A 36.0-ohm resistor, a 50.0-ohm resistor, and a 700-ohm resistor are connected together. What is the largest resistance obtainable? What is the smallest resistance obtainable?
We can call our 36.0-ohm resistor 𝑅 one, our 50.0-ohm resistor 𝑅 two, and our 700-ohm resistor 𝑅 three. We want to solve, first, for the largest resistance obtainable by combining these three in some way. We’ll call that 𝑅 sub max. And next, we want to solve for the smallest possible resistance that these three can be arranged to create. We’ll call that 𝑅 sub min.
To get started, let’s recall the two different ways that resistors can be combined in a circuit. In general resistors can be combined in a circuit either in series, that is all in a row, or in parallel, in separate branches that run alongside one another. Depending on which of these two types a set of resistors are arranged in, the way those resistors are combined mathematically changes.
In series, the total resistance, 𝑅 sub 𝑇, is equal to the linear sum of each of the individual resistors up to the last resistor, 𝑅 sub 𝑛, while in parallel one over the total resistance 𝑅 sub 𝑇 equals one over the first resistor value plus one over the second resistor value and so on up to one over the last resistor value, 𝑅 sub 𝑛.
One fact about adding resistors in parallel is that the total resistance, 𝑅 sub 𝑇, is always less than any of the individual resistors involved in the parallel circuit: 𝑅 sub one, 𝑅 sub two, up to the last resistor. This means that to solve for 𝑅 sub min, the least possible total resistance of these three resistors 𝑅 one, 𝑅 two, and 𝑅 three arranged together, we’ll use the parallel addition formulation.
The maximum possible resistance though, 𝑅 sub max, involves adding the resistors in series. So 𝑅 sub max equals 𝑅 one plus 𝑅 two plus 𝑅 three or 36.0 plus 50.0 plus 700 ohms. Altogether, to three significant figures, 𝑅 sub max is 786 ohms. That’s the highest total resistance we can create combining 𝑅 one, 𝑅 two, and 𝑅 three.
Now we’ll move to solving for 𝑅 sub min, where we’ll use the parallel addition formula. One over 𝑅 sub min equals one over 𝑅 one plus one over 𝑅 two plus one over 𝑅 three. If we multiply both sides of the equation by 𝑅 one times 𝑅 two times 𝑅 three and then rearrange to solve for 𝑅 sub min, we find it’s equal to the product of the three resistances divided by 𝑅 two times 𝑅 three plus 𝑅 one times 𝑅 three plus 𝑅 one times 𝑅 two.
When we plug in for these values and enter these values on our calculator, we find that, to three significant figures, 𝑅 min is 20.3 ohms. That’s the smallest total resistance these three resistors can be combined to make. And notice that it’s smaller than any one of them.