Interesting connection between a word problem and a physics formula

Mathematics Asked on January 5, 2022

I came across a math word problem about the time it takes workers to finish a job when they’re together vs. separately and I found its connection to a formula for the equivalent resistance of parallel resistors. I wanted to share it here.

I also wanted to ask for which types of problems and for what assumptions the type of logic detailed below can work. I think this assumes the time it takes the workers to finish a job varies linearly with the size of the job, and also that the voltage a resistor produces when a current is flowing through it varies linearly with the current (Ohm’s law basically which actually doesn’t hold for too high temperatures).

Problem for Workers

It takes $$T$$ days for $$1$$ job for $$n$$ workers together. Each worker $$i$$ does $$r_i$$ of a job in $$1$$ day ($$r_i$$ job / $$1$$ day).

Since it takes $$T$$ days for $$1$$ job when they’re working together, then in $$1$$ day together the workers can finish $$frac{1}{T}$$ of a job.

So we have $$sum_{i=0}^{n} r_i = frac{1}{T}$$

Now worker $$i$$ takes ($$1$$ day / $$r_i$$ job)

So to finish $$1$$ job it takes worker $$i$$, $$t_i = frac{1}{r_i}$$ days to do it.

So we have $$sum_{i=0}^{n} r_i = sum_{i=0}^{n} frac{1}{t_i} = frac{1}{T}$$

Problem for Resistors

($$G$$ is conductance and $$R$$ is resistance)

$$R_{eq}$$ volts are produced for $$1$$ amp of current with $$1$$ big equivalent resistor. Each small resistor $$i$$, when $$G_i$$ amps pass through it produces $$1$$ volt ($$G_i$$ amps / $$1$$ volt).

Since one large equivalent resistor produces $$R_{eq}$$ volts for $$1$$ amp of current then one large equivalent resistor also produces $$1$$ volt for $$frac{1}{R_{eq}}$$ amps of current.

So we have $$sum_{i=0}^{n} G_i = frac{1}{R_{eq}}$$

Now small resistor $$i$$ produces ($$1$$ volt / $$G_i$$ amps)

So when $$1$$ amp passes through it small resistor $$i$$ produces $$R_i = frac{1}{G_i}$$ volts

So we have $$sum_{i=0}^{n} G_i = sum_{i=0}^{n} frac{1}{R_i} = frac{1}{R_{eq}}$$

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