# What is the proper terminology for these two types of multiplication?

((QUESTION REWORKED))

First question on this site, and I apologize if this question has been answered. I searched and searched and the fact I don't know the basic terminology is hindering me from finding the answer.

This question stems from a current programming issue I am attempting to detail to others. In one instance we have the secondary variables being added together and then multiplied against the main variable. While in the other scenario, each one is multiplied and updates the main variable before the next. So you get very different numbers, depending on if you add the secondary together before multiplying or if you do them one at a time.

The variables: \begin{align*} x&=100 \\ a &= 2 & b &= 3 \\ c &= 4 & d &= 5 \end{align*}

The first formula is: $$y = d(c(b(a(x)))).$$

The second formula is: $$y = x(a+b+c+d).$$

What is the proper math terminology for both types?

Any help would be appreciated.

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The keyword you are looking for is "distributive". – user86828 Jul 18 '13 at 19:51
@user86828 I just changed the question. I wasn't giving the right formulas. Thanks – JClaspill Jul 18 '13 at 20:01
Welcome to MSE. Please use MathJax to format your mathematics. – dfeuer Jul 18 '13 at 20:17

I don't think there is specific terminology assigned to either form, rather the property is called the distributive property, and you are either adding first, or multiplying each term individually prior to adding.

Given your edits, the two different forms have nothing in common mathematically.

The first format indicates that $a$, $b$, $c$ and $d$ might be functions and each would perform an operation on the element inside the parentheses next to it. However, as they are listed as being variables and as having specific values, the first form is simply an explicit "multiply in this order" command, while the second form is an explicit "add these before multiplying" command.

If you were to speak what is written in each formula, they might sound like this:

(1) "$d$ times multiplication of $c$ and multiplication of $b$ and multiplication of $a$ and multiplication of $x$."

(2) "$x$ times sum of $a$, $b$, $c$ and $d$."

Note that the first form might be shortened using "pi" notation:

$$x\prod_{i=0}^n a_i$$

Where $a_0 = a$, $a_1 = b$, $a_2 = c$, and $a_3 = d$.

If a, b, c and d are given as percentages, note that the formula $x*a*b*c*d$ is not a correct way to calculate the compounding effect. Instead, it should be calculated like so:

$$x * {100 + a \over 100} * {100 + b \over 100} * {100 + c \over 100} * {100 + d \over 100}$$

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Hmm. I must not be representing this right still then. I think one is cumulative and the other is.. not. If that makes sense. And while they aren't common in mathematical terms, they are in programming. The example I have is how someone calculates multiple 'bonuses' to payroll. You can either add all the bonuses together and then multiply the base rate against it, or you can multiply each bonus % systematically. Both ways provide a vary different end result. And that is where I am stuck. I have code, but there is a huge communication barrier between HR and IT. Was hoping math could unify us – JClaspill Jul 18 '13 at 21:08
Okay, you are talking about banking terms. In this case, it is equivalent to the difference between "simple interest" and "compound interest". In your case, "simple interest" means adding the bonuses a, b, c, and d together to create a simple percentage bonus. "Compound interest" is then taking bonuses a, b, c and d and "compounding" them together creating a larger final bonus than in the "simple" case. Have I understood your problem correctly now? – abiessu Jul 19 '13 at 15:19