Algebra

A quick and easy introduction to algebra for answering important financial questions

You don’t need much algebra but knowing the basics will give you the ability to calculate the difference between a win and a lose.

Algebra is a way of abstracting the components of a problem so you can solve all variations of that problem. Unfortunately, it’s often taught in such abstract ways - ways not easily related to real problems - that inability is common.

What is algebra?

The word algebra came to English in the mid-16th century from Arabic “al-jabr” (الجبر meaning reunion of broken parts) from the title of a 9th century book by the Persian scholar Muḥammad ibn Mūsā al-Khwārizmī.

In its simplest form, it is a system of breaking down problems into parts (known & unknown quantities), representing those parts with symbols and using rules to manipulate them in order to obtain answers.

Why is algebra important?

Algebra allows you to generalize.

You know that 2 apples combined with 3 apples gives you 5 apples, 3 apples combined with 5 apples gives you 8 apples etc. so you can use the rule of addition indicated by the “+” operator to get the total number of apples from combining any two quantities represented by x1 & x2 using the formula x1+x2.

Realizing that this can be extended, if someone shows up with three boxes of oranges you can get the total using (x1+x2)+x3 and you can go on to perform more complex & useful calculations which can be easily translated for use with spreadsheets and computer programs.

Notation

In normal algebra, while add & subtract are always represented by + & - respectively, multiplication and division have alternatives to make the notation clearer:

Multiplication can be written x×y, x.y (dot product) or simply xy.

Division can be written x÷y or with a line as xy− or x/y.

Example

If you’re in the business of manufacturing “units” and you make \$5 profit per unit, do you accept an order for 20,000 units?

At face value, you’ll make \$100,000 but you’ll have to cover your overheads while your making the units and while waiting to get paid. You might also have to pay for the raw materials used to make the units before you get paid.

$X=\mathrm{nC}+\mathrm{mF}$

X will be the minimum amount of cash you need in advance where n is the number of units, C is the variable cost per unit, F is your monthly overheads and m is the number of months until you get paid:

DescriptionDataMath
Order size:$n$
Cost per unit:$C$
Monthly overheads:$F$
Months until paid:$m$
Result:$X$

“What Is Algebra?” at livescience.com.

“Algebra is great fun - you get to solve puzzles!” at mathsisfun.com.

“Examples for Algebra” at wolframalpha.com.

“What’s your favorite equation, formula, identity or inequality?” at mathoverflow.net.