Calculating % contrib. of variable growth to the growth of total, w product formulas

[KMarshall]

Hi, thank you for reading.

When looking at a number that is the product of two variables (A * B = C), I’m trying to figure out the contribution of the growth in one variable (A) to the growth in the product (C).

The way I was thinking about it (might be easier to follow along in excel if it helps):

Assume variable A grew 6% and variable B grew 8%, then the product C grows 14.48%, which is basically 6% + 8% + (6% * 8%). To figure out the contribution of A’s 6% growth to C’s growth of 14.48%, it would be [6% + (6% * 8%)*(6% / (6% + 8%)) ] / 14.48%.

In other words, 6.21% / 14.48%. It makes sense to me that the 6.21% would be higher than the initial 6% because of the compounding impact of using a product formula.

My problem seems to be with negative growth rates. For example, assume A grew -2%, B grew 5%, resulting in C growth of 2.90%. Using the above logic, the numerator I would use to figure out B’s contribution to C is 4.83%, below the original 5%.

First, am I thinking about this the right way? Second, can someone explain the logic of why the 4.83% would be less than 5%; this may seem silly, but I’m not sure if it makes sense the variables contribution would be punished.

It’s been years since I opened a math book, what can I read up on to help me with this? I would like to run the same exercise with three variables as well. Thank you again for reading.

 

[srmichael]

Hi, thank you for reading.

When looking at a number that is the product of two variables (A * B = C), I’m trying to figure out the contribution of the growth in one variable (A) to the growth in the product (C).

The way I was thinking about it (might be easier to follow along in excel if it helps):

Assume variable A grew 6% and variable B grew 8%, then the product C grows 14.48%, which is basically 6% + 8% + (6% * 8%). To figure out the contribution of A’s 6% growth to C’s growth of 14.48%, it would be [6% + (6% * 8%)*(6% / (6% + 8%)) ] / 14.48%.

In other words, 6.21% / 14.48%. It makes sense to me that the 6.21% would be higher than the initial 6% because of the compounding impact of using a product formula.

My problem seems to be with negative growth rates. For example, assume A grew -2%, B grew 5%, resulting in C growth of 2.90%. Using the above logic, the numerator I would use to figure out B’s contribution to C is 4.83%, below the original 5%.

First, am I thinking about this the right way? Second, can someone explain the logic of why the 4.83% would be less than 5%; this may seem silly, but I’m not sure if it makes sense the variables contribution would be punished.

It’s been years since I opened a math book, what can I read up on to help me with this? I would like to run the same exercise with three variables as well. Thank you again for reading.

Don't make this more complicated then it needs to be. Use that fact that is something grows by x% then the new value is (1 + x/100) and if it decreases by x% the new value is (1 - x/100)

In the first example, if A grew by 6% then the new A is 1.06A. Similarly, the new B is 1.08B. Thus, (1.06A)(1.08B) = 1.1448C, which means C grows by 14.48%.

In the second example, if A decreases by -2% then the new A is 0.98A. The new B is 1.05B. Thus, (0.98A)(1.05B) = 1.029C, which means C grows by 2.9%.

 

[Dale10101]

Some ideas

Hi. Am pretty new here too. Like you, am reviewing things previously learned. srmichael has already given you the skinny on the simple and elegant way to work your problem. I learned something from his explanation.

I see though that what you are after is not simply the straight forward way to solve your problem but where your error in thinking has occurred. It is much more difficult to get that sort of help. For example you present an equation:

"To figure out the contribution of A’s 6% growth to C’s growth of 14.48%, it would be [6% + (6% * 8%)*(6% / (6% + 8%)) ] / 14.48%."


Where did that come from? OK, what if it obvious to you … and you may be right. In that case someone here might follow your logic. But what if you are wrong? In that case it is impossible to know what you were thinking that led to your formulation. Just saying, in a friendly way, if you want a critique of your reasoning then some simple steps in which you build up to your conclusion would be helpful.

Like you I am reviewing things learned in the past so what follows is just my take. I have made a detailed response because I wanted to investigate certain things and to make a copy for my files.


Attached, on page one is a graphical representation of the problem in which I reason by thinking in terms of a rectangle. “a” and “b” are the sides, “c” = ab is the area of the rectangle.

From the graphs I can see what each the increase/decrease of “a” and “b” means in terms of additions and subtractions to “c”. I get the same results of 14.48% and 2.9% as you and srmichael.

On page 2 I derive the formula:

Delta_C = f + g + fg

Delta_C is the change in C as a percent.
f = the percent increase of "a".
g = the percent increase of "b".


This formula delivers the 14.48% and 2.9% results as well.

Looking at the formula for Delta_C, what you seem to be looking for is:

IF f = .06, g = .08

[(f+fg)/(f+g+fg)]* 14.48% = 6.48% as the part of the 14.48% increase in “c” due principally but not exclusively to f.


And, IF f = .05, g = -.02

[(f+fg)/(f+g+fg)]* 2.9% = 4.9% as the part of the 2.9% increase in “c” due principally but not exclusively by f.

WHAT !!! how can the 4.9% be a part of the 2.9%.

Hmmm, to see the answer one must look at the graphical interpretation on page 1. The 2.9% is the net result of competing terms. The 4.9% is the part of the 5.0% that f would contribute if it were not “punished” by “g” … bad, bad, boy g.

Anyway, I have learned something … I think … sometimes the math collective here sends me to the shower. Good luck, I like your thinking type approach as opposed to just slamming formulas around. It helps when you explain your steps using both words and equations.

If you want to study the attachments it helps to download the images and open them in a viewer like Faststone.

 

[KMarshall]

Thank you both, especially Dale10101 as that is exactly what I was looking for! I really appreciate all the effort that went in your reply (and pictures!).

I knew Δ c % = f + g + fg worked but your pictures showed WHY they worked and that helped a lot. I think an important point you made was the area A3 in your chart (below #2) showed the increase principally but not exclusively to either f or g. Is it possible to approximate the proportionate contriubtion though?

I suppose it's a bit silly, it's like asking...if a rectangle is 3 x 5 meters, how much of the area of 15 does each side contribute. But say you wanted to try. Would it be too simple to think think that the 3 meter length contributes (3 / (3+5)) = 37.5% to the area, and the 5 meters contributes (5/ (3+5) = 62.5%? I realize I'm mixing up basic arithmetic here, but do you guys have any suggestions?


On a different note, 3 variables would be f + g + h + fg + fh + gh + fgh. Fun.

 

[Dale10101]

Well

Thank you both, especially Dale10101 as that is exactly what I was looking for! I really appreciate all the effort that went in your reply (and pictures!).

I knew Δ c % = f + g + fg worked but your pictures showed WHY they worked and that helped a lot. I think an important point you made was the area A3 in your chart (below #2) showed the increase principally but not exclusively to either f or g. Is it possible to approximate the proportionate contriubtion though?

I suppose it's a bit silly, it's like asking...if a rectangle is 3 x 5 meters, how much of the area of 15 does each side contribute. But say you wanted to try. Would it be too simple to think think that the 3 meter length contributes (3 / (3+5)) = 37.5% to the area, and the 5 meters contributes (5/ (3+5) = 62.5%? I realize I'm mixing up basic arithmetic here, but do you guys have any suggestions?


On a different note, 3 variables would be f + g + h + fg + fh + gh + fgh. Fun.

Hmmm. I think you ask "back-word" questions. Which makes me smile because I do the same thing much to the "dis-comprehension" of most.

I often have ulterior motives like thinking of a bigger problem that my presented question is a piece of ... usually the tent pole breaks and I am left crawling out and stoking up the campfire and treating bruises ... but, ah well.

I am not a long hair here so you will please take my thoughts for what they worth.

I am not sure what you driving towards but your equations do not speak to me.

You ask what percent a side contributes to the area. A percent is a ratio, a ratio is a pure number, no dimension, yet you want x feet of a side divided by y square feel of area ... feet divided by square feet is not a pure number therefore you are asking for something that doesn't exist, a null concept.

The real question is, what is the question behind the question?

Again though, you want a ratio between a side and the area. In comparing the two sides I would again think of the rectangle, 3 feet high and 5 feet long. What are you asking then, how much of the area is contributed by a side that is two feet high, and how much by the length of 5 feet?

Neither side contributes any area, it is only when they are multiplied together that an area results.

Suppose you had only a length of fence 5 feet long, zero height, there is no area, the 5 feet contributes nothing to area, you are trying to compare things that are incommensurate.

You can compare the increase in area if one dimension is fixed and the other increased. In that case fixing the height at 3 feet and increasing the 5 foot length to 6 foot increases your area by 3 square feet. That you can see right? On the other hand keeping the length fixed and increasing the height by one foot increases the area by 5 square feet. Yes?

You could take a ratio of the two, say 5/3 and say increasing the height is 166% more effective in increasing the area then is increasing the length if that figure of merit is of any use.



You are apparently trying to think of something but applying the wrong concept.

Anyway, no more time right now, good luck.