Derivation of the "Combined Work Formula"

Before I get to my question, some background:

• Person $A$ can paint a fence at the rate $9 \frac{hour}{fence}$ (or equivalently $\frac{1}{9} \frac{fence}{hour}$)
• Person $B$ can paint a fence at the rate $5 \frac{hour}{fence}$ (or equivalently $\frac{1}{5} \frac{fence}{hour}$).

And one would be asked how long it takes them, combined, to paint $3$ fences. This is known as a "combined work" problem, and there exists an obscure formula to quickly solve the problem: $\frac{1}{t_A}+\frac{1}{t_B}=\frac{1}{t_{TOTAL}}$.

My question is, why is this formula true? It essentially says $rate_A+rate_B=rate_{total}$. But suppose I cannot remember the exact formula, and I'm attempting to solve the question I posed earlier. Suppose all I remember is $rate_A+rate_B=rate_{total}$. Then I could solve for $t_{total}$ in either \begin{equation} \frac{1 fence}{9 hour}+\frac{1 fence}{5 hour}=\frac{3 fence}{t_{total} hour} \end{equation} or \begin{equation} \frac{9 hour}{1 fence}+\frac{5 hour}{1 fence}=\frac{t_{total} hour}{3 fence} \end{equation}

Both of these are of the form $rate_A+rate_B=rate_{total}$. How should I know which of these is correct, and why?

Essentially, my question is, suppose I don't have this "combined work" formula memorized and want to derive it formally. Where should I start?

Thank you!

Solution 1:

I have always used direct relation ratios and dimensional analysis to attack problems more difficult than this, still works for simple problems like this. Here it goes:

$A$ takes 9 hours to paint a fence. In one hour he paints $\frac{1}{9}$ fence.

Similarly, $B$ takes 5 hours to paint a fence. In one hour he paints $\frac{1}{5}$ fence.

If both of them worked together, in one hour they would complete $\frac{1}{9}+\frac{1}{5} = 14/45$ fences/hour.

Now, with combined work, $14/45$ fence in one hour (fence/hour). One fence would take $\frac{45}{14}$ hours/fence which is the reciprocal, because: work output = rate $×$ time, or: $1 = \frac{14}{45} \times t$, and when you divide 1 by the rate you end up getting the reciprocal. Now 3 fences would take $\frac{3 × 45}{14}$. There goes your answer.

Hope it helps.

Thanks.

Satish.