## Problem
Tom and Rick can do a job in 2 hours.
Tom and Harry can do the same job in 3 hours.
Rick and Harry can do the same job in 4 hours.
How long will it take for all three of them do this job?
## Hints
The first intuition of most students is to write the following system of equations.
$
\begin{equation}
\begin{cases}
t+r=2
\\
r+h=3
\\
t+h=4
\end{cases}
\end{equation}
$
If we go on to solve this, we find
$
\begin{align}
2t+2r+2h=2+3+4
\\
t+r+h=\frac{9}{2}
\end{align}
$
Taking a step back, you might see the problem. This would imply adding somebody to work on this job results in the job taking *more* time, not less.
Unless that someone is just another bad project manager, something must be off with this calculation.
Now you might realize from here that what $\frac92$ represents is the time it would take for all three of them to do the job *three* times !
So it would make sense to divide that result by $3$, giving us $\frac96$, which, unfortunately, is still the wrong answer.
## Solution
The issue with the above approach lies in its modeling.
Rushing into translating words into equations can be misleading.
If we rephrase slightly we get a better intuition of what our system of equations should look like.
What "Tom and Rick can do a job in 2 hours" really means is that 2 hours of Tom's work and 2 hours of Rick's work gives us $1$ unit of work.
Let $t$, $r$ and $h$ represent respectively $1$ hour of Tom's work, $1$ hour of Rick's work and $1$ hour of Harry's work.
$
\begin{equation}
\begin{cases}
2t+2r=1
\\
3r+3h=1
\\
4t+4h=1
\end{cases}
\end{equation}
$
It now becomes obvious what $2t$ stands for: A percentage of one unit of work.
Let us call $x$ the time it will take for Tom, Rick and Harry to do the job. They will all be working the same amount of time, so we can write the following.
$
\begin{align}
xt+xh+xr=1
\\
x(t+h+r)=1
\\
x=\frac{1}{t+h+r}
\end{align}
$
To find out what $t+r+h$ stands for, let's rewrite our original system of equation.
$
\begin{equation}
\begin{cases}
t+r=\frac12
\\
r+h=\frac13
\\
t+h=\frac14
\end{cases}
\end{equation}
$
Combining those equations, we find
$
\begin{align}
2t+2r+2h=\frac12+\frac13+\frac14
\\
2(t+r+h)=\frac{26}{24}
\\
t+r+h=\frac{13}{24}
\end{align}
$
We can now solve for $x$.
$
\begin{align}
x=\frac{1}{t+h+r}
\\
x=\frac{1}{\frac{13}{24}}
\\
x=\frac{24}{13}
\end{align}
$
This is approximately $1.846$ hours, or $110.769$ minutes.
In standard form this gives us about 1 hour 51 minutes.
It follows that adding Harry to the Tom and Rick duo has only improved their performance by about 9 minutes.
Harry must therefore be the project manager.
___
*Source: [MindYourDecisions](https://www.youtube.com/watch?v=BUmk_h4nl8M)*