# ASK: Time for Alan & Brian to Paint House Together?

• MHB
• Monoxdifly
In summary, the purpose of "ASK: Time for Alan & Brian to Paint House Together?" is to determine the total time it will take for Alan and Brian to paint a house together, given their individual painting rates and the size of the house. To calculate this time, you can use the formula: Time = (House size) / (Alan's rate + Brian's rate). This formula can be used for any size house and painting rate as long as they are given in the same units. If Alan's painting rate is faster than Brian's, it will take them less time to paint the house together. The formula does not factor in breaks or overlapping work time, but these variables may affect the actual time it takes for them to complete
Monoxdifly
MHB
3 friends, Alan, Brian and Chester, paint a house. If Alan had to paint it on his own, it would take him one hour more than the time it would take for all three to paint it together. If Brian had to paint it on his own, it would take him five hours more than the time it would take for all three to paint it together, and Chester 8 hours more.
How much time would it take for Alan and Brian to paint it together?

If Alan had to paint it on his own, it would take him one hour more than the time it would take for all three to paint it together.
Would it be like this?
$$\displaystyle \frac1{A+1}=\frac1A+\frac1B+\frac1C$$?
Will it have something to do with quadratic equation?

Actually, the solution I came up with involved both a cubic & quadratic equation. There may be some clever, short solution, but I was unable to reason one out.

let $t$ be the number of hours it takes for all three painters to complete the job.

$\dfrac{1}{t+1} + \dfrac{1}{t+5} + \dfrac{1}{t+8} = \dfrac{1}{t}$

working this equation yielded the cubic equation

$t^3+7t^2-20=0$

using the rational root theorem resulted in the factorization

$(t+2)(t^2+5t-10)=0$

solving the quadratic factor yielded $t=\dfrac{-5+\sqrt{65}}{2} \approx 1.53 \text{ hrs}$, the time required to complete the job with all three working.

for just A and B ...

$\dfrac{1}{t+1}+\dfrac{1}{t+5} = \dfrac{1}{x}$, where $x$ is the time to complete the job with just A and B working.

Being lazy, I used my calculator ... $x \approx 1.82 \text{ hrs}$

Just curious, where did you get this problem?

skeeter said:
Just curious, where did you get this problem?
From another forum. I though it looked simple enough, so I saved it in case I needed it. Turned out I couldn't solve it, so I tried to retrace from which forum it was from, but Google search showed no result. Thanks for your help, though. :)

Monoxdifly said:
3 friends, Alan, Brian and Chester, paint a house. If Alan had to paint it on his own, it would take him one hour more than the time it would take for all three to paint it together. If Brian had to paint it on his own, it would take him five hours more than the time it would take for all three to paint it together, and Chester 8 hours more.
How much time would it take for Alan and Brian to paint it together?

If Alan had to paint it on his own, it would take him one hour more than the time it would take for all three to paint it together.
Would it be like this?
$$\displaystyle \frac1{A+1}=\frac1A+\frac1B+\frac1C$$?
Will it have something to do with quadratic equation?
Since you havn't said what "A", "B", and "C" represent it is impossible to say whether that equation is correct or not!

I will presume that "A" is the time, in hours, it would take Alan to do the job alone, that "B" is the time, in hours, it would take Brian to do the job alone, and that "C" is the time, in hours, it would take Chester to do the job alone. The rate at which each works will be 1 "job" divided by the time it takesto do job. Alan works at the rate of $$\displaystyle \frac{1}{A}$$ job/hour, Brian works at the rate of $$\displaystyle \frac{1}{B}$$ job/hour, and Chester works at the rate of $$\displaystyle \frac{1}{C}$$ job/hour.

When people work together their rates add. Working together, their rate will be $$\displaystyle \frac{1}{A}+ \frac{1}{B}+ \frac{1}{C}=\frac{BC+AC+ AB}{ABC}$$ job/hour and the time it would take for all three to do the job working together is
$$\displaystyle \frac{1}{\frac{BC+ AC+ AB}{ABC}}= \frac{ABC}{BC+ AC+ AB}$$

If Alan had to paint it on his own, it would take him one hour more than the time it would take for all three to paint it together.
So $$\displaystyle A= \frac{ABC}{BC+AC+AB}+ 1$$.

If Brian had to paint it on his own, it would take him five hours more than the time it would take for all three to paint it together
So $$\displaystyle B= \frac{ABC}{BC+AC+AB}+ 5$$.

and Chester 8 hours more.
So $$\displaystyle C= \frac{ABC}{BC+AC+AB}+ 8$$.

Solve those three equations for A, B, and C.

Since this has been here since January and I just can't leave it undone:

We have the three equations

$A= \frac{ABC}{AB+ AC+ BC}+ 1$

$B= \frac{ABC}{AB+ AC+ BC}+ 5$

$C= \frac{ABC}{AB+ AC+ BC}+ 8$
We can write those as

$A- 1= \frac{ABC}{AB+ AC+ BC}$

$B- 5= \frac{ABC}{AB+ AC+ BC}$ and

$C- 8= \frac{ABC}{AB+ AC+ BC}$
So A- 1= B- 5= C- 8

From A- 1= B- 5, A= B- 4.

From B- 5= C- 8, C= B+ 3.
So we can write $B- 5= \frac{ABC}{AB+ AC+ BC}$ as

$B- 5= \frac{B(B- 4)(B+3)}{B(B- 4)+ (B- 4)(B+ 3)+ B(B+ 3)}$.

$(B-5)B(B- 4)+ (B-5)(B+4)(B+3)+ (B-5)B(B+ 3)= B(B- 4)(B+3)$.
Multiply that out and each side will have a term of $B^3$ which will cancel leaving a quadratic equation to solve for B.

## 1. What is "ASK: Time for Alan & Brian to Paint House Together" about?

"ASK: Time for Alan & Brian to Paint House Together" is a computer program that simulates two people, Alan and Brian, working together to paint a house. It allows users to input various factors such as the size of the house, the painting speed of each person, and breaks they may take, and then calculates how long it will take for them to finish painting the house together.

## 2. How accurate is the simulation?

The simulation is based on realistic factors and calculations, so it is relatively accurate. However, it is important to keep in mind that it is still a simulation and may not account for all variables and situations that may occur in real life.

## 3. Can I customize the factors in the simulation?

Yes, the program allows users to input their own values for factors such as house size, painting speed, and breaks. This allows for a more personalized and accurate simulation for your specific situation.

## 4. Is this program only for painting houses?

No, the program can be used for any task that involves two people working together and can be adjusted to fit different scenarios. However, it is primarily designed for tasks that involve a similar level of physical labor, such as painting a house.

## 5. Can this simulation be used for team building or productivity purposes?

Yes, this simulation can be used as a tool for team building or productivity exercises. It can help teams understand the importance of communication, coordination, and efficiency in completing tasks together. It can also be used to identify and address potential issues or challenges that may arise when working with others.

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