# Total Percentage Change - Why does this work?

Algebraically this works, but I'm looking to understand (1) why it works and (2) if there is a simpler formula.

Problem: I want to get total % change over time.

Facts:

Example Data -

Jan 2013 - 14.29%

May 2013 - 25%

Oct 2013 - 20%

**Solution Methodology:**

1. Take a random baseline number to plugin: 100

2. Calculate the actual numbers:

Jan 2013 - 14.29% Increase = 114.29

May 2013 - 25% Increase = 142.8625

Oct 2013 - 20% Increase = 171.435

3. Calculate Total % Difference:

(171.435 - 100) / 100 = 71.435%

4. Plug in variables to determine formula to do this:

Baseline Number = X

A = Jan 2013 Increase % = (x)(A + 1) = (xA + x)

B = May 2013 Increase % = (xA + x)(B + 1) = (xAB + xA + xB + x)

C = Oct 2013 Increase % = (xAB + xA + xB + x)(C + 1) = (xABC + xAC + xBC + xC + xAB + xA + xB + x) = x(ABC + AC + BC + C + AB + A + B + 1)

Calculate % Change =
(x(ABC + AC + BC + C + AB + A + B + 1) - x) / x


Final Algebraic Formula = ABC + AC + BC + C + AB + A + B

Other Notes Interestingly enough if I only looked to get the Total % Difference of May 2013 and Jan 2013 the algebraic formula becomes: AB + A + B

There is a chance I'm overcomplicating this or am calling it the wrong thing.

• Try not to have a lopsided view twitter.com/grenouf/status/724263408888107008 of algebraic experssions. It may be that you have a preference for the sum-of-products form. Try to also get used to expressing your formulas in product-of-sums form e.g. $x(A+1)(B+1)(C+1)$ – John Joy May 18 '16 at 13:54
Example: starting with $100$, a $A=10\%$ increase is equivalent to $100*1.10$, where the percentage $10\%$ is divided by $100$ and added to $1$.
We add $1$ to include the initial amount.
When multiple interests are compounded (applied sequentially), you are simply multiplying the respective decimal values together. Your final algebraic formula works as a result of multiplication's distributive property, but it can be written more generally as $$(\prod_{i=1}^n (A_i+1)) - 1$$ Where $A$ represents the interest rate, and $\prod$ means multiply all of the following together, for $i$ starting at $1$, and incrementing by $1$ until it reaches $n$.