SOLUTION: If painter A can paint a wall in one hour and painter B can paint the same wall in two hours, how long does it take for painter A and B to paint the same wall together?

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Question 1210372: If painter A can paint a wall in one hour and painter B can paint the same wall in two hours, how long does it take for painter A and B to paint the same wall together?
Found 4 solutions by ikleyn, mccravyedwin, josgarithmetic, math_tutor2020:
Answer by ikleyn(52772)   (Show Source): You can put this solution on YOUR website!
.
If painter A can paint a wall in one hour and painter B can paint the same wall in two hours,
how long does it take for painter A and B to paint the same wall together?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


This problem is so simple that I will solve it mentally using reasoning, ONLY. 

Working together, they will spend equal time.


Since painter A works twice as fast as painter B, painter A will make twice greater job than painter B.


Hence, painter A will make 2/3 of the job, while painter B will make 1/3 of the job.


To make 2/3 of the job, painter A needs 2/3 of an hour, i.e. 40 minutes.


Hence, working together, both painters will complete the job in 2/3 of an hour, or in 40 minutes.

Solved.

Simple problems should be solved with simple methods, isn't it ?


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This problem is interesting and deserves a thorough discussion.

From one hand side, it is a typical joint-work word problem.

But a typical joint-work word problem usually tells about small pieces of a whole job
that can be done by participants in a certain time interval.

Therefore, been a typical problem, this given problem, from the other hand side, is unusual,
because it tells about grand pieces of a whole job, like the whole wall, which can be painted in one hour.

And if a student is non-experienced, this immediately puts the student in a dead end.

Some tutors may come explaining that a student can work with grand pieces exactly in the same
way as he/she works with small pieces.

But I do not believe that an average/typical beginner student will be satisfied and will accept such an explanation.
Such explanations can be good for tutors or for teachers (because formally they complete their job doing this way),
but a student will unlikely be satisfied and unlikely will accept such a solution.

Again, explanations with grand pieces are good for teacher and for tutors, especially if they want to shock a student.
Yes, the shocking method is sometimes a good method for teaching, especially for advanced students.
But an average young student will remain unsatisfied. His soul will ask for a more standard solution/explanation
in terms he/she is accustomed to: in terms of "small pieces".

Here is an example on how to build/(to create) such a solution/explanation for this given problem.

Let's make our reasoning not in terms of hours, but in terms of minutes.

Then the problem can be equivalently re-phrased this way 
  
    If painter A can paint a wall in 60 minutes and painter B can paint the same wall in 120 minutes,
    how long does it take for painters A and B to paint the same wall together?

Surely, now this modified problem looks like an absolutely standard even for a beginner.
Now we pronounce a standard mantra 

    painter A makes 1/60 of the job in one minute,

    painter B makes 1/120 of the job in one minute.

    And working together, they will make  

          +  =  +  =  =   of the job per minute.


   From it, we conclude immediately that the necessary time 
   to complete the job working together is 40 minutes.

This way, the problem is solved within the usual/traditional framework.

I wrote these explanations to show that sometime it is useful
to take a look to a problem from the other/unexpected point/angle of view.


XXXXXXXXXXXX   S e v e r a l   d a y s   l a t e r   XXXXXXXXXXXX


And now I invite you to laugh with me at the AI solution to a similar problem.

Today, 8/13/2025, I posted this similar problem to Google 

    If painter A can paint a wall in half an hour and painter B can paint the same wall in half an hour,
    how long does it take for painters A and B to paint the same wall together?


Below is copy-paste of the AI response

            - - - - - - - S T A R T - - - - - - - - -

AI Overview

If painter A can paint a wall in 30 minutes (1/2 hour) and painter B can paint the same wall in 30 minutes (1/2 hour),
together they would take 15 minutes (1/4 hour) to paint the wall.

Explanation:

Rate of painting: Painter A paints 1/2 of the wall in 1 minute, and painter B also paints 1/2 of the wall in 1 minute.

Combined rate: Together, they paint 1/2 + 1/2 = 1 wall in one minute.

Time to paint the whole wall: Since they paint 1 wall in 1 minute, they can paint the whole wall together in 1/(1/2 + 1/2) = 1/1 = 15 minutes.

https://www.google.com/search?q=If+painter+A+can+paint+a+wall+in+half+an+hour+and+painter+B+can+paint+the+same+wall+in+half+an+hour%2C+how+long+does+it+take+for+painter+A+and+B+to+paint+the+same+wall+together%3F&rlz=1C1CHBF_enUS1071US1071&oq=If+painter+A+can+paint+a+wall+in+half+an+hour+and+painter+B+can+paint+the+same+wall+in+half+an+hour%2C+how+long+does+it+take+for+painter+A+and+B+to+paint+the+same+wall+together%3F&gs_lcrp=EgZjaHJvbWUyBggAEEUYOdIBCTE5MDZqMGoxNagCCLACAfEFFpskuz7iUCjxBRabJLs-4lAo&sourceid=chrome&ie=UTF-8

            - - - - - - - F I N I S H - - - - - - - - -

8/13/2025


Notice that the answer of the AI is correct, but his intermediate reasoning and calculations are wrong.

In other words, in this case, the AI, without having a prototype of the solution / (without having a TEMPLATE)
behaves exactly like a dumb student: he can't figure out what to do and how to get the answer correctly.
This only confirms my reasoning above about the average beginner student.

The Google AI and such a student behave identically and suffer identically.
Proved experimentally.

Ha - ha - ha.


///////////////////////////////////////////


This story has interesting continuation.

On September 2, 2025, I submitted this problem to Google AI again.

This time it produced correct, right, perfect and ideal solution.

It tells me that AI needs very much to have an adequate, attentive and qualified teacher/tutor/mentor.

Then the AI learns and increases its level and becomes adequate.

I am glad to see this progress.



Answer by mccravyedwin(406)   (Show Source): You can put this solution on YOUR website!
Another way to look at it is what they COULD do in the same time.

Painter A can paint a wall in 60 minutes and painter B can paint the same wall in
120 minutes.  Therefore,

painter A COULD paint 2 walls in 120 minutes while painter B COULD paint only 1
wall in that same 120 minutes, so in 120 minutes together they COULD paint 3
walls, so to paint only 1 wall it would take them only 1/3 of 120 minutes or 40
minutes.

Edwin
Answer by josgarithmetic(39615)   (Show Source): You can put this solution on YOUR website!
PAINTER        RATE         TIME        WORK
    A           1            1           1
    B           1/2          2           1
   A&B         1+1/2         ?           1

x, the time for A and B together to do the one job of "paint the wall"




or about 40 minutes
Answer by math_tutor2020(3816)   (Show Source): You can put this solution on YOUR website!

Let's say the wall has an area of 300 square feet.
This value can be changed to anything you want, and the final answer will remain the same.

When working alone painter A can paint this wall in one hour, so his or her rate is 300/1 = 300 sq ft per hour.
The formula to use would be: speed = amountDone/time
Painter B has a rate of 300/2 = 150 sq ft per hour when working alone.

When the two work together, assuming neither worker slows down the other, we add their unit rates.
A+B = 300+150 = 450
Their combined rate is 450 sq ft per hour.

rate*time = amountDone
(450 sq ft per hr)*(x hours) = 300 sq ft
450*x = 300
x = 300/450
x = (2*150)/(3*150)
x = 2/3
It takes 2/3 of an hour if the two work together.
This is equivalent to 40 minutes because (2/3)*60 = 40.


Answer: 40 minutes
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