2 4 rational equations word-problems

Rational Equations Word-Problems
Frank Ma © 2011

We look at the following applications of rational equations.

Problems from the Multiplication–Division Operations

All the multiplicative formulas of the form AB = C may be written
as A = .
C
B

as A = . This divisional form leads to rational equations.
C
B

C
B
The calculation of “per unit” is a good example:.

C
B
The calculation of “per unit” is a good example:
Total amount
Per unit amount =
Number of units

C
B
Total amount
Per unit amount =
Number of units
Cost per Unit

C
B
Total amount
Per unit amount =
Number of units
Cost per Unit
Total cost
Per unit cost =
Number of units

C
B
Total amount
Per unit amount =
Number of units
Cost per Unit
Total cost
Per unit cost =
Number of units
For example, a group of 5 people rent a taxi that cost $20
so each person’s share is 20/5 or $4 per person.

C
B
Total amount
Per unit amount =
Number of units
Cost per Unit
Total cost
Per unit cost =
Number of units
However if one person backs out, then the cost goes up to
s = 20/4 = $5 per person.

C
B
Total amount
Per unit amount =
Number of units
Cost per Unit
Total cost
Per unit cost =
Number of units
s = 20/4 = $5 per person. Each remaining person has to pay $1 more.

C
B
Total amount
Per unit amount =
Number of units
Cost per Unit
Total cost
Per unit cost =
Number of units
s = 20/4 = $5 per person. Each remaining person has to pay $1 more. Let’s formulate this as a word problem and solve it using rational equations.

A table is useful in organizing repeated calculations for comparing the results.

Example A. A taxi is rented by a group of x people for $20 and the cost is shared equally. If there is one less person in the group then each of the remaining people has to pay $1 more. What is x?

Total Cost
No. of people

Total Cost
No. of people
20
x
20
(x – 1)

Total Cost
No. of people
20
x
20
(x – 1)
Let’s compare the two different per/person costs.
20
20
(x – 1)
x

Total Cost
No. of people
20
x
20
(x – 1)
20
20
(x – 1)
x
cost more
cost less

Total Cost
No. of people
20
x
20
(x – 1)
20
20
(x – 1)
x
cost more
cost less
($1 more)

Total Cost
No. of people
20
x
20
(x – 1)
20
20
–
= 1
(x – 1)
x
cost more
cost less
($1 more)

Total Cost
No. of people
20
x
20
(x – 1)
20
20
]
[
x (x – 1)
–
= 1
clear the denominators
by LCM
(x – 1)
x

Total Cost
No. of people
20
x
20
(x – 1)
x
(x – 1)
x (x – 1)
20
20
]
[
x (x – 1)
–
= 1
by LCM
(x – 1)
x

Total Cost
No. of people
20
x
20
(x – 1)
x
(x – 1)
x (x – 1)
20
20
]
[
x (x – 1)
= 1
by LCM
–
(x – 1)
x
20x – 20(x – 1) = x(x – 1)

Total Cost
No. of people
20
x
20
(x – 1)
x
(x – 1)
x (x – 1)
20
20
]
[
x (x – 1)
= 1
by LCM
–
(x – 1)
x
20x – 20(x – 1) = x(x – 1)
20x – 20x + 20 = x2 – x

Total Cost
No. of people
20
x
20
(x – 1)
x
(x – 1)
x (x – 1)
20
20
]
[
x (x – 1)
= 1
by LCM
–
(x – 1)
x
20x – 20(x – 1) = x(x – 1)
set one side as 0
20x – 20x + 20 = x2 – x
0 = x2 – x – 20

0 = x2 – x – 20

0 = x2 – x – 20
0 = (x – 5)(x + 4)

0 = x2 – x – 20
0 = (x – 5)(x + 4)
x = 5 or x = –4.

0 = x2 – x – 20
0 = (x – 5)(x + 4)
x = 5 or x = –4.
Hence there are 5 people.

0 = x2 – x – 20
0 = (x – 5)(x + 4)
x = 5 or x = –4.
Rate–Time–Distance

0 = x2 – x – 20
0 = (x – 5)(x + 4)
x = 5 or x = –4.
Let R = rate (mph), T = time (hours) and D = Distance (m)
then RT = D or that T = . .
D
R

0 = x2 – x – 20
0 = (x – 5)(x + 4)
x = 5 or x = –4.
D
R
We hiked a 6–mile trail from A to B at a rate of 3 mph so the trip took 6/3 = 2 hours.

0 = x2 – x – 20
0 = (x – 5)(x + 4)
x = 5 or x = –4.
D
R
We hiked a 6–mile trail from A to B at a rate of 3 mph so the trip took 6/3 = 2 hours. Going back from B to A our rate was
2 mph so the return took 6/2 = 3 hours.

0 = x2 – x – 20
0 = (x – 5)(x + 4)
x = 5 or x = –4.
D
R
2 mph so the return took 6/2 = 3 hours. In a table,

0 = x2 – x – 20
0 = (x – 5)(x + 4)
x = 5 or x = –4.
D
R
2 mph so the return took 6/2 = 3 hours. In a table,
Let’s turn this example into a rational–equation word problem.

Example B. We hiked 6 miles from A to B at a rate of x mph. For the return trip our rate was 1 mph faster. It took us 5 hours for the entire round trip. What is x?

The rate of the return is “1 mph faster then x” so it’s (x + 1).

Put these into the table

Now we made use the information about the times.

6
6
(x + 1)
x
number of
hrs for returning
number of
hrs for going

6
6
(x + 1)
x
(total trip 5 hrs)
number of
hrs for returning
number of
hrs for going

6
6
= 5
+
(x + 1)
x
(total trip 5 hrs)
number of
hrs for returning
number of
hrs for going

6
6
use the cross
multiplication method
= 5
+
(x + 1)
x

6
6
use the cross
= 5
+
(x + 1)
x
6(x + 1) + 6x
= 5
x(x + 1)

6
6
use the cross
= 5
+
(x + 1)
x
6(x + 1) + 6x
= 5
x(x + 1)
12x + 6
5
=
x(x + 1)
1

6
6
use the cross
= 5
+
(x + 1)
x
6(x + 1) + 6x
= 5
x(x + 1)
12x + 6
5
=
Cross again. You finish it.
x(x + 1)
1

6
6
use the cross
= 5
+
(x + 1)
x
6(x + 1) + 6x
= 5
x(x + 1)
12x + 6
5
=
Cross again. You finish it.
x(x + 1)
1
(Remember that x = 2 mph.)

Job–Rates

Job–Rates
If we can eat 6 pizzas in 2 hours then our
pizza–eating rate =

Job–Rates
6 pizzas
2 hours

Job–Rates
6 pizzas
3 (piz/hr)
=
2 hours

Job–Rates
6 pizzas
3 (piz/hr)
=
2 hours
2 pizzas
6 hours

Job–Rates
6 pizzas
3 (piz/hr)
=
2 hours
2 pizzas
1
(piz/hr)
=
6 hours
3

Job–Rates
6 pizzas
3 (piz/hr)
=
2 hours
2 pizzas
1
(piz/hr)
=
6 hours
3
This type of rates (in per unit of time) are called job–rates.

Job–Rates
6 pizzas
3 (piz/hr)
=
2 hours
2 pizzas
1
(piz/hr)
=
6 hours
3
We note the following about job–rates.

Job–Rates
6 pizzas
3 (piz/hr)
=
2 hours
2 pizzas
1
(piz/hr)
=
6 hours
3
* In all such problems, a complete job to be done may be viewed as a pizza needing to be eaten and it’s set to be 1.

Job–Rates
6 pizzas
3 (piz/hr)
=
2 hours
2 pizzas
1
(piz/hr)
=
6 hours
3
Hence if it takes 3 hrs to paint a room, to mow a lawn, or to fill a pool, then the job–rate for each is
1 job
3 hours

Job–Rates
6 pizzas
3 (piz/hr)
=
2 hours
2 pizzas
1
(piz/hr)
=
6 hours
3
Hence if it takes 3 hrs to paint a room, to mow a lawn, or to fill a pool, then the job–rate for each is
1 job
1
(room), (lawn), or (pool) / hr
=
3 hours
3

* The job–rate may be computed even if only a portion of the job is completed.

* The job–rate may be computed even if only a portion of the job is completed. For example, if it took us 6 hrs to paint 2/3 of the room then the job–rate is
2/3 room
6 hours

2
1
2/3 room
=
*
3
6
6 hours

1
2
1
2/3 room
(room/hr)
=
=
*
9
3
6
6 hours
3

1
2
1
2/3 room
(room/hr)
=
=
*
9
3
6
6 hours
3
* The reciprocal of a job–rate is the amount of time it would take to complete one job.

1
2
1
2/3 room
(room/hr)
=
=
*
9
3
6
6 hours
3
1
(lawn/hr),
Hence if the job–rate is
9
9
then it would take
= 9 hrs to complete the entire lawn.
1

1
2
1
2/3 room
(room/hr)
=
=
*
9
3
6
6 hours
3
1
(lawn/hr),
9
9
then it would take
1
Example C. We can paint 3/4 of a wall in 4 ½ hrs, what is the job–rate? How long would it take to paint the entire wall?

1
2
1
2/3 room
(room/hr)
=
=
*
9
3
6
6 hours
3
1
(lawn/hr),
9
9
then it would take
1
3/4
The job–rate is
4½

1
2
1
2/3 room
(room/hr)
=
=
*
9
3
6
6 hours
3
1
(lawn/hr),
9
9
then it would take
1
3/4
3
2
3/4
The job–rate is
=
=
*
9/2
4
9
4½

1
2
1
2/3 room
(room/hr)
=
=
*
9
3
6
6 hours
3
1
(lawn/hr),
9
9
then it would take
1
1
3/4
3
2
3/4
The job–rate is
(wall / hr)
=
=
=
*
6
9/2
4
9
4½
2
3

1
2
1
2/3 room
(room/hr)
=
=
*
9
3
6
6 hours
3
1
(lawn/hr),
9
9
then it would take
1
1
3/4
3
2
3/4
The job–rate is
(wall / hr)
=
=
=
*
6
9/2
4
9
4½
2
3
The reciprocal of this job–rate is 6 (hr / wall) or it would take 6 hours to paint the entire wall.

Example D. Pipe A can fill a full tank of water in 3 hours, pipe B can fill ¾ of the (same) tank of water in 1½ hr.
a. What is unit of the fill–rate and what is the rate of each pipe?

The unit of the rate in question is tank/hr.

1
The rate of A = tank/hr.
3

1
3
3/4
The rate of B =
=
1½

1
3
3/4
3/4
=
The rate of B =
=
3/2
1½

1
3
3/4
1
3/4
=
The rate of B =
=
tank/hr.
3/2
2
1½

1
3
3/4
1
3/4
=
The rate of B =
=
tank/hr.
3/2
2
1½
b. How much time would it take for pipe B to fill a full tank alone?

1
3
3/4
1
3/4
=
The rate of B =
=
tank/hr.
3/2
2
1½
b. How much time would it take for pipe B to fill a full tank alone?
We reciprocate the rate of B “½ (tank/hr)” and get 2 hr/tank
or that it would take 2hrs for pipe B to fill a full tank.

c. What is the combined rate if both pipes are used and how long would it take to fill the entire tank if both pipes are used?

The combined rate is the sum of the rates
1
1
+
3
2

5
1
1
=
tank/hr.
+
6
3
2

5
1
1
=
tank/hr.
+
6
3
2
The reciprocal of the combined rate is 6/5 therefore it would take 1 1/5 hr if both pipes are used.

5
1
1
=
tank/hr.
+
6
3
2
We note that in the above example, it takes pipe B one hour less to fill the tank then pipe A does and that together they got the job done in 6/5 hr.

5
1
1
=
tank/hr.
+
6
3
2
We note that in the above example, it takes pipe B one hour less to fill the tank then pipe A does and that together they got the job done in 6/5 hr. Let’s treat this as a word problem.

Example E. Pipe B takes one hour less to fill a full tank of water then pipe A does and together they got the job done in 6/5 hr. How long will it take pipe A to fill the tank alone?
(What should x be?)

(What should x be?)
Set x = number of hours for A to fill the tank

(What should x be?)
So the number of hours for B to fill the tank is (x – 1)

(What should x be?)
It takes 6/5 hr to use both. .

(What should x be?)
It takes 6/5 hr to use both.
Put these into a table.

(What should x be?)
Put these into a table. The rates for A and B add to the combined rate 5/6 so we get a rational equation in x.

(What should x be?)
1
5
1
+
=
6
x
(x – 1)

(What should x be?)
1
5
1
]
[
+
x (x – 1)
=
6
x
(x – 1)

(What should x be?)
6(x – 1)
6x
x (x – 1)
1
5
1
]
[
+
6x (x – 1)
=
6
x
(x – 1)

(What should x be?)
6(x – 1)
6x
x (x – 1)
1
5
1
]
[
+
6x (x – 1)
=
6
x
(x – 1)
6x – 6 + 6x = 5x2 – 5x

(What should x be?)
6(x – 1)
6x
x (x – 1)
1
5
1
]
[
+
6x (x – 1)
=
6
x
(x – 1)
6x – 6 + 6x = 5x2 – 5x
you finish this...
0 = 5x2 – 17x + 6
(Question: Why is the other solution no good?)

Ex. A. For problems 1 – 9, fill in the table, set up an equation and solve for x. (If you can guess the answer first, go ahead. But set up and solve the problem algebraically to confirm it.)
Total Cost
No. of people
1. A group of x people are to share 6 slices of pizzas equally.
If there is one less person in the group then each person would get one more slice. What is x?
If there are three less people in the group then each person would get one more slice. What is x?
If one more person joins in to share the pizza then each person would get three less slices. What is x?

If two more people want to partake then each person would get one less slice. What is x?
If four people in the group failed to show up then each person would get four more slices. What is x?
7. A group of x people are to share the cost of $24 to rent a van equally. If two more people want to partake then each person would pay $4 less. What is x?
8. A group of x people are to share the cost of $24 to rent a van equally. If four more people join in the group then each person would pay $1 less. What is x?
9. A group of x people are to share the cost of $24 to rent a van equally. If there are two less people in the group then each person would pay $1 more. What is x?

HW C. For problems 10 – 16, fill in the table, set up an equation and solve for x.
10. We hiked 6 miles from A to B at a rate of x mph. For the return trip our rate was 1 mph slower. It took us 5 hours for the entire round trip. What is x?
11. We hiked 6 miles from A to B at a rate of x mph. For the return trip our rate was 3 mph faster. It took us 3 hours for the entire round trip. What is x?
12. We hiked 6 miles from A to B at a rate of x mph. For the return trip our rate was 1 mph faster. It took us 9 hours for the entire round trip. What is x?

13. We hiked 3 miles from A to B at a rate of x mph. Then we hiked 2 miles from B to C at a rate that was 1 mph slower.
It took us 2 hours for the entire round trip. What is x?
14. We hiked 6 miles from A to B at a rate of x mph. Then we hiked 3 miles from B to C at a rate that was 1 mph faster.
It took us 3 hours for the entire round trip. What is x?
15. We ran 8 miles from A to B. Then we ran 5 miles from B to C at a rate that was 1 mph faster. It took one more hour to get from A to B than from B to C. What was the rate we ran from A to B?
16. We piloted a boat 12 miles upstream from A to B. Then we piloted it downstream from B back to A at a rate that was
4 mph faster. It took one more hour to go upstream from A to B than going downstream from B to C. What was the rate of the boat going up stream from A to B?

For problems 17–22, don’t set up equations. Find the answers directly and leave the answer in fractional hours.
17. Its takes 3 hrs to complete 2/3 of a job.
What is the job–rate?
How long would it take to complete the whole job?
How long would it take to complete ½ of the job?
18. Its takes 4 hrs to complete 3/5 of a job.
What is the job–rate?
How long would it take to complete the whole job?
How long would it take to complete 1/3 of the job?
19. Its takes 4 hrs for A to complete 3/5 of a job and 3 hr for B to do 2/3 of the job. Complete the table.
What is their combined job–rate?
How long would it take for A & B to complete the whole job togrther?

20. Its takes 1/2 hr for A to complete 2/5 of a job and 3/4 hr for B to do 1/6 of the job. Complete the table.
What is their combined job–rate?
How long would it take for A & B to complete the whole job together?
21. It takes 3 hrs for pipe A to fill a
pool and 5 hrs for pipe B to drain the pool. Complete the table.
If we use both pipes to fill and drain simultaneously, what is their combined job–rate?
Will the pool be filled eventually?
If so, how long would it take for
pipes A & B to filled the pool?

22. A leak in our boat will fill the boat in 6 hrs at which time the boat sinks.
We have a pump that can empty a filled boat in 9 hrs. How much time do we have before the boat sinks?
For 23 –26, set up the x and equations. Fill in the table.
23. It take A one more hour than B to
do the job alone. It takes them 2/3 hr to do the job together.
24. It take B two more hours than B todo the job alone. It takes them 3/4 hr to do the job together.
25. It take B two hour less than A to do the job alone.
It takes them 4/3 hr to do the job together.
26. It take A three more hours than B to do the job alone.
It takes them 2 hr to do the job together.

2 4 rational equations word-problems

by math123b on Aug 28, 2011

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2 4 rational equations word-problems — Presentation Transcript