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CHAPTER 1
BASIC BUSINESS MATH
INTRODUCTION
Whether their jobs are in finance, human resources, marketing or any other function,
businesspeople need sound math skills. This package will teach many essential basic and advanced
quantitative methods to those who need to develop them or others who need to brush them up,
whether they are currently in business, about to enter business school or still in college. After you
read this text and complete the software, you will have the quantitative skills necessary to perform
operations critical to many business functions.
Before you start doing anything, however, it would be a good idea to have a pen or pencil
and plenty of paper available. You probably will also need a calculator to follow the examples in
the text or complete the problems on the computer.
FUNDAMENTAL MATHEMATICAL CONCEPTS
Basic Equations and Transposing Terms
An equation is a mathematical statement in which two expressions are set equal to each
other. Here is a simple example:
15 + 19 = 34
After performing whatever mathematical operations (addition, subtraction, multiplication,
or division) are required on each side of the equation (addition on the left-hand side of the equation
only in this instance), we find that the numbers on both sides of the “equal” sign are, in fact, equal.
As a matter of fact, you can subtract or multiply or do whatever you want to the numbers on one
side of the equation (except divide them by zero) and, as long as you do the same thing to the
numbers on the other side of the equation as well, the result on both sides will be equal. This idea
comes in very handy, especially were we not to supply one of the numbers in the equation, and to
put in its place an x (or unknown variable, often shortened to unknown) instead:
15 + x = 34
Since we are naturally inquisitive, we would like to find out what x is (often called solving for x).
The easiest way to do this is to isolate x from the rest of the equation, to get it alone on its own side
of the equal sign and get everything else together on the other side. To do this requires a process
called transposition, which means changing a term’s sign and moving it to the other side of the
equation. This is accomplished by performing the same mathematical function on both sides of
the equation, which we know we are allowed to do from the paragraph above. The mathematical
function we use depends on how we intend to isolate the unknown. For instance, in our example
above, to isolate x, we would want to eliminate the 15 from the left side of the equation. In order to
do that, we have to subtract 15 from the left side of the equation, which means subtracting it from
the right side as well.
15 + x -15 = 34 - 15
After we do the subtraction, the simplified equation looks like this:
x = 19
We have thus solved for the unknown variable x in the equation. This is called an algebraic
equation because of the fact that we have solved for an unknown. There is a more detailed process
for transposition which we will discuss shortly. First, however, we have to introduce you to
some other things you might run into which tend to make some equations more complex and thus
transposition much more useful.
Coefficients and Exponents
Often an unknown variable presented to us has a number in front of it (such as 5 x ). This
number is called the variable’s coefficient and is multiplied with the unknown variable (5 x means
multiply 5 by x or add x to itself four times, x + x + x + x + x ) when you work out the expression.
Other times, you may have a situation where there is a number or an unknown variable in superscript
above and to the right of another number. An example of this is:
4 =?
With expressions like this, the large number is referred to as the base and the smaller number
as the exponent (often called the power ). When you see an exponent, you know that the base is
going to be multiplied by itself a certain number of times. The exact number of times the base is
multiplied by itself is the exponent minus one if the exponent is positive. The case of the negative
exponent is covered in the next paragraph. In this example, since the exponent (4) is positive, one
would multiply 6 by itself three times (4 - 1 = 3). Upon doing this, we would see that (6x 6 x 6 x6)
or 6 4 = 1296. When the exponent is one, it means to multiply the base by itself (1-1) = 0 times, so
the value of a base with an exponent of one is just the value of the base itself. When the exponent
is zero for any number, the expression has a value of one.
You may find instances where there is an unknown variable in the exponent rather than the
base. When the unknown variable is in the exponent, the procedure gets somewhat complicated.
Usually, the easiest way to solve for this is by using a calculator or a computer. While we will go
over an extended set of calculator functions later in this chapter, here is an example of the hand
calculations necessary to solve for an expression with an unknown as the exponent:
Monomials and Polynomials
A monomial is an expression that contains only one term (such as 7x 2 ), which often may
contain an unknown variable with both a coefficient and an exponent, although a term can be a
simple number like 7 or 9 as well. A polynomial is an expression that has more than one term, all
or some of which are unknown variables with non-zero coefficients and exponents. Binomials and
trinomials are examples of polynomials, because they both have more than one term: binomials
have two terms (such as 7x^4 + 10x^2 ), and trinomials have three (such as 7x^4 + 10x^2 + 8). Note that
in all of these examples, all like terms have been consolidated; the expression 7x 2
a trinomial but a binomial, because it can be consolidated as 11x^2 + 5.
There are also some rules regarding adding, subtracting, multiplying and dividing two or
more monomials when they are in the form of unknown variables with coefficients and/or exponents
attached to them. You can add and subtract two or more unknown variables, as long as they are
the same variable and have the same exponent, by adding or subtracting the coefficients just as you
would with simple numbers (i.e. you could combine 5x + 9x to equal 14x, but you could not add 5x
- 9y, nor could you add 5x^2 + 9x without knowing what x was).
For multiplication and division, things get somewhat more complicated. When you divide
one monomial by another, it does not matter whether the coefficients or powers are the same,
but the unknown variable should be in order to simplify it. When you multiply two monomials,
however, it does not matter if the coefficients, powers or even the unknown variables are the same.
You can multiply two monomials no matter what their unknown variables and exponents are.
2
x = 16
2
(^ x^ x^^1 x^ ) = 16
(^1 x^ x^1 )
2
1 = 16 x
( )^1 (^) x = 16
2 x 16 4 16
2 =
x = 4
Coefficients get multiplied the same way they would be if they were just numbers separate
from the unknown variables. When the same unknown variables are multiplied together, exponents
of those unknown variables actually get added together.
Example:
3x^4 x 4x^5 = 12x^9
5x^7 x 9x 9 = 45x^16
When multiplication of different unknown variables is performed, the coefficients will get multiplied
normally just like simple numbers even though they are attached to different variables. The
unknown variables and the exponents assigned to each of them stay independent when multiplying
them together.
Example:
3x 8 x 9y 5 = 27x 8 y 5
Division of monomials is achieved in a similar fashion. Coefficients of different variables
are divided just as they would be if they were simple numbers. When you have the same unknown
variable in the numerator as in the denominator, the exponent of the variable in the denominator is
subtracted from the exponent of the same variable in the numerator.
Example:
When you have different unknown variables in the numerator and the denominator, neither
the unknown variables nor their exponents can be divided by each other, but the coefficients can
be.
Example:
30 x
13
10 x
4 3 x
9 =
16 x
4
4 y^2
4 x
4
y 2
Simultaneous Equations
Often you may have two or more variables to solve given two or more equations with which
to solve them (you cannot solve for two variables with less than two equations). This requires
solving for the unknowns in all of the equations, which are referred to as simultaneous equations.
Here is an example:
3x = 14 - 5y 7x + 2y = 23
To solve for y in terms of x in this equation, we follow this procedure:
Process Example
- Arrange the equation so that like terms are 3 x + 5y = 14 - 5y + 5y
in the same column. 7 x + 2y = 23
- Multiply both sides of one equation so that the 2 x (3 x + 5y = 14), or
coefficients of one of the unknowns will have 6x + 10y = 28
the same absolute value (this means the + or - sign
is irrelevent) as the coefficient of the same unknowns
in all of the other equations (sometimes you have 5 x (7 x + 2y = 23), or
to multiply each equation by something in order to 35x + 10y = 115
achieve this, such as in this example).
- To eliminate the unknown whose coefficients have 35 x + 10y = 115
the same absolute value: -(6 x +10y = 28)
a. Add them together if their signs are unlike. 29x = 87 b. Subtract one from the other if they have the same sign.
- Divide both sides by the coefficient of the
other unknown to find its value:
x = 3
- Find the value of the other unknown by putting 3 x = 14 - 5y
the value of the variable you now know into the 3(3) = 14 - 5y
equations, and follow the process of transposition 9 = 14 - 5y
discussed earlier, if necessary: -5 = -5y
y = 1
- Check the common solution in each of the other original 7 x + 2y = 23
equations: 7(3) + 2(1) = 23
23 = 23
That’s it! You can now solve for two variables with two or more equations.
29 x 29
OTHER IMPORTANT MATHEMATICAL SKILLS
Rounding Rules
Often, when you solve for a variable or perform a mathematical procedure, the number you
come up with may be long and unwieldy, with many numbers to the right of the decimal point.
Sometimes you can express this as a fraction, but other times, such as on a spreadsheet, a certain
number of decimal places is necessary. For example:
7x = 23
x = 3.
Since in many cases it is unnecessary to include the entire number, rounding is used. How
many decimal points the user rounds to is dependent on the good judgment of that person, but rules
govern how the number itself is rounded. Say, for example, that the person determining the number
above wants to round to the third decimal place (to the thousandth). Thus:
x = 3.
One rounds up or down depending on how close the number is to the whole digit in the
decimal place to which the user is rounding. Since in this example, the number was closer to
3.286 than 3.285, the number rounded to was 3.286. When a whole number or a decimal is exactly
halfway between two possible answers, round it off to the larger. Thus, if x had equalled 3.2855 in
the above example, you would round to 3.286, but if it had equalled 3.28549, you would round to
3.285.
Parentheses, Brackets and Braces
If you saw the expression 5 x 8 + 19 and did not have any indication in what order to
perform the functions, you might come up with one of the two very different answers of 59 (if
you multiplied the 5 and the 8 first and then added the 19), or 135 (if you added the 8 and 19 first
and then multiplied the sum by 5). Without any guidelines as to which operation, addition or
multiplication, to perform first, you have to guess which one should be done first. You often need
a way to indicate to your readers in which order to perform mathematical operations when there
are two or more in your expressions. Using parentheses to identify those operations that should be
done first is the easiest way to achieve clarity and understanding with your readers. For example,
if the writer had simply written the expression (5 x 8) + 19, it would cut down on the guessing, and
the reader would know the answer is 59.
Sometimes as expressions become more complicated, there will be operations within
operations that need to be separated. If you used parentheses exclusively, things might get
confusing very quickly. Instead, there is a hierarchy of notations that can be used. The
order in which they should be used is this: parentheses “( )” around those operations that
7 x
Raisng a Number to a Power Enter number Press {yx} key Enter exponent Press {=} key
Finding a Square Root of a Number Press { } or { x 1/2} key Enter number Press {=} key
Find the nth Root of a Number Enter n Press { } or { y 1/ x } key Enter number Press {=} key
Finding the Inverse of a Number Enter number Press { x -1} or { } key Press {=} key
FUNCTIONAL BUSINESS MATH
Right about now, you are probably saying, “Well, I’ve learned all this stuff, but where has
it gotten me? What are the practical applications? How is this going to help me be a better
manager?” Though currently you may doubt it, the algebra and other skills you have learned so far
are necessities for many different functional areas of business. Let’s look at a few applications of
these principles in business in order to see the truth of this statement.
Fixed and Variable Costs
A central tenet to many business fields, among them managerial accounting, operations and
marketing, is the idea of fixed and variable costs. Quite simply, a variable cost is one that is
incurred by a company depending on how much is produced. For example, rubber is a variable cost
in a tire factory; if the factory does not produce any tires, it will not need to buy any rubber. The
total cost of the rubber varies with the level of production, whereas the variable cost for each unit
is unchanged. Often certain labor, such as the wages of workers on an assembly line, is considered
a variable cost when the need for that labor is dependent on whether a product is built or not; this
is called direct labor and is assumed to only be needed and used when a product is built. On the
other hand, labor that is not dependent on a product being built is referred to as indirect labor.
Examples of indirect labor include the salaries of the accounting staff: their job is not linked to
the production of any specific product in a factory, but to all of them. For this reason, accounting
staff salary and other indirect labor is referred to as a fixed cost. A fixed cost is one that will be
incurred regardless of the level of production. Examples include rent on a factory and the cost of
new equipment. Regardless of whether the factory produces one unit or a million, the fixed costs
will still be incurred. Collectively, a company’s fixed and variable costs make up its total costs.
Managers like knowing how much things are going to cost before incurring those
costs and having to dip into the company’s coffers in order to pay for them. The problem is
x
x 1
that often total costs will change as the number of units produced changes. The reason for this is
that the variable costs change as the production level increases or decreases. However, when a
manager knows what both the variable costs per unit and the fixed costs will be, he or she can set
up a simple algebraic equation with one unknown variable and substitute different unit amounts of
production into that variable to find what the total costs will be for any production level. Putting
those words into an equation form, that is:
Total Costs = (Variable Costs Per Unit x Units of Production) + Fixed Costs
Let’s say the manager finds out that the fixed costs for a certain project will be $1 million and
the variable cost per unit is $5. He or she can set up an equation like this:
Total Costs = ($5 x x) + $1,000,
where x is the unknown for which the manager can substitute different unit levels of production.
Now suppose this manager wants to find out what the total costs will be for 100,000 units. To do
this, he or she substitutes 100,000 for the x in the equation:
Total Costs = ($5 x 100,000) + $1,000,000 = $1,500,
The total costs for the project will be $1,500,000. This may seem relatively easy, but it gets somewhat more complicated when all you have is
a sheet with the costs and have to determine which are fixed and which are variable. Let’s look at
a situation where this is true.
Example: Tubetime Industries builds televisions sold in the American market. Management
is thinking about creating a new television that will fit on a watch. The president of Tubetime has to
meet with the board tomorrow morning and introduce the proposal to fund this project. She needs
to know how much it is going to cost so that she can ask the board for the right amount of money.
The production manager gives her the following costs:
Material $75/television Direct Labor $40/television New Manufacturing Equipment $1,500, Modifications to Factory $3,000,
produce). The contribution margin is defined as the amount the company receives from the sale
of its product minus the product’s variable costs, and thus how much can be contributed to fixed
costs.
For example, say a manufacturer of oak desks can produce its basic model for $125 (including
materials, labor, and all other variable costs), and it can sell the model to a chain of office furniture
stores for $175. The contribution margin would be:
$175 - $125 = $ Thus, for every basic desk they sell, the company makes a $50 contribution towards its fixed
costs. Please note that the contribution margin is the amount the company receives, not necessarily
the amount for which it is sold to the end consumer. The office furniture store might sell the
desk to the consumer for $250, but the amount used to figure out the contribution margin is $175,
how much the company receives from its sale of the product to the office furniture store. This is
especially true when there are many intermediaries between the producer and the end user, and
when a company has several distribution channels through which to sell its product.
The Concept of Breakeven
Breakeven is the volume (in units or dollars) of sales needed to cover fixed costs after the
variable costs have been subtracted. Exactly what those fixed costs are may vary from company
to company, product to product. More often than not, a product must not only make a contribution
to those fixed costs which are directly involved in its production (such as machinery purchased
specifically to produce that product), but an allocation of costs relating to the rest of the company,
which may seem largely unrelated, i.e. administrative cost at headquarters, salaries of the officers
of the firm, etc. Otherwise, the product will be dropped. Contribution margin per unit is used to
determine breakeven level, the formula for which is shown below.
Breakeven Level =
Let’s examine the use of breakeven with a simple example. Let’s suppose a company is
making a product and has invested $100,000 on a slick advertising campaign and $75,000 each on
five slick salespeople (annual salary, including benefits and support). These are the only fixed costs
that the company has assigned to the product, and the contribution margin per unit is $50. To cover
those fixed costs, the company would like to determine how many units it would have to sell.
[$100,000 + (5 x $75,000)] = $475,000 (total fixed costs)
= 9,500 (breakeven level in units)
Fixed Costs Contribution Margin Per Unit
The company would therefore have to sell 9,500 units to break even or cover its fixed
costs.
Typically, the concept of breakeven is used in an incremental sense (i.e. how many additional
units would an advertising campaign have to sell above what would normally sell in order to be
worth spending the company’s money). In the example above, if this company did not have a
salesforce before and was not considering using an advertising campaign, 9,500 units would be the
number of additional sales necessary for this investment to be worthwhile.
GRAPHING
In the previous sections, you have learned a great deal about equations, monomials and
polynomials and how to apply them to real business problems. You found out what total costs and
breakeven levels were for a certain level of production. In the real world, however, you may want
to examine the costs associated with many production levels, and you might wonder if there is a
better way to do this than working out the same equation multiple times. In fact, there is. You can
make your work much simpler by building a graph and plotting a line or curve on that graph to
represent an equation. Depending on how precisely the line and the graph are drawn, you can then
estimate to some degree of accuracy at what level one parameter will be for a given level of another.
For example, you could see how total costs would vary with changes in the level of production by
plotting the equation of the line on a graph and then finding the point on the line which corresponds
with the total cost for a given level of production. Graphing is also an easy way to communicate
plans, ideas and historical trends to others in your office or elsewhere.
Basics of Graphing
What we will be using in the following few pages are technically called Cartesian Coordinate
Systems, but for simplicity we will call them “graphs”. Physically, a graph generally represents
two perpendicular lines, with numbers marked alongside each of those lines, such as in the example
on the next page.
Figure 1- An L-Shaped Graph Form
Plotting a Line
Now that we know what the form of a graph looks like, we should understand what we have
to put on it in order to make it useful to us. The basic unit of all graphing is the point. The point is
a location in space which can be defined by a set of coordinates : the coordinates are the numbers
on the axes which correspond to the point. In a two-dimensional graph, which are identified by
having two axes such as those above, a point is a location on a plane, or two-dimensional surface,
which can be defined by two coordinates: ( x,y ). The x coordinate of a point is called the abscissa ;
the coordinate tells its straight line distance from the y , or vertical, axis, and on which side of that
axis it sits. The y coordinate of a point is called the ordinate ; the number that is the coordinate
tells how many units away it is from the x , or horizontal, axis, and on which side of that axis it sits.
Using this information, the user of a two-dimensional graph can find in what quadrant a point sits
and then find or define that point.
A line is a series of points adjacent to one another. If you know you are graphing a line, you
do not have to graph all of the points on that line to get a line. You can estimate what a line will
look like just by finding two points far enough apart, plotting them on a graph, connecting them
with a straight line and then continuing that line as far as you would like beyond those two points.
You can determine two points using a linear equation , an equation whose graph is a straight line.
The basic form of a linear equation is given below:
y = a + bx
1
2
3
(^1 2 )
y
x
I
Several texts show this equation as y = mx + b ; we prefer y = a + bx. An equation is linear
as long as the exponent of x is 1 (which is usually denoted simply by x ). The x is often referred
to as the independent variable (a term you will see later in the regression analysis chapter), the y
as the dependent variable. The b is called the coefficient of x just like it was before, and its value
determines the slope of the line, or the line’s vertical rise divided by its horizontal run. The slope
can be denoted as:
where means “the change in the variable that follows between two points”. While positive
slopes (the slopes of positive coefficients) curve upward, left to right, negative slopes (the slopes
of negative coefficients) curve downward, left to right. The farther away from zero the coefficient
is, the steeper the slope. A line with a slope of zero or no slope is horizontal. The value given to
a , the constant, shows the location at which the line that is graphed from this equation crosses the
y-axis (where x = 0). It is called the y-intercept. If the constant is negative (such as y = - 3 + 5 x ),
then it is a negative intercept, and is below the x -axis. A linear equation has only one coefficient,
and only one y -intercept. If an equation has an x with an exponent of any other number than one, it
is not a linear equation, although a may be represented occasionally as ax^0 , which is the same thing
as saying a , as for example 9 x^0 , which equals 9.
Once you are given a linear equation, you can graph its line just by substituting in different
values for x and through that generating different values of y. We can use a fixed/variable cost
example from our functional business math section in order to demonstrate this.
Example:
The fixed costs for a certain project are $10 (it is a very inexpensive project) and the variable
costs are $2 per unit. We want to build an equation and then graph the line. We can easily determine
the equation to be:
y = 10 + 2 x
where y is the total costs, and
x is the units of production.
Now, in order to build our graph, we have to compile the information that we already know.
We know that the slope is 2 (the value of the coefficient, b ), and that the y -intercept is 10 (the value
of a ). At the y -intercept, x = 0, so we already know one of our two points (0,10). We can find
another point just by picking an arbitrary x far enough away from the y -intercept (0,10) to draw a
reasonably distinguishable line. When x = 20, for example, y = 50, so we could graph (20,50) and
connect it to (0,10), as below, then extend the line as far upwards as we wanted.
y
x
b =
Putting these two equations together on the same graph, they look like this:
Figure 1- Graph of Two Equations
You can see that the point at which the two lines intersect is (5,21). You therefore know that
these two equations are equal when x = 5 and y = 21. Graphing is helpful when it is either difficult
to solve for simultaneous equations or you do not like solving for them.
So far in this chapter, you have learned to set up a problem as an equation and then graph
it. Now we will examine what happens as that equation becomes more complex and exponents are
added to the variables.
5
10
15
**-
-**
-15 -10 -5 5 10 15
y
x
-
-20 20
20
y =16+ x -------
y =11+2 x --------
----------point of
intersection
Quadratic Equations
Previously, we said that a polynomial is an expression which contains two or more terms
(like 5 x^2 + 9). A quadratic equation is an equation containing a polynomial on one of its sides that
has one or more terms containing the unknown variable x , where one of those x ’s has an exponent
of two and none of the other x ’s has an exponent above two or below zero. The polynomial is set
equal to a number or variable on the other side of the equation. If this description confuses you, a
quadratic equation can be denoted this way:
y = ax^2 + bx + c
where x and y are variables,
a and b are coefficients, and c is the constant.
While b and c can be any number, including zero, fractions, and negative numbers, a can
be any number but zero. When you are given a quadratic equation, often it will be in a form where
you can simplify it. For example, y = 3 x 2
- 18 x + 3 can be simplified to
y = 3( x^2 + 6 x + 1), and then the whole equation can be simplified by dividing both sides by 3. If
you have fractions in the coefficients or constants, you can multiply both sides of the equation
to simplify them. For example, can be simplified by multiplying both sides of the
equation by 4, which results in 4 y = 2 x 2
- x + 20. While a quadratic equation can only contain x ’s up to the exponent of two, a cubic equation
(such as y = 4 x 3
- 8 x - 7) contains an x with an exponent of three and no x ’s with exponents
below zero, which can be denoted as:
y = ax
+ bx
+ cx + d
Though we will preoccupy ourselves mainly with quadratic equations in this chapter, it is
important to know that these other sorts of equations exist as well and can be graphed.
While we spent some time in the past section discussing y -intercepts in graphs of linear
equations, in the graph of a quadratic equation of the form y = ax^2 + bx + c sometimes you may
have two x-intercepts (where the graph crosses the x -axis) and sometimes you may have none at
all. The reason for this is that an x in a polynomial can take on several values and still result in the
same y : for instance, in the equation y = 5 x 2
- 3, x can take on a value of 2 or -2 and y will still equal
- Thus, you have two x ’s for every y in an equation of the form y = ax^2 + bx + c , and the result is
a curve called a parabola. A parabola is symmetric around an invisible line that cuts it in half; this
is called its axis of symmetry. The point at which the axis of symmetry intersects the parabola is
called its vertex.
Let’s look at a basic example. The equation y = 4 + 2 x^2 is a quadratic equation even though
it may not look like it: it follows the form for quadratic equations given earlier, except in this case
the coefficient b is zero, so bx completely disappears. The graph of this equation is the upper one
in Figure 1-5:
y =^1 2
x^2 +^1 4 x + 5
