# New Method for Calculating Parallel Resistors

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#### paulfr

##### New Member
I teach HS Physics part of which is Electric Circuits,
I have discovered a rule / technique for parallel resistors that I never encountered
in my teaching, or in all of my 30+ years in electronics engineering, nor in any textbook on Circuits.

It is what I call " The N + 1 Rule "

We all know the Reciprocal Rule
1 / RT = 1/R1 + 1/R2 + 1/R3 ..... + 1/Rn

AND
we know that for 2 resistors, this becomes the Product over the Sum of the 2 R's

BUT
The N+1 Rule is this
1/ Find N = the ratio of the two R's
2/ Add 1 to it to get N + 1
3/ Divide the largest R by N+1

E.g.
4 and 20 ohms
N = 20/4 = 5
N+1 = 6
RT = Rtotal = 20/6 or 10/3
Check
Product = 80
Sum= 24
RT = 80/24 = 10/3

It is quite useful when the numbers are large and thus the Product is very large.
No need to remember it and then do long division.
e.g.
300 and 50 becomes much easier and thus faster with N+1 than with Product-Sum.
300/50 = 6 ==> Rtotal = 300/7
Check with Product Sum Rule
300 (50) / 300 + 50 = 15000/350 = 300/7

It works even when N is not an integer.
e.g.
500 and 300 ohms
500/300 + 1 = 5/3 + 1 = 8/3
Rtotal = 500 / (8/3) = 1500/8
Check with Product Sum Rule
500(300) / (500 + 300) = 150000 / 800 = 1500/8

Have any of you ever seen this ???

Just curious and wondering why it is not in all the textbooks on Circuits.

#### crutschow

##### Well-Known Member
That may be easier if you are doing the calculation by hand, but with a calculator, it's very easy to do the reciprocal of the sum of the reciprocals.

#### gophert

##### Well-Known Member
I teach HS Physics part of which is Electric Circuits,
I have discovered a rule / technique for parallel resistors that I never encountered
in my teaching, or in all of my 30+ years in electronics engineering, nor in any textbook on Circuits.

It is what I call " The N + 1 Rule "

We all know the Reciprocal Rule
1 / RT = 1/R1 + 1/R2 + 1/R3 ..... + 1/Rn

AND
we know that for 2 resistors, this becomes the Product over the Sum of the 2 R's

BUT
The N+1 Rule is this
1/ Find N = the ratio of the two R's
2/ Add 1 to it to get N + 1
3/ Divide the largest R by N+1

E.g.
4 and 20 ohms
N = 20/4 = 5
N+1 = 6
RT = Rtotal = 20/6 or 10/3
Check
Product = 80
Sum= 24
RT = 80/24 = 10/3

It is quite useful when the numbers are large and thus the Product is very large.
No need to remember it and then do long division.
e.g.
300 and 50 becomes much easier and thus faster with N+1 than with Product-Sum.
300/50 = 6 ==> Rtotal = 300/7
Check with Product Sum Rule
300 (50) / 300 + 50 = 15000/350 = 300/7

It works even when N is not an integer.
e.g.
500 and 300 ohms
500/300 + 1 = 5/3 + 1 = 8/3
Rtotal = 500 / (8/3) = 1500/8
Check with Product Sum Rule
500(300) / (500 + 300) = 150000 / 800 = 1500/8

Have any of you ever seen this ???

Just curious and wondering why it is not in all the textbooks on Circuits.

It is interesting and I have never seeen it before. I don't know that it is easier. Also, it adds a layer of mystery so there is no visibility to the process so the extra level of "abstraction" turns a calculation and a process into a memorization trick. memorization is the preferred method of learning (and teaching?) for some people but I prefer to know what I am doing down to the nuts and bolts.

#### DerStrom8

##### Super Moderator
This is pretty simple to prove as being equivalent to the reciprocal rule using basic algebra:

$Rtot = \frac{1}{(\frac{1}{A})+(\frac{1}{B})} = \frac{1}{\frac{(\frac{B}{A})}{B}+\frac{1}{B}} = \frac{1}{\frac{(\frac{B}{A})+1}{B}} = \frac{B}{(\frac{B}{A})+1}$

Notice that the last part of the equation is the mathematical representation of your N+1 rule. It does not matter whether resistance A or B is higher when doing it algebraically. Only when doing it mentally and ensuring the ratio N is correct.

Sometimes seeing formulas in a different light will help clarify the concepts in peoples' minds. This equivalent representation could certainly prove useful to some people.

#### rjenkinsgb

##### Well-Known Member
My quick method uses somewhat different approach, usually for a mental approximation.

Find a common multiple of both (or all) resistors, or nearly so for an approximation; eg. for 10K and 100K, 100K is a common multiple.
for 22K, 33K and 68K, 200K is near enough for an approximation.

Add the number of that multiple it would take in parallel to make each resistor value:

10K & 100K = 10 + 1 = 11 x 100K; work that out: 11 x 100k (or 100k / 11) = about 9.1K (actual value 9.091K).

22, 33 68 = 9 + 6 + 3 = 18 x 200K; 200K / 18 = 100K / 9 = ~11K (actual value 11.054K).

You can get a good approximation without a calculator & it's not limited to two resistors (or equally, capacitors in series).

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