If every resistor below has resistance R, what is the equivalent resistance of the combination, in terms of R?

Hey, guys. Hope you try this practice problem. Let's check it out. So here we're gonna find the equivalent resistance of this combination if all the resistors have resistance R. So RRRR. And I wanna know what is the equivalent resistance in terms of R. In terms of r means that the equivalent resistance will be something like some coefficient, some number, some fraction times r and we wanna know what is that. K? I wanna do this in 2 ways. I wanna show you sort of the long way and how you can also take advantage of shortcuts to solve this. Now the first thing you can do here is you can combine these 2 r's because they're in series and you can combine these 2 r's because they're in series. You can't really do anything else because none of these because you don't have 2 branches that have a single resistor. This branch has 2, this branch has 1, that's good, but this branch has 2. And you cannot combine in parallel unless you have 2 opposite sides that have lone resistors. Okay? So what I can do first is I can combine these in series. So in series you just add the 2. So r+r is 2r. So I'm gonna get 2r at the top then there's a single r in the middle and then you got 2 r at the bottom. Okay? Now if I wanna combine this, I can do it with the general equation that works for all of them, not the shortcut. I can do that. And I can say that one over r equivalence is gonna be 1 over the first resistor plus 1 over the second resistor plus 1 over the third resistor. Okay? And I'm just gonna write them here, 2r, r and 2 r. And remember, you want a common denominator. So what I'm gonna do is I'm gonna multiply this here by a 2 and then I gotta do it to the top as well so that they're all 2 r's. Okay? So if you do this, you get 1+2+ 1, that's 4 divided by 2 r, 2 r, 2 r. So that's just 2 r. You can simplify this to be a 2. This is a one, so you're gonna get 2 over r. Are we done? Not really. You gotta flip. Remember, we're solving for our equivalence. You have to flip this. I can flip this and then I can flip this, which means I end up with r equivalence at the top here equals r over 2. Or another way to write this is that it's half times r. In other words, the number that goes in front of the r and the answer is half. So the equivalent resistance of this entire thing, it's the same thing as if you had half of a resist or resistance with half of the resistance of any of them. K? So this was the sort of the full way of doing it. What about with a shortcut? So this is the full way. And now what about with the shortcut? The reason I'm showing you both of these, is so that you're you're pretty familiar with this stuff. You may remember that if you have 2 resistors if you have 2 resistors, 2 or more resistors, any number of resistors in parallel that are the same resistance, you can combine them. K? So the equivalent resistance when you have same resistances in parallel is just r over n. R is the resistance and n is the number of resistors. There are 2 resistors that are the same. The r is not the same but the 2 r's are the same. So I can write 2 r, which is the resistance that is the same divided by 2 because there's 2 of them. The r's the two's cancel rather and I get an r. What this means is that I can merge the 2 blue resistors into a single r resistor. Okay? So you can think of this as this 2 goes up here or the 2 at the top goes down there and they merge and they become 1. But there's still this original r somewhere over here. Okay? So this is how you simplify, the first step you can take in simplifying. Notice that I could have tried to merge the 2 r with the r, but it's easier if I can just merge resistors that are the same first. And now look what happens. I'm back with just 2 resistors and they also happen to be the same. So I can use the same shortcut again, and I can say r equivalents when you have the same is r over n. The resistance of these guys is just r and n is 2. So the final answer is r over 2 just as it was here. So you can see that this stuff works. That's it. So to sort of summarize here, you look at the 2 of the 2 and you say, okay, I'm gonna get half of that. So then it becomes just this r here. And then I got a r and an r, so the answer is r over 2. K? So there's a better a best path that you can follow in merging these things. That's the first one. Let's keep going.

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If every resistor below has resistance R, what is the equivalent resistance of the combination, in terms of R?

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