Derivation of the "Combined Work Formula"

Question

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!

Answer 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.

Answer 2

You have to think about what happens when they both paint part of the fence. They paint for the same number of hours, so the appropriate thing to do is add how much each one can paint in one hour. If you gave each of them some fences to paint, and the painting was sequential, A would paint his fences at $9$ hours each, then B would paint his at $5$ hours each.

Answer 3

the question reminds me of another problem

If Bob can drive from Boston to New York in 4 hours, and Fred can do the same trip in 5 hours, how long would it take them to do the trip together?

Ok, well maybe not exactly the same.

Although both formulas are technically rates, one is a rate of production or frequency, while the other is a Period. I would use one formula if the fences were painted simultaneously, and the other if they were painted sequentially.