Maths / Electronics question...

[Andy in Germany]

I'm currently trying to practice the maths for parallel resistance, and the "worked examples" are frankly quite entertaining. Could someone with a more mathematical brain explain?

The equation given is:

1 / r (total) = 1 / r1 + 1 / r2 + 1/ r3

However my experience with maths is that I probably won't understand it: I just need to learn to use the equation and the number on the other end is usually the right one. However, this doesn't make sense to me because what use is knowing "1 / r (total)"?

The problem set for me was:

r1 = 9Ω
r2 = 9Ω
r3 = 9Ω

Find r (Total)

1 / r (total) = 1 / r1 + 1 / r2 + 1/ r3

= 0,1111 + 0,1111 + 0,1111 = 0,3333 Ω

So 1/ r (total) / 1 is 0,3333 Ω

I wasn't sure what to do with this so I looked up the solution. It said:

r Total = 93 = 3Ω

This didn't help, so I asked AI an AI programme. This is what AI said:

For three resistors in parallel, use:

1/r total=1 / r1+1 / r2+1 / r3

Example:

R1=6 Ω,
R2=3 Ω,
R3=2 Ω

Substitute the values:

1 / r total=1/6 +1/3 +1/2

Okay, not sure how that should help, but I got:

1/r total = 0,1666 + 0,1333 + 0,5 = 0,7993Ω

But the "example" seems to go off the deep end:

Using a common denominator:

1 / r total =1/6+2/6+3/6= 1Ω

Where did the "common denominator" come from, and why is it a 6? To me it looks like it substitutes a stack of random numbers to get the right answer.

Can anyone explain how to consistently use the equation to get an accurate answer? I am aware this is a sign of discalculia; I don't understand how it works and probably won't, but I've found that given the formula, with care I can get the right answer, even if it is like pulling the lever on a fruit machine.

I just need to pass the exam, have the certificate, and can get on with my real job, so any assistance is welcome.

 

[Ming the Merciless]

The first bit you did correctly

1/r = 1/9 + 1/9 + 1/9 = (1+1+1) / 9 = 3/9 = 1/3

But that is 1/r not r. To get r you have to take the reciprocal. Thus r = 1 / (1/3) = 3. The other way to adjust is to multiply both sides by r. Thus (1/r) * r = (1/3) * r. Cancelling terms you get 1 = 1/3 r. Multiply both sides by 3 and you get 3 = r.

 

[T4tomo]

the denominator is the bottom number of the fraction, to add fractions, if they all have the same bottom number you just add the top numbers.

so in your example all of 6, 3 and 2 are factors of 6.

if you multiply r1 r2 & r3 together you will find a common denominator, but not necessarily the lowest. in this example it then becomes 36.

hence you rewrite 1/6 +1/3 +1/2 as 6/36 + 12/36 +18/36 = 36/36 =1 so 1/r =1 and r=1

but if you'd use 6 as the LCD then it 1/6 + 2/6 + 3/6 which still = 1!

 

[T4tomo]

Example 2
if it was say 3. 4 & 7 ohms
1/3+1/4+1/7 - so need a number they are all factors of. 3*4*7=84, so
i.e 28/84 + 21/84 + 7/84 = 56/84 or 2/3 therefore r=3/2 or 1.5ohms.

 

[Andy in Germany]

The first bit you did correctly

1/r = 1/9 + 1/9 + 1/9 = (1+1+1) / 9 = 3/9 = 1/3

But that is 1/r not r. To get r you have to take the reciprocal. Thus r = 1 / (1/3) = 3. The other way to adjust is to multiply both sides by r. Thus (1/r) * r = (1/3) * r. Cancelling terms you get 1 = 1/3 r. Multiply both sides by 3 and you get 3 = r.

How do I know what the reciprocal is?

 

[sungod]

How do I know what the reciprocal is?

the reciprocal of n is simply 1/n

unless n is a convenient value, use a calculator etc.

 

[Dogtrousers]

I think you're asking two separate questions here:
1) How do I calculate resistances in parallel? This is physics, but uses maths
2) What's all this about lowest common denominator? This is just maths,

Now, knowing (2) actually makes it easier to tackle (1) but you don't HAVE to use it. If you have access to a calculator you can use that.

your example of 6, 3 and 2 you could just use your calculator or, better still, a spreadsheet to add up 1/6, 1/3 and 1/2 which will give you 1/r you then take the reciprocal of the result (one divided by the result) to give you r.

So 1/6 is 0.1666, 1/3 is 0.1333, and 1/2 is 0.5. You need the sum of these fractions.
Add them together you get 1/r = 0.9999 - that's a rounding error because I've truncated the numbers above. Take it to be 1.
Now divide 1 by your sum of the fractions. 1 divided by 1 is 1 so your answer is 1.

Learning the maths of how to deal with fractions is useful, and it does make it easier and more elegant once you've learned it. But if you're having trouble with two things at once, just brute-force it with a calculator or a spreadsheet.

Knowing about lowest common denominators is only helpful when you have nice co-operative numbers. If your values were 3.2Ω, 8.46Ω and 1.03Ω then all the lowest common denominators in the world wouldn't help you.

So in short, my advice is: Rely on your calculator for this. Write everything down step by step.

Consider learning more about handling fractions as a separate exercise if that's what floats your boat.

 

[Andy in Germany]

Example 2
if it was say 3. 4 & 7 ohms
1/3+1/4+1/7 - so need a number they are all factors of. 3*4*7=84, so
i.e 28/84 + 21/84 + 7/84 = 56/84 or 2/3 therefore r=3/2 or 1.5ohms.

I can follow up until you added 28/84, then to mee it looks like we are back to random numbers. I can see where the 84 comes from but not the 28 or the 21, or the 7,

I understand how 56 is 1/3 of 84 (with help from a calculator) just not how we get 56 in the first place.

What I really need is a step by step instruction to follow. As said above, from experience I probably won't understand the equation, but if I have a set of steps to follow then I can be confident of getting the correct number at the end.

If that means "Add the top row and multiply the bottom, then divide the bottom by the total of numbers on top" or whatever that's what I'll do, then I'll practice until the exam, and forget it a day later.

 

[Dogtrousers]

How do I know what the reciprocal is?

One divided by the thing. Reciprocal of a half (0.5) is one divided by 0.5 = 2.

Smarty pants people with more maths would just say, well a half is 1/2 so the reciprocal turns it upside down giving 2/1. But if I were you I would just take my calculator and do one divided by the thing. Or maybe it has a reciprocal button

 

[sungod]

putting things into words rather than equations may help see what's going on

the rule for calculating the effective resistance of a number of resistors that are wired in parallel, is to take the reciprocal of each resistor's value, add up all the reciprocals, then take the reciprocal of that total, this gives you the effective resistance

as an aside, should you ever need it, the same rule is used to calculate the capacitance of a number of capacitors in series

 

[Ming the Merciless]

How do I know what the reciprocal is?

Simple. The reciprocal of a/b is b/a. It is the number that when multiplied by the original returns 1.

Thus 4/5, the reciprocal is 5/4; 45/123 the reciprocal is 123/45; 1/4 the reciprocal is 4/1 = 4.

If you can stick with fractions then the maths you will do for this is only ever multiplying integers together. Then swapping the numbers in the intermediate result either side of the division symbol. Which keeps things simple and easy to do without a calculator.

 

[Andy in Germany]

I think you're asking two separate questions here:
1) How do I calculate resistances in parallel? This is physics, but uses maths
2) What's all this about lowest common denominator? This is just maths,

Now, knowing (2) actually makes it easier to tackle (1) but you don't HAVE to use it. If you have access to a calculator you can use that.

That's my general approach.

your example of 6, 3 and 2 you could just use your calculator or, better still, a spreadsheet to add up 1/6, 1/3 and 1/2 which will give you 1/r you then take the reciprocal of the result (one divided by the result) to give you r.

So 1/6 is 0.1666, 1/3 is 0.1333, and 1/2 is 0.5. You need the sum of these fractions.
Add them together you get 1/r = 0.9999 - that's a rounding error because I've truncated the numbers above. Take it to be 1.
Now divide 1 by your sum of the fractions. 1/1 is 1 so your answer is 1.

Followed this and remarkably got the same answer, so far so good...

Learning the maths of how to deal with fractions is useful, and it does make it easier and more elegant once you've learned it. But if you're having trouble with two things at once, just brute-force it with a calculator or a spreadsheet.

This is the nub of the problem, which is quite hard to explain. Brute force and calculator are the order of the day. Apart from anything else, I've never needed to calculare parallel resistances before, while working with bicycles, and I doubt I'll ever need to again.

Knowing about lowest common denominators is only helpful when you have nice co-operative numbers. If your values were 3.2Ω, 8.46Ω and 1.03Ω then all the lowest common denominators in the world wouldn't help you.

I thought that may be the case: in my experience there are no nice cooperative numbers.

So in short, my advice is: Rely on your calculator for this. Write everything down step by step.

Exactly my plan...

I'll attack the equation on paper later and take pictures, then develop a checklist...

 

[Ming the Merciless]

Some other outcomes to check your result against. Always worth knowing a ballpark for what you expect result to be, if using a calculator.

When resistors are connected in parallel, the total resistance is always less than the smallest individual resistor. Additionally, for identical resistors in parallel, the total resistance can be calculated by dividing the resistance of one resistor by the number of resistors.

 

[Dogtrousers]

Some other outcomes to check your result against. Always worth knowing a ballpark for what you expect result to be, if using a calculator.

When resistors are connected in parallel, the total resistance is always less than the smallest individual resistor. Additionally, for identical resistors in parallel, the total resistance can be calculated by dividing the resistance of one resistor by the number of resistors.

Very much this.

When chaining together calculations - in a spreadsheet or with a calculator it's easy to make a minor error that gives you an insane result. But if it's all just a sea of numbers you might not notice that your result is in fact plain daft. Happens to me embarrassingly often. Decimal points in the wrong place and so on.

So having rules of thumb like Ming's above is important for ballpark /sanity checking.

 

[Andy in Germany]

Some other outcomes to check your result against. Always worth knowing a ballpark for what you expect result to be, if using a calculator.

When resistors are connected in parallel, the total resistance is always less than the smallest individual resistor. Additionally, for identical resistors in parallel, the total resistance can be calculated by dividing the resistance of one resistor by the number of resistors.

I've spotted that: I can calculate series resistance and there's a different formula for parallel up to two resistors, but this one is proving troublesome.