17 applications of proportions and the rational equations

Rational Equations Word-Problems

Frank Ma © 2011

Rational Equations Word-ProblemsWe look at the following applications of rational equations.


Problems from the Multiplication–Division OperationsWe look at the following applications of rational equations.



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




as A = . This divisional form leads to rational equations. The calculation of “per unit” is a good example:.

C B





Total amount Number of units

The calculation of “per unit” is a good example: Per unit amount =

C B






Cost per Unit


C B






Cost per Unit


C B

Total cost Number of units Per unit cost =






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.

Cost per Unit


C B







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.However if one person backs out, then the cost goes up tos = 20/4 = $5 per person.

Cost per Unit


C B







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.However if one person backs out, then the cost goes up tos = 20/4 = $5 per person. Each remaining person has to pay $1 more.

Cost per Unit


C B







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.However if one person backs out, then the cost goes up tos = 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.

Cost per Unit


C B


Rational Equations Word-ProblemsA 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?

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




Total Cost No. of People Cost per Person =

20

20

Total CostNo. of people





20 x

20 (x – 1)






20 x

20 (x – 1)


20x

20(x – 1)





20 x

20 (x – 1)


20x

20(x – 1)

Let’s compare the two different per/person costs. 20

(x – 1)20 x





20 x

20 (x – 1)


20x

20(x – 1)

Let’s compare the two different per/person costs.

cost more cost less

20(x – 1)

20 x





20 x

20 (x – 1)


20x

20(x – 1)


cost more cost less ($1 more)

20(x – 1)

20 x




20 x

20 (x – 1)


20x

20(x – 1)

20(x – 1)

20 x


– = 1

cost more cost less ($1 more)





20 x

20 (x – 1)


20x

20(x – 1)

20(x – 1)


– = 1[ ] x (x – 1) clear the denominators by LCM


20 x




20 x

20 (x – 1)


20x

20(x – 1)

20(x – 1)

20 x


– = 1[ ] x (x – 1) clear the denominators by LCM

x (x – 1) x (x – 1)





20 x

20 (x – 1)


20x

20(x – 1)

20(x – 1)


= 1[ ] x (x – 1) clear the denominators by LCM

x (x – 1) x (x – 1)

20x – 20(x – 1) = x(x – 1)


20 x –




20 x

20 (x – 1)


20x

20(x – 1)

20(x – 1)



x (x – 1) x (x – 1)

20x – 20(x – 1) = x(x – 1)

20x – 20x + 20 = x2 – x


20 x –




20 x

20 (x – 1)


20x

20(x – 1)

20(x – 1)



x (x – 1) x (x – 1)

20x – 20(x – 1) = x(x – 1)


0 = x2 – x – 20set one side as 0

20 x –

20x – 20x + 20 = x2 – x

Rational Equations Word-Problems0 = x2 – x – 20

Rational Equations Word-Problems0 = x2 – x – 200 = (x – 5)(x + 4)

Rational Equations Word-Problems0 = x2 – x – 200 = (x – 5)(x + 4)x = 5 or x = –4.

Rational Equations Word-Problems0 = x2 – x – 200 = (x – 5)(x + 4)x = 5 or x = –4. Hence there are 5 people.


Rate–Time–Distance

0 = x2 – x – 200 = (x – 5)(x + 4)x = 5 or x = –4. Hence there are 5 people.


Let R = rate (mph), T = time (hours) and D = Distance (m)

then RT = D or that T = . .DR






We hiked a 6–mile trail from A to B at a rate of 3 mph so the trip took 6/3 = 2 hours.






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.






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. In a table,

Distance (m) Rate (mph) Time = D/R (hr)

Going

Return








Going 6 3 2

Return








Going 6 3 2

Return 6 2 3








Going 6 3 2

Return 6 2 3

Let’s turn this example into a rational–equation word problem.

0 = x2 – x – 200 = (x – 5)(x + 4)x = 5 or x = –4.


Hence there are 5 people.

Rational Equations Word-ProblemsExample 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).


D = Distance (m)

R = Rate (mph)

Time = D/R (hr)

Go

Return

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


D = Distance (m)

R = Rate (mph)

Time = D/R (hr)

Go 6

Return 6



D = Distance (m)

R = Rate (mph)

Time = D/R (hr)

Go 6 x

Return 6



D = Distance (m)

R = Rate (mph)

Time = D/R (hr)

Go 6 x 6/x

Return 6



D = Distance (m)

R = Rate (mph)

Time = D/R (hr)

Go 6 x 6/x

Return 6 x + 1



D = Distance (m)

R = Rate (mph)

Time = D/R (hr)

Go 6 x 6/x

Return 6 x + 1 6/(x + 1)



D = Distance (m)

R = Rate (mph)

Time = D/R (hr)

Go 6 x 6/x

Return 6 x + 1 6/(x + 1)


Now we use the information about the times.


D = Distance (m)

R = Rate (mph)

Time = D/R (hr)

Go 6 x 6/x

Return 6 x + 1 6/(x + 1)


6(x + 1)

6 x

number of

hrs for going


number of

hrs for returning


D = Distance (m)

R = Rate (mph)

Time = D/R (hr)

Go 6 x 6/x

Return 6 x + 1 6/(x + 1)


6(x + 1)

6 x


(total trip 5 hrs)number of

hrs for going

number of

hrs for returning


D = Distance (m)

R = Rate (mph)

Time = D/R (hr)

Go 6 x 6/x

Return 6 x + 1 6/(x + 1)


6(x + 1)

6 x


+ = 5

(total trip 5 hrs)number of

hrs for going

number of

hrs for returning


D = Distance (m)

R = Rate (mph)

Time = D/R (hr)

Go 6 x 6/x

Return 6 x + 1 6/(x + 1)


6(x + 1)

6 x


+ = 5 use the cross multiplication method


D = Distance (m)

R = Rate (mph)

Time = D/R (hr)

Go 6 x 6/x

Return 6 x + 1 6/(x + 1)


6(x + 1)

6 x


+ = 5

6(x + 1) + 6xx(x + 1) = 5

use the cross multiplication method


D = Distance (m)

R = Rate (mph)

Time = D/R (hr)

Go 6 x 6/x

Return 6 x + 1 6/(x + 1)


6(x + 1)

6 x


+ = 5

6(x + 1) + 6xx(x + 1) = 5

5 1 = x(x + 1)

12x + 6



D = Distance (m)

R = Rate (mph)

Time = D/R (hr)

Go 6 x 6/x

Return 6 x + 1 6/(x + 1)


6(x + 1)

6 x


+ = 5

6(x + 1) + 6xx(x + 1) = 5

5 1 = x(x + 1)

12x + 6 Cross again. You finish it.



D = Distance (m)

R = Rate (mph)

Time = D/R (hr)

Go 6 x 6/x

Return 6 x + 1 6/(x + 1)


6(x + 1)

6 x


+ = 5

6(x + 1) + 6xx(x + 1) = 5

5 1 = x(x + 1)

12x + 6 Cross again. You finish it.(Remember that x = 2 mph.)


Rational Equations Word-ProblemsJob–Rates


If we can eat 6 pizzas in 2 hours then our

pizza–eating rate =



6 pizzas 2 hours pizza–eating rate =



6 pizzas 2 hours pizza–eating rate = = 3 (piz/hr)





pizza–eating rate =





2 pizzas 6 hours pizza–eating rate =





2 pizzas 6 hours pizza–eating rate = = (piz/hr) 1

3






3

These types of rates (in per unit of time) are called job–rates.






3

These types of rates (in per unit of time) are called job–rates. We note the following about job–rates.






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.






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 job3 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 job3 hours = (room), (lawn), or (pool) / hr 1

3

Rational Equations Word-Problems* The job–rate may be computed even if only a portion of the job is completed.

Rational Equations Word-Problems* 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 room6 hours


2/3 room6 hours = 2

3 1 6

*


2/3 room6 hours = (room/hr)1

9 2 3

1 6

* =3



9 2 3

1 6

* =

* The reciprocal of a job–rate is the amount of time it would take to complete one job.

3



9 2 3

1 6

* =


(lawn/hr), 1 9 Hence if the job–rate is

9 1 then it would take = 9 hrs to complete the entire lawn.

3



9 2 3

1 6

* =




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?

3



9 2 3

1 6

* =





The job–rate is 3/44½

3



9 2 3

1 6

* =





The job–rate is 3/44½ = 3/4

9/2 = 3 4

2 9

*

3



9 2 3

1 6

* =





The job–rate is 3/44½ (wall / hr)= 1

6 =3/49/2 = 3

42 9

*2 3

3



9 2 3

1 6

* =





The job–rate is 3/44½ (wall / hr)= 1

6 =3/49/2 = 3

42 9

*2 3

The reciprocal of this job–rate is 6 (hr / wall) or it would take 6 hours to paint the entire wall.

3

Rational Equations Word-ProblemsExample 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?


a. What is unit of the fill–rate and what is the rate of each pipe?

Pipes A = Amount (tank) T = Time (hr) Rate = A/T (tank/hr)

A

B


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.


A

B




A 1 3

B



1 3 The rate of A = tank/hr.


A 1 3 1/3

B





A 1 3 1/3

B 3/4 1 ½ = 3/2 1/2




The rate of B = 3/41½ =


A 1 3 1/3

B 3/4 1 ½ = 3/2 1/2



1 3

=

The rate of A = tank/hr.

The rate of B = 3/41½ = 3/4

3/2


A 1 3 1/3

B 3/4 1 ½ = 3/2 1/2



1 3

= 12


The rate of B = tank/hr. 3/41½ = 3/4

3/2


A 1 3 1/3

B 3/4 1 ½ = 3/2 1/2



1 3

= 12



3/2

b. How much time would it take for pipe B to fill the tank alone?


A 1 3 1/3

B 3/4 1 ½ = 3/2 1/2



1 3

= 12



3/2

b. How much time would it take for pipe B to fill the 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.


A 1 3 1/3

B 3/4 1 ½ = 3/2 1/2

Rational Equations Word-ProblemsPipes A = Amount (tank) T = Time (hr) Rate = A/T (tank/hr)

A 1 3 1/3

B 3/4 1 ½ = 3/2 1/2

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


A 1 3 1/3

B 3/4 1 ½ = 3/2 1/2


1 3 + 1

2


5 6 =


tank/hr.


A 1 3 1/3

B 3/4 1 ½ = 3/2 1/2


1 3 + 1

2


5 6 =


tank/hr.


A 1 3 1/3

B 3/4 1 ½ = 3/2 1/2

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


1 3 + 1

2


5 6 =


tank/hr.


A 1 3 1/3

B 3/4 1 ½ = 3/2 1/2



1 3 + 1

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 6 =


tank/hr.


A 1 3 1/3

B 3/4 1 ½ = 3/2 1/2



1 3 + 1

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.

Rational Equations Word-ProblemsExample 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?)


Set x = number of hours for A to fill the tank

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?)




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




So the number of hours for B to fill the tank is (x – 1) It takes 6/5 hr to use both. .



Pipes Time (hr)

Rate (tank/hr)


So the number of hours for B to fill the tank is (x – 1) It takes 6/5 hr to use both. Put these into a table.



Pipes Time (hr)

Rate (tank/hr)

A

B

A & B





Pipes Time (hr)

Rate (tank/hr)

A x

B (x – 1)

A & B 6/5





Pipes Time (hr)

Rate (tank/hr)

A x 1/x

B (x – 1) 1/(x – 1)

A & B 6/5 5/6





Pipes Time (hr)

Rate (tank/hr)

A x 1/x

B (x – 1) 1/(x – 1)

A & B 6/5 5/6


So the number of hours for B to fill the tank is (x – 1) It takes 6/5 hr to use both. . 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.



Pipes Time (hr)

Rate (tank/hr)

A x 1/x

B (x – 1) 1/(x – 1)

A & B 6/5 5/6


So the number of hours for B to fill the tank is (x – 1) It takes 6/5 hr to use both. .

5 6 = 1

x + 1(x – 1)

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.


5 6 = 1

x + 1(x – 1) clear the denominatorsx (x – 1)] [


Pipes Time (hr)

Rate (tank/hr)

A x 1/x

B (x – 1) 1/(x – 1)

A & B 6/5 5/6


So the number of hours for B to fill the tank is (x – 1) It takes 6/5 hr to use both. . 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.


5 6 = 1

x + 1(x – 1) clear the denominators6x (x – 1)

6x] [


Pipes Time (hr)

Rate (tank/hr)

A x 1/x

B (x – 1) 1/(x – 1)

A & B 6/5 5/6




It takes 6/5 hr to use both. .

6(x – 1) x (x – 1)


5 6 = 1


6x] [


Pipes Time (hr)

Rate (tank/hr)

A x 1/x

B (x – 1) 1/(x – 1)

A & B 6/5 5/6





6(x – 1) x (x – 1)

6x – 6 + 6x = 5x2 – 5x


5 6 = 1


6x] [


Pipes Time (hr)

Rate (tank/hr)

A x 1/x

B (x – 1) 1/(x – 1)

A & B 6/5 5/6





6(x – 1) x (x – 1)

6x – 6 + 6x = 5x2 – 5x0 = 5x2 – 17x + 6

(Question: Why is the other solution no good?)

you finish this...

Rational Equations Word-ProblemsEx. 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.)

2. A group of x people are to share 6 slices of pizza equally. If there are three less people in the group then each person would get one more slice. What is x?3. A group of x people are to share 6 slices of pizza equally. If one more person joins in to share the pizza then each person would get three less slices. What is x?

1. A group of x people are to share 6 slices of pizza equally. If there is one less person in the group then each person would get one more slice. What is x?

Total Cost No. of People Cost per Person = Total CostNo. of people

Rational Equations Word-Problems5. A group of x people are to share 12 slices of pizza equally. If two more people want to partake then each person would get one less slice. What is x?

6. A group of x people are to share 12 slices of pizza equally. 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 three 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?


Distance (m)

Rate (mph) Time = D/R (hr)

1st trip

2nd trip10. 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?

HW C. For problems 10 – 16, fill in the table, set up an equation and solve for 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?

Rational Equations Word-Problems13. 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?


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?

For problems 17–22, don’t set up equations. Find the answers directly and leave the answer in fractional hours.

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?

Time Rate

A

B

A & B

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 together?


Time Rate

A

B

A & B

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?

Time Rate

A

B

A & B

21. It takes 3 hrs for pipe A to fill apool 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 the pool to fill?

Rational Equations Word-ProblemsTime Rate

leak

pump

both

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, given the information, how many hours it would take for A or B to do the job alone.

23. It takes A one more hour than B to do the job alone. It takes them 2/3 hr to do the job together.

24. It takes A two more hours than B to do the job alone. It takes them 3/4 hr to do the job together.

25. It takes B two hour less than A to do the job alone. It takes them 4/3 hr to do the job together.

26. It takes A three more hours than B to do the job alone. It takes them 2 hr to do the job together.

Education

17 applications of proportions and the rational equations