real world problems ma.912.a.5.7 solve real-world problems involving rational equations (mixture,...
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Real World Problems
MA.912.A.5.7 Solve real-world problems involving rational equations (mixture, distance, work, interest, and ratio).
Work Problems
Work = Rate Time
€
•Rate = Work Time
Time = Work Rate
Work = 1 (entire job)
Work Problems
1. John and Ana must mow the lawn before
they can go swimming. Working alone,
John would take 30 minutes and Ana would
take 45 minutes. If they both work together
how long would it take to do the job?
Work Problems1. John and Ana must mow the lawn before they
can go swimming. Working alone, John would take 30 minutes and Ana would take 45 minutes. If they both work together how long would it take to do the job?
€
Let x = # of min John and Ana
have been working together.
* Identify the question being asked and define the variable.
Work Problems1. John and Ana must mow the lawn before they
can go swimming. Working alone, John would take 30 minutes and Ana would take 45 minutes. If they both work together how long would it take to do the job?
John’s Rate:
Ana’s Rate:
€
1
30job /min
€
1
45job /min
Rate = Work Time
Work Problems1. John and Ana must mow the lawn before they
can go swimming. Working alone, John would take 30 minutes and Ana would take 45 minutes. If they both work together how long would it take to do the job?
Amount ofwork completedBy John in x minutes:
€
1 job
30 min• x min
€
1 job
45 min• x min
Amount ofwork completedBy Ana in x minutes:
Work Problems1. John and Ana must mow the lawn before they can go
swimming. Working alone, John would take 30 minutes and Ana would take 45 minutes. If they both work together how long would it take to do the job?
€
1
30x +
1
45x =1
€
90
€
3x +2x = 90
5x = 90
x =18
John and Anawould take 18 minutes to mow the lawnif they were to work together.
Work Problems
2. Nancy must mow the lawn before
she can go to the movies. Working alone,
she would take 50 minutes. Her friend
Joe decides to help, and they complete the
Job in 30 minutes. How long would it have
taken Joe to mow the lawn on his own?
Work Problems2. Nancy must mow the lawn before she can go to the
movies. Working alone, she would take 50 min. Her friend Joe decides to help, and they complete the Job in 30 minutes. How long would it have taken Joe to mow the lawn on his own?
* Identify the question being asked and define the variable.
€
Let x = # of min Joe would take to
complete the entire job on his own.
Work Problems
2. Nancy must mow the lawn before she can go to the movies. Working alone, she would take 50 minutes. Her friend Joe decides to help, and they complete the
Job in 30 minutes. How long would it have taken Joe to mow the lawn on his own?
Nancy’s Rate: Joe’s Rate:
€
1
50job /min
€
1
xjob /min
Work Problems
Amount of work completed by Nancy in 30 minutes:
€
1 job
50 min• 30 min
€
1 job
x min• 30 min
Amount of work completed by Joe in 30 minutes:
2. Nancy must mow the lawn before she can go to the movies. Working alone, she would take 50 minutes. Her friend Joe decides to help, and they complete the
Job in 30 minutes. How long would it have taken Joe to mow the lawn on his own?
Work ProblemsAmount of work completed by Nancy in 30 minutes:
€
1 job
50 min• 30 min
€
1 job
x min• 30 min
Amount of work completed by Joe in 30 minutes:
€
3
5job
€
30
xjob
€
+
€
=1
Work Problems
€
3
5+30
x=1
€
5x
€
3x +150 = 5x
Joe would take 75 minutes to mow the lawn on his own.
Distance = Rate Time
€
•
Rate = Distance Time
Time = Distance Rate
Solving Real World Problems
(Average Speed)
3. Suppose you are traveling through the water
at x km/h against a current of 5 km/h your
avg speed (rate) would be ______, where x is
your speed in still water.
You want to complete a 10 km trip in 3 hours.
Write an equation to solve for your avg speed.
€
d = rt
10 = (x −5)3
€
x −5
€
x = 81
3km /h
Upstream-Downstream Problems
Upstream-Downstream Problems
3. A boat moves through the water at x km/h.
It makes a journey 20 km upstream against
a current of 3 km/h and then returns with the
current. How fast must the boat travel through
the water in order to complete the trip in
5 hours?
Upstream-Downstream Problems
3. A boat moves through the water at x km/h. It makes a journey 20 km upstream against a current of 3 km/h and then returns with the current. How fast must the boat travel through the water in order to complete the trip in 5 hours?
* Identify the question being asked and define the variable.
€
Let x = The speed at which the boat
travels in still water.
Upstream-Downstream Problems3. A boat moves through the water at x km/h. It makes a journey 20 km upstream against a current of 3 km/h and then returns with the current. How fast must the boat travel through the water in order to complete the trip in 5 hours?
Net Speed Upstream:
Net Speed Downstream:
€
(x − 3) km /h
€
(x + 3) km /hagainst the current
with the current
Upstream-Downstream Problems3. A boat moves through the water at x km/h. It makes a journey 20 km upstream against a current of 3 km/h and then returns with the current. How fast must the boat travel through the water in order to complete the trip in 5 hours?
Number of Hours Upstream:
€
20 km
(x − 3) km /h
Number of Hours Downstream:
€
20 km
(x + 3) km /h
€
if d = rt then t =d
r
Upstream-Downstream Problems3. A boat moves through the water at x km/h. It makes a journey 20 km upstream against a current of 3 km/h and then returns with the current. How fast must the boat travel through the water in order to complete the trip in 5 hours?
Total time for round trip:
€
20
(x − 3)hours
€
+20
(x + 3)hours
€
20
x − 3+20
x + 3= 5
Upstream-Downstream Problems
€
20
x − 3+20
x + 3= 5
€
x − 3( ) x + 3( )
€
20 x + 3( ) +20 x − 3( ) = 5 x + 3( ) x − 3( )
20x +60 +20x −60 = 5 x 2 −9( )
40x = 5x 2 − 45
0 = 5x 2 − 40x − 45
Upstream-Downstream Problems
€
5x 2 − 40x − 45 = 0
5(x 2 −8x −9) = 0
5 x −9( ) x +1( ) = 0
x = 9 or x = −1
The boat must travel at a rate of 9 km/h