SOLUTION: Quiboy can clean the barn in 6 hours, and Quiboy and Rod together can clean the same barn in 3 hours. How long would it take for Rod to clean the barn by himself?

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Question 1198854: Quiboy can clean the barn in 6 hours, and Quiboy and Rod together can clean the same barn in 3 hours. How long would it take for Rod to clean the barn by himself?
Found 4 solutions by ikleyn, greenestamps, math_tutor2020, MathTherapy:
Answer by ikleyn(52848)   (Show Source): You can put this solution on YOUR website!
.
Quiboy can clean the barn in 6 hours, and Quiboy and Rod together can clean
the same barn in 3 hours. How long would it take for Rod to clean the barn by himself?
~~~~~~~~~~~~~ 

The combined rate of work of two individuals is    of the job per hour,

while the rate of work of one of them is    of the job per hour.



Hence, the rate of work of the other individual is the difference

     -  =  -  =   of the job per hour.



It means that the second individual can make this job in 6 hours, working alone.    ANSWER

Solved.

---------------

It is a standard and typical joint work problem.

There is a wide variety of similar solved joint-work problems with detailed explanations in this site.  See the lessons
    - Using Fractions to solve word problems on joint work
    - Solving more complicated word problems on joint work
    - Selected joint-work word problems from the archive


Read them and get be trained in solving joint-work problems.

Also,  you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this textbook under the topic
"Rate of work and joint work problems"  of the section  "Word problems".


Save the link to this online textbook together with its description

Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson

to your archive and use it when it is needed.



Answer by greenestamps(13203)   (Show Source): You can put this solution on YOUR website!


Tutor @ikleyn has provided a typical formal algebraic solution.

Without algebra, it might make sense to you that, since the two together take half as much time as Quiboy alone, it would also take Rod the same 6 hours to do the job alone.

If it is hard for you to see that solution using logical reasoning, here is another method for solving the problem.

In 6 hours, Quiboy can do the job alone.

In those same 6 hours, the two of them together could do the job twice, because it takes them 3 hours to do the job once.

So in 6 hours Quiboy could do the job 1 time, while in 6 hours the two together could do the job twice; that means in those same 6 hours Rod could do the job once.

ANSWER: Rod alone would take 6 hours to clean the barn alone.
Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

The jump from 6 hours to 3 hours is "times 1/2".

This strongly suggests the two people work at the same rate, and take the same amount of time.
This is because they split the work in half (thereby cutting their worktime in half as well).

--------------------------------------

We can think of it like saying the task is to move 60 bales of hay.
Quiboy needs 6 hours to do the job by himself.
His unit rate is 60/6 = 10 bales per hour.

When working together, the two men can do the job in 3 hours.
Their combined rate is 60/3 = 20 bales per hour.
This of course assumes neither worker hinders the other.

Quiboy offers 10 bales per hour, and the total combined rate is 20 bales per hour.
Rod offers the remaining 20-10 = 10 bales per hour.
His rate is the same as Quiboy. So he takes the same amount of time.

--------------------------------------

If you wish to use algebra, then,
x = time (in hours) needed for Rod to do the job alone

1/6 = Quiboy's unit rate
1/x = Rod's unit rate
1/3 = combined unit rate

1/6 + 1/x = 1/3
x/(6x) + 6/(6x) = 1/3
(x+6)/(6x) = 1/3
3(x+6) = 6x*1
3x+18 = 6x
18 = 6x-3x
18 = 3x
3x = 18
x = 18/3
x = 6

Or you could say
1/6 + 1/x = 1/3
6x * ( 1/6+1/x ) = 6x*( 1/3 )
6x*(1/6) + 6x*(1/x) = 6x*( 1/3 )
x + 6 = 2x
6 = 2x-x
6 = x
x = 6


In my opinion an algebraic approach is a bit overkill.
Despite that, it's still handy to know this pathway and get practice.

--------------------------------------

Answer: 6 hours
Answer by MathTherapy(10555)   (Show Source): You can put this solution on YOUR website!

Quiboy can clean the barn in 6 hours, and Quiboy and Rod together can clean the same barn in 3 hours. How long would it take for Rod to clean the barn by himself?
Let time Rod takes to do job, alone, be R
Then Rod can complete  of job in 1 hour
As Quiboy takes 6 hours to complete job alone,  of job can be completed in 1 hour
As both take 3 hours to complete job, working together, we get: 
                                                                6 + R = 2R ---- Multiplying by LCD, 6R
                                                                    6 = 2R - R

                         Time Rod takes to complete job, alone, or

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