time
With moving
570
r + 1.5
sidewalk
Without the moving
sidewalk
420
r
5
11)
Allen paddles 55 meters per minute in still water. He paddles 106 meters
upstream and 228 meters downstream in a total time of 6 minutes. What is the
speed (rate) of the current?
Distance
Rate
Time
Upstream
106
55 - c
106
55 c
Downstream
228
55 + c
228
55 + c
time upstream + time downstream totals 6 min.
106
228
+
=6
55 c 55 + c
LCD = (55 c)(55 + c)
106
228
(55 c)(55 + c)
+ (55 c)(55 + c)
= 6(55 c)(55 + c)
55 c
55 + c
106(55 + c) + 228(55 c) = 6(3025 c 2 )
5830 + 106c + 12540 228c = 18150 6c 2
18370 122c = 18150 6c 2
6c 2 122c + 220 = 0
2(3c 2 61c + 110) = 0
product = 330, sum = 61
55 and 6
3c 2 6c 55c + 110
I showed the product/sum
method of factoring the
trinomial, since the leading
coefficient was not a 1.
3c(c 2) 55(c 2)
(c 2)(3c 55) = 0
c2=0
c=2
3c 55 = 0
3c = 55
c = 18
1
3
Current could be flowing at 2 mps or 18
1
3
mps.
6