Basic Business Math - Study Notes
MariusFaillotDevarre
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Dec 21, 2013
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About This Presentation
A basic understanding of decimals and percentages is key to any businessperson, whether tallying costs for warehouse supplies or estimating resource allocation.
Therefore learn to use decimals, addition, subtraction, multiplication, and division; and to solve problems involving percentages.
Also, ...
Slide Content
Basic Business Math
Study Notes
Entry Level
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Study Notes
Entry Level
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Table of Contents
1. Importance of Basic Math in Business .................................................................................. 4
2. Defining Number Concepts ................................................................................................... 4
3. Estimating Whole Numbers in Business ................................................................................ 6
4. Adding and Subtracting Fractions ......................................................................................... 7
5. Multiplying Fractions............................................................................................................. 9
6. Dividing Fractions ................................................................................................................ 11
7. Performing Operations with Fractions ................................................................................ 12
8. Solving Simple Equations .................................................................................................... 13
9. The Order of Operations ..................................................................................................... 15
10. Applying the Rules of Order ............................................................................................ 16
11. Place Values .................................................................................................................... 18
12. Project Cost Estimate ...................................................................................................... 19
13. Business Finance ............................................................................................................. 20
14. Glossary ........................................................................................................................... 22
2 | Pa g e B a s i c B u s i n e s s M a t h
Table of Contents
1. Importance of Basic Math in Business .................................................................................. 4
2. Defining Number Concepts ................................................................................................... 4
3. Estimating Whole Numbers in Business ................................................................................ 6
4. Adding and Subtracting Fractions ......................................................................................... 7
5. Multiplying Fractions............................................................................................................. 9
6. Dividing Fractions ................................................................................................................ 11
7. Performing Operations with Fractions ................................................................................ 12
8. Solving Simple Equations .................................................................................................... 13
9. The Order of Operations ..................................................................................................... 15
10. Applying the Rules of Order ............................................................................................ 16
11. Place Values .................................................................................................................... 18
12. Project Cost Estimate ...................................................................................................... 19
13. Business Finance ............................................................................................................. 20
14. Glossary ........................................................................................................................... 22
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1. Importance of Basic Math in Business
i. Calculate Production Costs
Before you formally establish your business, you must estimate the cost to manufacture or
acquire your product or perform your service. Adding all expenses associated with making
or buying items helps you realize if you can be competitive with other companies and
profitable enough to sustain your business and make a reasonable income. In addition to
the standard costs of production, such as materials and machinery, add accompanying
expenses, such as shipping, labor, interest on debt, storage and marketing. The basis to your
business plan is an accurate representation of how much you will spend on each item.
ii. Measure Profits
If you want to determine the net profit for a certain time period, you will need to subtract
returns, costs to produce an item and operating expenses from your total amount of sales,
or gross revenue, during that time. Discounts on products, depreciation on equipment and
taxes also must be calculated and subtracted from revenue. To arrive at your net profit, add
any interest you earned from credit extended to customers, which is reflected as a percent
of the amount each person owes. Your net profit helps you understand if you are charging
enough for your product and selling an adequate volume to continue to operate your
business or even expand.
iii. Analyze Finances
To analyze the overall financial health of your business, you will need to project revenue and
expenses for the future. It's important to understand the impact to your accounting records
when you change a number to reflect an increase or decrease in future sales. Estimating
how much an employee affects revenue will indicate if you can afford to add to your staff
and if the profits realized will be worth the expense. If a competitor starts selling a cheaper
product, you may need to calculate the amount by which your volume must increase if you
reduce prices. You may need to know if you can afford to expand your operations to
improve sales. Using basic business math to understand how these types of actions impact
your overall finances is imperative before taking your business to the next level.
2. Defining Number Concepts
Many jobs require specific mathematical skills while others require you to think precisely
and reason logically—skills gained from using math.
Here, you will learn about three types of numbers that you are likely to see on the job. They
are:
1. Whole numbers
A good way to understand whole numbers is to use a number line. The numbers to the left
4 | Pa g e B a s i c B u s i n e s s M a t h
1. Importance of Basic Math in Business
i. Calculate Production Costs
Before you formally establish your business, you must estimate the cost to manufacture or
acquire your product or perform your service. Adding all expenses associated with making
or buying items helps you realize if you can be competitive with other companies and
profitable enough to sustain your business and make a reasonable income. In addition to
the standard costs of production, such as materials and machinery, add accompanying
expenses, such as shipping, labor, interest on debt, storage and marketing. The basis to your
business plan is an accurate representation of how much you will spend on each item.
ii. Measure Profits
If you want to determine the net profit for a certain time period, you will need to subtract
returns, costs to produce an item and operating expenses from your total amount of sales,
or gross revenue, during that time. Discounts on products, depreciation on equipment and
taxes also must be calculated and subtracted from revenue. To arrive at your net profit, add
any interest you earned from credit extended to customers, which is reflected as a percent
of the amount each person owes. Your net profit helps you understand if you are charging
enough for your product and selling an adequate volume to continue to operate your
business or even expand.
iii. Analyze Finances
To analyze the overall financial health of your business, you will need to project revenue and
expenses for the future. It's important to understand the impact to your accounting records
when you change a number to reflect an increase or decrease in future sales. Estimating
how much an employee affects revenue will indicate if you can afford to add to your staff
and if the profits realized will be worth the expense. If a competitor starts selling a cheaper
product, you may need to calculate the amount by which your volume must increase if you
reduce prices. You may need to know if you can afford to expand your operations to
improve sales. Using basic business math to understand how these types of actions impact
your overall finances is imperative before taking your business to the next level.
2. Defining Number Concepts
Many jobs require specific mathematical skills while others require you to think precisely
and reason logically—skills gained from using math.
Here, you will learn about three types of numbers that you are likely to see on the job. They
are:
1. Whole numbers
A good way to understand whole numbers is to use a number line. The numbers to the left
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of zero are negative and the ones to the right of zero are positive. Zero is neither positive
nor negative.
Whole numbers are just the counting numbers and zero: . . . -4, -3, -2, -1, 0, 1, 2, 3, 4, . . .
The set of whole numbers goes on without end. Whole numbers:
Represent whole units—Zero and negative numbers—both of which are categories
of whole numbers—can be difficult concepts. Basically, zero is a number that is
neither positive nor negative. A negative number is a number that is less than zero.
Can be 0—Zero is a whole number that represents no units. It can be multiplied, but
the result is always zero. You could even have one million groups of zero, and they
would still add up to zero. You can't divide by zero.
Can be positive or negative, with the exception of 0—Negative numbers are less
than zero. They're like subtracted units—a debt or a deficit. For example, when you
borrow money, and spend it, you are left with a debt—a negative number. You have
to re-pay it before you can have positive cash flow.
2. Fractions
Think of whole numbers as one or more complete units. If whole numbers are divided or
split, the parts become fractions of the whole numbers. For example, if a whole number is
divided into five parts, each part becomes a fraction: 1/5. Two (of the five) parts would be
2/5, three (of the five) parts would be 3/5, and so on.
The number on top of the bar is called the numerator. It represents the number of parts of
the whole. The number underneath the bar is called the denominator. It tells you how many
parts the whole is divided into.
There are two different types of fractions: proper fractions and improper fractions. In order
to solve problems involving fractions, you will need to understand and use both types. They
can be defined as follows:
Proper fraction—This is a fraction in which the numerator is smaller than the
denominator—3/16, 1/4, 2/3, 1/2, 7/8, for example.
Improper fraction—This is a fraction in which the numerator is equal to or larger
than the denominator—6/3, 7/4, 9/8, 5/2, 11/5, 1/1, for example.
Improper fractions are improper because they are equal to or greater than the whole
number one. For example, 6/6 would equal one, because it represents all six parts of a six-
part whole. 7/6 is six parts, plus one extra one, written as 1/6. So the improper fraction 7/6
could be written as the mixed number 1 1/6. Mixed numbers consist of a whole number
followed by a proper fraction.
3. Mixed numbers
You use mixed numbers when you need to count whole units and parts at the same time.
For example: If you have three full cans of soda and one half-full can of soda, you write it
like this: 3 1/2, and say it like this three and one half. It's really 3 + 1/2. That's why we say
the and.
5 | Pa g e B a s i c B u s i n e s s M a t h
of zero are negative and the ones to the right of zero are positive. Zero is neither positive
nor negative.
Whole numbers are just the counting numbers and zero: . . . -4, -3, -2, -1, 0, 1, 2, 3, 4, . . .
The set of whole numbers goes on without end. Whole numbers:
Represent whole units—Zero and negative numbers—both of which are categories
of whole numbers—can be difficult concepts. Basically, zero is a number that is
neither positive nor negative. A negative number is a number that is less than zero.
Can be 0—Zero is a whole number that represents no units. It can be multiplied, but
the result is always zero. You could even have one million groups of zero, and they
would still add up to zero. You can't divide by zero.
Can be positive or negative, with the exception of 0—Negative numbers are less
than zero. They're like subtracted units—a debt or a deficit. For example, when you
borrow money, and spend it, you are left with a debt—a negative number. You have
to re-pay it before you can have positive cash flow.
2. Fractions
Think of whole numbers as one or more complete units. If whole numbers are divided or
split, the parts become fractions of the whole numbers. For example, if a whole number is
divided into five parts, each part becomes a fraction: 1/5. Two (of the five) parts would be
2/5, three (of the five) parts would be 3/5, and so on.
The number on top of the bar is called the numerator. It represents the number of parts of
the whole. The number underneath the bar is called the denominator. It tells you how many
parts the whole is divided into.
There are two different types of fractions: proper fractions and improper fractions. In order
to solve problems involving fractions, you will need to understand and use both types. They
can be defined as follows:
Proper fraction—This is a fraction in which the numerator is smaller than the
denominator—3/16, 1/4, 2/3, 1/2, 7/8, for example.
Improper fraction—This is a fraction in which the numerator is equal to or larger
than the denominator—6/3, 7/4, 9/8, 5/2, 11/5, 1/1, for example.
Improper fractions are improper because they are equal to or greater than the whole
number one. For example, 6/6 would equal one, because it represents all six parts of a six-
part whole. 7/6 is six parts, plus one extra one, written as 1/6. So the improper fraction 7/6
could be written as the mixed number 1 1/6. Mixed numbers consist of a whole number
followed by a proper fraction.
3. Mixed numbers
You use mixed numbers when you need to count whole units and parts at the same time.
For example: If you have three full cans of soda and one half-full can of soda, you write it
like this: 3 1/2, and say it like this three and one half. It's really 3 + 1/2. That's why we say
the and.
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Numbers are not all the same. Fractions, mixed numbers, and whole numbers all possess
different characteristics that you must understand in order to complete mathematical
problems.
Understanding number concepts is a first step to mastering basic math. Whole numbers,
fractions, and mixed numbers are the building blocks that enable you to perform numerous
important business calculations.
3. Estimating Whole Numbers in Business
Not all numbers are whole numbers, but whole numbers are easy to work with, and they're
all that’s required in many business situations.
Place value
Estimating requires you to understand the concept of place value. Look at the example
below to find out more about place value.
The number 123 consists of three digits. The value of each digit depends on its place, or
position, in the number. The digit 1 is in the hundreds place. The digit 2 is in the tens place.
And the digit 3 is in the ones place. Each place has a value of 10 times the place to its right.
Therefore, there is one set of 100, plus two sets of 10, plus three sets of 1 in the number
123.
When you round a number, how many numbers you change depends upon whether you
want the number rounded to the nearest one, the nearest ten, the nearest hundred, or
whatever place value you choose. Look at the number just to the right of that place value to
determine whether you round up or down.
Generally, you round up if the digit to the right of your place value is 5 or above, and down if
the place value is 4 or below. For example, if you wanted to round 150 to the nearest
hundred, the answer would be 200.
Quick estimate
When an employee needs a ballpark figure to make a fast business decision, he often
doesn't want to be bothered with odd numbers or fractions. He will round whole numbers
to get a quick estimate. When you need to estimate a size, distance, weight, time, or any
other measurement, you start off by rounding the numbers to be included in your
calculation.
For example, if you need to know about how much room is left in a 55-gallon drum, and you
know you've put in 11 gallons, 13 gallons and 17 gallons, you would round those volumes to
10 gallons, 10 gallons, and 20 gallons for a quick estimate. Adding those rounded numbers,
you would estimate that there was 40 gallons in the tank, so you have room for about 15
more. That gives close approximations of the exact figures, which are 41 and 14,
respectively.
6 | Pa g e B a s i c B u s i n e s s M a t h
Numbers are not all the same. Fractions, mixed numbers, and whole numbers all possess
different characteristics that you must understand in order to complete mathematical
problems.
Understanding number concepts is a first step to mastering basic math. Whole numbers,
fractions, and mixed numbers are the building blocks that enable you to perform numerous
important business calculations.
3. Estimating Whole Numbers in Business
Not all numbers are whole numbers, but whole numbers are easy to work with, and they're
all that’s required in many business situations.
Place value
Estimating requires you to understand the concept of place value. Look at the example
below to find out more about place value.
The number 123 consists of three digits. The value of each digit depends on its place, or
position, in the number. The digit 1 is in the hundreds place. The digit 2 is in the tens place.
And the digit 3 is in the ones place. Each place has a value of 10 times the place to its right.
Therefore, there is one set of 100, plus two sets of 10, plus three sets of 1 in the number
123.
When you round a number, how many numbers you change depends upon whether you
want the number rounded to the nearest one, the nearest ten, the nearest hundred, or
whatever place value you choose. Look at the number just to the right of that place value to
determine whether you round up or down.
Generally, you round up if the digit to the right of your place value is 5 or above, and down if
the place value is 4 or below. For example, if you wanted to round 150 to the nearest
hundred, the answer would be 200.
Quick estimate
When an employee needs a ballpark figure to make a fast business decision, he often
doesn't want to be bothered with odd numbers or fractions. He will round whole numbers
to get a quick estimate. When you need to estimate a size, distance, weight, time, or any
other measurement, you start off by rounding the numbers to be included in your
calculation.
For example, if you need to know about how much room is left in a 55-gallon drum, and you
know you've put in 11 gallons, 13 gallons and 17 gallons, you would round those volumes to
10 gallons, 10 gallons, and 20 gallons for a quick estimate. Adding those rounded numbers,
you would estimate that there was 40 gallons in the tank, so you have room for about 15
more. That gives close approximations of the exact figures, which are 41 and 14,
respectively.
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Degree of accuracy
What place value you round a number to depends upon the degree of accuracy you need,
and upon the size of the number. Usually, a higher number can be rounded to a higher place
value.
Estimating whole numbers
Rounding is an important part of estimating in the workplace environment, but it's not the
only step. When you need to calculate requirements, output, time, distance, volume, or
area, you need to apply other math skills.
The steps for estimating whole numbers include:
Rounding up, if place value is 5 or above—Rounding is an essential step that makes
estimating a short-cut to useful data. You need to determine the place value to
round to, depending upon the precision needed for the estimate, based on how the
estimate will be used.
Rounding down, if place value is 4 or below—Rounding is an essential step that
makes estimating an easier, faster process than using mixed numbers or odd
numbers.
Identifying the process to be used in your equation—You have to determine
whether your estimate involves addition, multiplication, subtraction, or division.
Adding, multiplying, subtracting, or dividing—Doing the math is the last step after
you decide what mathematical process to use and round off the numbers that are
being included in the estimate.
Estimating whole numbers to a given place value is a great way to save time and work when
precise calculations aren't required, so you can make timely, reasonable business decisions.
4. Adding and Subtracting Fractions
Fractions are different from whole numbers in one important way: They are parts of a
whole. For example, the fraction one-quarter simply means one part of a unit that has been
divided into four parts. This is expressed as one-quarter.
Denominator and numerator
The number of parts a whole is divided into, called the denominator, is shown by the
number under the bar. The number of parts in the fraction—the number above the bar—is
called the numerator.
Reading fractions out loud
When you read fractions out loud, the general rule is to substitute the word over for the bar
(/). So 23/8 should be read as twenty-three over eight. However, fractions with
denominators between 2 and 9 have designated names: half, thirds, quarters, fifths, sixths,
sevenths, eighths, and ninths. So you would say five-sixths, not five over six, for example.
Also, if the denominator is a power of 10 (10, 100, 1,000, for example) you always say the
decimal name: one-tenth, four-hundredths, twenty-six thousandths.
7 | Pa g e B a s i c B u s i n e s s M a t h
Degree of accuracy
What place value you round a number to depends upon the degree of accuracy you need,
and upon the size of the number. Usually, a higher number can be rounded to a higher place
value.
Estimating whole numbers
Rounding is an important part of estimating in the workplace environment, but it's not the
only step. When you need to calculate requirements, output, time, distance, volume, or
area, you need to apply other math skills.
The steps for estimating whole numbers include:
Rounding up, if place value is 5 or above—Rounding is an essential step that makes
estimating a short-cut to useful data. You need to determine the place value to
round to, depending upon the precision needed for the estimate, based on how the
estimate will be used.
Rounding down, if place value is 4 or below—Rounding is an essential step that
makes estimating an easier, faster process than using mixed numbers or odd
numbers.
Identifying the process to be used in your equation—You have to determine
whether your estimate involves addition, multiplication, subtraction, or division.
Adding, multiplying, subtracting, or dividing—Doing the math is the last step after
you decide what mathematical process to use and round off the numbers that are
being included in the estimate.
Estimating whole numbers to a given place value is a great way to save time and work when
precise calculations aren't required, so you can make timely, reasonable business decisions.
4. Adding and Subtracting Fractions
Fractions are different from whole numbers in one important way: They are parts of a
whole. For example, the fraction one-quarter simply means one part of a unit that has been
divided into four parts. This is expressed as one-quarter.
Denominator and numerator
The number of parts a whole is divided into, called the denominator, is shown by the
number under the bar. The number of parts in the fraction—the number above the bar—is
called the numerator.
Reading fractions out loud
When you read fractions out loud, the general rule is to substitute the word over for the bar
(/). So 23/8 should be read as twenty-three over eight. However, fractions with
denominators between 2 and 9 have designated names: half, thirds, quarters, fifths, sixths,
sevenths, eighths, and ninths. So you would say five-sixths, not five over six, for example.
Also, if the denominator is a power of 10 (10, 100, 1,000, for example) you always say the
decimal name: one-tenth, four-hundredths, twenty-six thousandths.
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Improper fraction
If the numerator is equal to or greater than the denominator, the fraction is always
expressed as numerator over denominator, thirty-three over four, for example. So the
fraction 1/4 simply means that you have one part of a whole that has been divided into four
parts.
Adding and subtracting fractions
When adding and subtracting fractions, the first thing you deal with is determining the
denominator. If the denominators (the numbers under the bar) are the same in the
numbers you are adding or subtracting, your first step is to identify that denominator in
your answer by writing it under the bar. Then you can add or subtract the numerators (the
numbers above the bar) as you would any other numbers, and place the result above the
bar.
For example, if you're subtracting 1/4 from 3/4, just subtract 1 from 3, then put the result,
2, over the common denominator which is 4.
Change an improper fraction into a mixed number
When your addition or subtraction is finished—if the result is an improper fraction—you just
have to convert your results back into mixed numbers again. For example, consider 30/8.
You can change this improper fraction back into a mixed number by using five steps:
1. Divide the denominator into the numerator to determine the whole number—
Eight goes into 30 three times 3 x 8 = 24.
2. Calculate the remainder by multiplying the whole number by the denominator,
then subtracting the product from the numerator—Since 8 goes into 30 three
times, 3 becomes the whole number in the mixed number— 3 ?/8.
3. The remainder becomes the new numerator in the fraction—Since 8 x 3 only equals
24, you've got 6 left over that you must add in to reach 30. This is called a remainder.
The remainder becomes the numerator of the fraction part of the mixed number.
4. Write the mixed number—Now you know 3 is the whole number in the mixed
number, and you have a remainder of 6. That means 30/8 gets changed to 3 (the
whole number) and 6 (the remainder)/8. So, the new mixed number will be 3 6/8.
5. Simplify the fraction—Any time the numerator and denominator in a fraction are
divisible by the same number, you should do that math, and use the quotients as the
numerator and denominator. In the fraction 6/8, for example, 6 and 8 are both
divisible by 2, so the fraction can be simplified to 3/4.
Common denominator
You cannot add or subtract fractions that have different denominators. You must find a
common denominator. For example, say you have to add 1/3 and 1/4, you need to find the
lowest number that is divisible (can be divided) by both denominators (4 and 3). Often the
easiest way to do this is to simply multiply the denominators. In this example, the common
denominator would be 12 (4 x 3 = 12).
8 | Pa g e B a s i c B u s i n e s s M a t h
Improper fraction
If the numerator is equal to or greater than the denominator, the fraction is always
expressed as numerator over denominator, thirty-three over four, for example. So the
fraction 1/4 simply means that you have one part of a whole that has been divided into four
parts.
Adding and subtracting fractions
When adding and subtracting fractions, the first thing you deal with is determining the
denominator. If the denominators (the numbers under the bar) are the same in the
numbers you are adding or subtracting, your first step is to identify that denominator in
your answer by writing it under the bar. Then you can add or subtract the numerators (the
numbers above the bar) as you would any other numbers, and place the result above the
bar.
For example, if you're subtracting 1/4 from 3/4, just subtract 1 from 3, then put the result,
2, over the common denominator which is 4.
Change an improper fraction into a mixed number
When your addition or subtraction is finished—if the result is an improper fraction—you just
have to convert your results back into mixed numbers again. For example, consider 30/8.
You can change this improper fraction back into a mixed number by using five steps:
1. Divide the denominator into the numerator to determine the whole number—
Eight goes into 30 three times 3 x 8 = 24.
2. Calculate the remainder by multiplying the whole number by the denominator,
then subtracting the product from the numerator—Since 8 goes into 30 three
times, 3 becomes the whole number in the mixed number— 3 ?/8.
3. The remainder becomes the new numerator in the fraction—Since 8 x 3 only equals
24, you've got 6 left over that you must add in to reach 30. This is called a remainder.
The remainder becomes the numerator of the fraction part of the mixed number.
4. Write the mixed number—Now you know 3 is the whole number in the mixed
number, and you have a remainder of 6. That means 30/8 gets changed to 3 (the
whole number) and 6 (the remainder)/8. So, the new mixed number will be 3 6/8.
5. Simplify the fraction—Any time the numerator and denominator in a fraction are
divisible by the same number, you should do that math, and use the quotients as the
numerator and denominator. In the fraction 6/8, for example, 6 and 8 are both
divisible by 2, so the fraction can be simplified to 3/4.
Common denominator
You cannot add or subtract fractions that have different denominators. You must find a
common denominator. For example, say you have to add 1/3 and 1/4, you need to find the
lowest number that is divisible (can be divided) by both denominators (4 and 3). Often the
easiest way to do this is to simply multiply the denominators. In this example, the common
denominator would be 12 (4 x 3 = 12).
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Next, you must convert the original fractions to new fractions, using the new common
denominator. To do this, divide the original denominator into the new denominator, then
multiply the original numerator by the result to arrive at the new numerator. In the
example, the fraction 1/3 would convert to 4/12. To convert the fraction one-quarter (1/4)
to twelfths, you would divide four into 12, then multiply the result (3) times the original
numerator (1) to get three over twelve (3/12).
Now, you've got two fractions with the same denominators, so you can just add them: 4/12
+ 3/12 = 7/12.
Adding mixed numbers
Now you're ready to consider problems with numbers that include both whole numbers and
fractions—2 3/8 + 1 7/8, for example. These are called mixed numbers. When adding, you
can treat this as two separate problems. First add the whole numbers, (2 + 1), then add the
fractions (3/8 + 7/8), then add the two sums (3 + 1 2/8). The answer, of course, is 4 2/8, or
simplified, 4 1/4.
Subtracting mixed numbers
Your first step to subtract mixed numbers is to convert them to improper fractions.
Generally, we think of a fraction as something less than one. However, mixed numbers can
also be expressed as fractions, too. Because they are more than one, they are called
improper fractions.
To convert a mixed number to an improper fraction, you first multiply the denominator by
the whole number, then add the numerator to the result. For example, in the mixed number
2 3/8, you multiply the denominator 8 by the whole number 2. Then you take that result
(16) and add it to the numerator (3). That sum, 19, becomes the new numerator, so the
improper fraction is 19/8. The denominator, 8, does not change. The whole number is gone,
because it is now included in the improper fraction.
The steps for adding and subtracting fractions and mixed numbers basically involve making
the fractions similar so they are convenient to work with, doing the math, and then
converting the results into the number that is easiest to communicate and understand.
5. Multiplying Fractions
Multiplying fractions can be an important skill. For example, a warehouse may hire a
temporary worker for 5 1/2 hours, and pay her 10 1/4 dollars per hour. To figure out how
much the company will owe the worker at the end of the day, the warehouse manager will
need to multiply 5 1/4 x 10 1/4. If you can multiply and divide whole numbers, you can
multiply fractions.
The steps in the process are as follows:
multiply the numerators
multiply the denominators
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Next, you must convert the original fractions to new fractions, using the new common
denominator. To do this, divide the original denominator into the new denominator, then
multiply the original numerator by the result to arrive at the new numerator. In the
example, the fraction 1/3 would convert to 4/12. To convert the fraction one-quarter (1/4)
to twelfths, you would divide four into 12, then multiply the result (3) times the original
numerator (1) to get three over twelve (3/12).
Now, you've got two fractions with the same denominators, so you can just add them: 4/12
+ 3/12 = 7/12.
Adding mixed numbers
Now you're ready to consider problems with numbers that include both whole numbers and
fractions—2 3/8 + 1 7/8, for example. These are called mixed numbers. When adding, you
can treat this as two separate problems. First add the whole numbers, (2 + 1), then add the
fractions (3/8 + 7/8), then add the two sums (3 + 1 2/8). The answer, of course, is 4 2/8, or
simplified, 4 1/4.
Subtracting mixed numbers
Your first step to subtract mixed numbers is to convert them to improper fractions.
Generally, we think of a fraction as something less than one. However, mixed numbers can
also be expressed as fractions, too. Because they are more than one, they are called
improper fractions.
To convert a mixed number to an improper fraction, you first multiply the denominator by
the whole number, then add the numerator to the result. For example, in the mixed number
2 3/8, you multiply the denominator 8 by the whole number 2. Then you take that result
(16) and add it to the numerator (3). That sum, 19, becomes the new numerator, so the
improper fraction is 19/8. The denominator, 8, does not change. The whole number is gone,
because it is now included in the improper fraction.
The steps for adding and subtracting fractions and mixed numbers basically involve making
the fractions similar so they are convenient to work with, doing the math, and then
converting the results into the number that is easiest to communicate and understand.
5. Multiplying Fractions
Multiplying fractions can be an important skill. For example, a warehouse may hire a
temporary worker for 5 1/2 hours, and pay her 10 1/4 dollars per hour. To figure out how
much the company will owe the worker at the end of the day, the warehouse manager will
need to multiply 5 1/4 x 10 1/4. If you can multiply and divide whole numbers, you can
multiply fractions.
The steps in the process are as follows:
multiply the numerators
multiply the denominators
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10 | Pa g e B a s i c B u s i n e s s M a t h
simplify the fraction.
Multiplying the numerators and the denominators
Fractions get multiplied just like whole numbers, except that it's basically two problems—
one above the bar (multiplying the numerators), and one below the bar (multiplying the
denominators).
In this problem—2/3 x 1/4, for example, the numerators, 2 and 1, are above the bar, the
denominators, 3 and 4, are below the bar. The answer in a multiplication problem is called
the product. Multiplying the numerators (2 x 1) gives us 2, which is the numerator in the
product. Multiplying the denominators (3 x 4) gives us 12, which is the denominator in the
product. So the product is 2/12.
Simplifying the fraction
When you multiply fractions, you will often end up with a final product that can be
simplified. To simplify a fraction, say 6/12, you need to find a number that will divide into
both the numerator and the denominator. Then, do that math, and use the quotients as the
numerator and denominator. For example, 6 and 12 are both divisible by 6. Six divided by
six equals one, and twelve divided by six equals two.
Doing that division, you can simplify the fraction 6/12 to 1/2. Repeat this process until there
are no more numbers that will divide into both the numerator and the denominator.
When you multiply fractions, you need to reduce the answer to its lowest terms. This means
you must make sure there is no number, except 1, that can be divided evenly into both the
numerator and the denominator.
Canceling
You can also simplify the multiplication process by canceling before multiplying. Canceling is
a way to put a fraction into its lowest terms before you do the multiplication.
You cancel by dividing one numerator (any one) and one denominator (any one) by the
same number (any number). For example, if you're multiplying 3/4 x 5/9, the result is
(3x5)/(4x9). Both 3 (the first numerator) and 9 (the second denominator) are divisible by 3.
Dividing both numbers by 3 simplifies the equation to (1x5)/(4x3). When the numerator and
denominator are the same number, the fraction can be simplified to the number 1. For
example, 3/3 = 1.
When canceling, show the numerators being multiplied above the bar, and the
denominators being multiplied below the bar. Any one numerator and any one denominator
that are divisible by the same number can be canceled.
Multiplying mixed numbers
Sometimes, you're not just multiplying fractions, but mixed numbers containing both whole
numbers and fractions. The first step to multiply mixed numbers is to convert them into
improper fractions. For example, consider the problem 3 7/8 x 1/2 = ? The mixed number, 3
7/8 would be converted to 31/8 by multiplying 8 (the denominator) by 3 (the whole
number), and then adding 7 (the numerator) to produce a new numerator for multiplication
10 | Pa g e B a s i c B u s i n e s s M a t h
simplify the fraction.
Multiplying the numerators and the denominators
Fractions get multiplied just like whole numbers, except that it's basically two problems—
one above the bar (multiplying the numerators), and one below the bar (multiplying the
denominators).
In this problem—2/3 x 1/4, for example, the numerators, 2 and 1, are above the bar, the
denominators, 3 and 4, are below the bar. The answer in a multiplication problem is called
the product. Multiplying the numerators (2 x 1) gives us 2, which is the numerator in the
product. Multiplying the denominators (3 x 4) gives us 12, which is the denominator in the
product. So the product is 2/12.
Simplifying the fraction
When you multiply fractions, you will often end up with a final product that can be
simplified. To simplify a fraction, say 6/12, you need to find a number that will divide into
both the numerator and the denominator. Then, do that math, and use the quotients as the
numerator and denominator. For example, 6 and 12 are both divisible by 6. Six divided by
six equals one, and twelve divided by six equals two.
Doing that division, you can simplify the fraction 6/12 to 1/2. Repeat this process until there
are no more numbers that will divide into both the numerator and the denominator.
When you multiply fractions, you need to reduce the answer to its lowest terms. This means
you must make sure there is no number, except 1, that can be divided evenly into both the
numerator and the denominator.
Canceling
You can also simplify the multiplication process by canceling before multiplying. Canceling is
a way to put a fraction into its lowest terms before you do the multiplication.
You cancel by dividing one numerator (any one) and one denominator (any one) by the
same number (any number). For example, if you're multiplying 3/4 x 5/9, the result is
(3x5)/(4x9). Both 3 (the first numerator) and 9 (the second denominator) are divisible by 3.
Dividing both numbers by 3 simplifies the equation to (1x5)/(4x3). When the numerator and
denominator are the same number, the fraction can be simplified to the number 1. For
example, 3/3 = 1.
When canceling, show the numerators being multiplied above the bar, and the
denominators being multiplied below the bar. Any one numerator and any one denominator
that are divisible by the same number can be canceled.
Multiplying mixed numbers
Sometimes, you're not just multiplying fractions, but mixed numbers containing both whole
numbers and fractions. The first step to multiply mixed numbers is to convert them into
improper fractions. For example, consider the problem 3 7/8 x 1/2 = ? The mixed number, 3
7/8 would be converted to 31/8 by multiplying 8 (the denominator) by 3 (the whole
number), and then adding 7 (the numerator) to produce a new numerator for multiplication
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purposes. The new numerator goes over the original denominator, so the problem becomes
31/8 x 1/2, which equals 31/16. That improper fraction can then be simplified, so your final
answer is 1 15/16.
Once you prepare your multiplication problem by converting mixed numbers and canceling,
just multiply the numerators (the numbers above the line) and the denominators (the
numbers below the line). The result is your answer.
Although multiplying fractions can be cumbersome, understanding the steps can increase
your credibility and help improve job results.
6. Dividing Fractions
The process of dividing fractions is very similar to the process of multiplying fractions. It can
be a useful skill in many situations. For example, you might have 16 containers, each holding
1/3 of a gallon of resin. You want to put this material into a drum that holds 4 1/2 gallons.
To find out how many of the containers you can empty into the drum, you need to divide 4
1/2 by 1/3.
If you can multiply and divide whole numbers, you can divide fractions. The steps in the
process of dividing fractions are as follows:
1. Convert to improper fractions
When you divide fractions, you first need to convert both fractions to improper fractions, if
necessary. For example, let’s say you want to divide 6 7/8 by 2 13/16. To start, you first have
to convert both mixed numbers to the improper fractions 55/8 and 45/16.
Do this by multiplying the denominator by the whole number, and then adding the
numerator. Put the sum over the original denominator to create an improper fraction that
has the same value as the mixed number.
2. Invert the denominator fraction (the divisor)
After converting mixed numbers to improper fractions, your next step is to find the
reciprocal of the divisor. The divisor is the fraction that you are dividing into the other. The
fraction being divided is called the dividend. For example, in the problem 55/8 divided by
11/3, 11/3 is the divisor. Finding the reciprocal of 11/3 is easy. All you have to do is invert it,
or flip it over, to make the fraction 3/11—3/11 is the reciprocal of 11/3.
3. Multiply the fractions
Dividing fractions is basically a process of multiplication. For example, say you're cutting
lengths of sheet metal into strips, and you need to divide 6 7/8 inches by 2 13/16 inches.
4. Simplify the quotient (the answer in a division problem)
Follow these steps to calculate the quotient:
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purposes. The new numerator goes over the original denominator, so the problem becomes
31/8 x 1/2, which equals 31/16. That improper fraction can then be simplified, so your final
answer is 1 15/16.
Once you prepare your multiplication problem by converting mixed numbers and canceling,
just multiply the numerators (the numbers above the line) and the denominators (the
numbers below the line). The result is your answer.
Although multiplying fractions can be cumbersome, understanding the steps can increase
your credibility and help improve job results.
6. Dividing Fractions
The process of dividing fractions is very similar to the process of multiplying fractions. It can
be a useful skill in many situations. For example, you might have 16 containers, each holding
1/3 of a gallon of resin. You want to put this material into a drum that holds 4 1/2 gallons.
To find out how many of the containers you can empty into the drum, you need to divide 4
1/2 by 1/3.
If you can multiply and divide whole numbers, you can divide fractions. The steps in the
process of dividing fractions are as follows:
1. Convert to improper fractions
When you divide fractions, you first need to convert both fractions to improper fractions, if
necessary. For example, let’s say you want to divide 6 7/8 by 2 13/16. To start, you first have
to convert both mixed numbers to the improper fractions 55/8 and 45/16.
Do this by multiplying the denominator by the whole number, and then adding the
numerator. Put the sum over the original denominator to create an improper fraction that
has the same value as the mixed number.
2. Invert the denominator fraction (the divisor)
After converting mixed numbers to improper fractions, your next step is to find the
reciprocal of the divisor. The divisor is the fraction that you are dividing into the other. The
fraction being divided is called the dividend. For example, in the problem 55/8 divided by
11/3, 11/3 is the divisor. Finding the reciprocal of 11/3 is easy. All you have to do is invert it,
or flip it over, to make the fraction 3/11—3/11 is the reciprocal of 11/3.
3. Multiply the fractions
Dividing fractions is basically a process of multiplication. For example, say you're cutting
lengths of sheet metal into strips, and you need to divide 6 7/8 inches by 2 13/16 inches.
4. Simplify the quotient (the answer in a division problem)
Follow these steps to calculate the quotient:
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Convert the dividend to an improper fraction—To convert 6 7/8 to an improper
fraction, multiply the denominator 8 and the whole number 6. Then add the
numerator 7. The sum, 55, is the numerator of the improper fraction. The
denominator stays unchanged. So, 6 7/8 as an improper fraction is 55/8.
Convert the denominator to an improper fraction—To convert 2 13/16 to an
improper fraction, multiply the denominator 16 and the whole number 2. Then add
the numerator 13. The sum, 45, is the numerator of the improper fraction. The
denominator stays unchanged. So, 2 13/16 as an improper fraction is 45/16.
Invert the denominator fraction—Before you can solve this division problem, you
need to convert the denominator fraction (45/16) into its reciprocal. Just invert the
fraction to find the reciprocal (flip the fraction upside-down). The reciprocal of 45/16
is 16/45. So the new equation is 55/8 x 16/45.
Use canceling to simplify the problem—Using canceling, the equation 55/8 x 16/45,
can be simplified to 11/1 x 2/9. Five goes into 55 eleven times and into 45 nine
times. So, in the equation, change the 55 to 11, and 45 to 9. Eight goes into 8 one
time, and into 16 two times. So change the 8 to 1, and the 16 to 2.
Multiply the numerator fraction by the reciprocal of the denominator fraction—
Solve the problem by multiplying the numerators (11 and 2). The result, 22, will be
the numerator in your answer. Also, we'll multiply the denominators (1 and 9). That
gives you 9, which will be the denominator in your answer. This process gives you a
quotient of 22/9.
Reduce the quotient to its lowest term—Finally, you need to reduce your quotient
of 22/9 to the lowest term. In this case, all you can do is convert it to the mixed
number 2 4/9. Divide the denominator into the numerator to get the whole number,
and put the remainder above the bar. The denominator remains the same.
It's important to be aware of all the steps when you need to divide fractions. Sometimes
certain steps will not be necessary. If you're not working with mixed numbers, for example,
you don't have to convert them into improper fractions. Sometimes canceling won't work,
or quotients can't be further reduced. Still, being aware of every step in the process will
keep division problems as simple as possible, and help ensure accuracy.
Understanding the step-by-step process of dividing fractions—including the tricky concepts,
like mixed numbers and improper fractions—is a useful skill that helps you be a more
educated employee and a sharper manager.
7. Performing Operations with Fractions
When you're working with fractions, you also need to make sure you're using the right
operation—addition, subtraction, multiplication, or division. That might seem simple, but it
isn't always clear which operation you should use.
Addition
When you have several sets of similar units, and you're trying to figure out how many you
have altogether, you need to use addition. For example, if you have a half-pound of wax at
12 | Pa g e B a s i c B u s i n e s s M a t h
Convert the dividend to an improper fraction—To convert 6 7/8 to an improper
fraction, multiply the denominator 8 and the whole number 6. Then add the
numerator 7. The sum, 55, is the numerator of the improper fraction. The
denominator stays unchanged. So, 6 7/8 as an improper fraction is 55/8.
Convert the denominator to an improper fraction—To convert 2 13/16 to an
improper fraction, multiply the denominator 16 and the whole number 2. Then add
the numerator 13. The sum, 45, is the numerator of the improper fraction. The
denominator stays unchanged. So, 2 13/16 as an improper fraction is 45/16.
Invert the denominator fraction—Before you can solve this division problem, you
need to convert the denominator fraction (45/16) into its reciprocal. Just invert the
fraction to find the reciprocal (flip the fraction upside-down). The reciprocal of 45/16
is 16/45. So the new equation is 55/8 x 16/45.
Use canceling to simplify the problem—Using canceling, the equation 55/8 x 16/45,
can be simplified to 11/1 x 2/9. Five goes into 55 eleven times and into 45 nine
times. So, in the equation, change the 55 to 11, and 45 to 9. Eight goes into 8 one
time, and into 16 two times. So change the 8 to 1, and the 16 to 2.
Multiply the numerator fraction by the reciprocal of the denominator fraction—
Solve the problem by multiplying the numerators (11 and 2). The result, 22, will be
the numerator in your answer. Also, we'll multiply the denominators (1 and 9). That
gives you 9, which will be the denominator in your answer. This process gives you a
quotient of 22/9.
Reduce the quotient to its lowest term—Finally, you need to reduce your quotient
of 22/9 to the lowest term. In this case, all you can do is convert it to the mixed
number 2 4/9. Divide the denominator into the numerator to get the whole number,
and put the remainder above the bar. The denominator remains the same.
It's important to be aware of all the steps when you need to divide fractions. Sometimes
certain steps will not be necessary. If you're not working with mixed numbers, for example,
you don't have to convert them into improper fractions. Sometimes canceling won't work,
or quotients can't be further reduced. Still, being aware of every step in the process will
keep division problems as simple as possible, and help ensure accuracy.
Understanding the step-by-step process of dividing fractions—including the tricky concepts,
like mixed numbers and improper fractions—is a useful skill that helps you be a more
educated employee and a sharper manager.
7. Performing Operations with Fractions
When you're working with fractions, you also need to make sure you're using the right
operation—addition, subtraction, multiplication, or division. That might seem simple, but it
isn't always clear which operation you should use.
Addition
When you have several sets of similar units, and you're trying to figure out how many you
have altogether, you need to use addition. For example, if you have a half-pound of wax at
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13 | Pa g e B a s i c B u s i n e s s M a t h
your workstation, seven and a half pounds in the supply closet, and ten and a quarter
pounds in the warehouse, how much do you have altogether? That would be an addition
math problem because you're trying to figure out how many units (pounds) you have in
total.
Subtraction
On the other hand, if you know how much wax you have in total, but want to know how
much would be left if you use some, you would need to subtract to find the difference.
Multiplication
If you have a per-unit value and you need to know how much you need for a number of
units, you multiply.
For example, if you know you need 1/24 gallons of coloring agent for each gallon, and
you've got to make 22 3/4 gallons, you would multiply to figure out how much coloring
agent you need.
Division
If you need to know how many units of one thing are included in another, you need to
divide. For example, if you've got 175 drums that each hold 50 3/4 gallons of waste, and
your truck holds 5,000 gallons, how many drums can you empty into the truck?
Understanding how to work with fractions is just part of the issue when using math in the
workplace. You also need to choose the right process to get the information you need from
the numbers you have. Today’s workplace often requires higher education and advanced
skills. Employees with good math skills have a definite advantage over their coworkers.
In order to handle everyday workplace problems and make the best decisions, it is useful to
understand fractions and how to add, subtract, multiply, and divide fractions.
8. Solving Simple Equations
An equation is a mathematical statement that two expressions are equal. An equation
always includes an equal sign (=). For example, 75 + 25 = 100 is a very simple equation. An
equation is an analysis tool that lets you map out the relationship between known
quantities (numbers) and an unknown quantity. Stating that relationship in an equation
enables you to establish the value of the unknown quantity.
To solve business problems involving mathematics, you need to translate them into the
language of mathematics.
Converting a business problem into an equation
There are four steps to convert a business problem into an equation as follows:
1. Determine what you need to find out (the variable)—The first step is to make sure
you are clear about what information you need. Do you need to know how many
13 | Pa g e B a s i c B u s i n e s s M a t h
your workstation, seven and a half pounds in the supply closet, and ten and a quarter
pounds in the warehouse, how much do you have altogether? That would be an addition
math problem because you're trying to figure out how many units (pounds) you have in
total.
Subtraction
On the other hand, if you know how much wax you have in total, but want to know how
much would be left if you use some, you would need to subtract to find the difference.
Multiplication
If you have a per-unit value and you need to know how much you need for a number of
units, you multiply.
For example, if you know you need 1/24 gallons of coloring agent for each gallon, and
you've got to make 22 3/4 gallons, you would multiply to figure out how much coloring
agent you need.
Division
If you need to know how many units of one thing are included in another, you need to
divide. For example, if you've got 175 drums that each hold 50 3/4 gallons of waste, and
your truck holds 5,000 gallons, how many drums can you empty into the truck?
Understanding how to work with fractions is just part of the issue when using math in the
workplace. You also need to choose the right process to get the information you need from
the numbers you have. Today’s workplace often requires higher education and advanced
skills. Employees with good math skills have a definite advantage over their coworkers.
In order to handle everyday workplace problems and make the best decisions, it is useful to
understand fractions and how to add, subtract, multiply, and divide fractions.
8. Solving Simple Equations
An equation is a mathematical statement that two expressions are equal. An equation
always includes an equal sign (=). For example, 75 + 25 = 100 is a very simple equation. An
equation is an analysis tool that lets you map out the relationship between known
quantities (numbers) and an unknown quantity. Stating that relationship in an equation
enables you to establish the value of the unknown quantity.
To solve business problems involving mathematics, you need to translate them into the
language of mathematics.
Converting a business problem into an equation
There are four steps to convert a business problem into an equation as follows:
1. Determine what you need to find out (the variable)—The first step is to make sure
you are clear about what information you need. Do you need to know how many
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14 | Pa g e B a s i c B u s i n e s s M a t h
items to buy? How much resin to add to an epoxy mixture? How many hours it will
take to drive to a client's office?
2. Identify the information you know (the constants)—In order to calculate an answer,
you need information to relate it to. For example, if you're trying to calculate your
profit margin, you will need to know how much you bought goods for and how much
you sold them for. Those are the constants you need to calculate the variable.
3. Decide which operation(s) you can use to find the answer—Decide which operation
or operations you will apply to your known numbers (constants) to find the solution
(variable). For example, if you have six 50 gallon drums of waste material, you will
have to multiply to determine how much you have in total.
4. Write an equation to express the problem—Your last step is to write the problem as
an equation, so you can solve it. For example, if you have six 50 gallon drums of
waste material, and you will multiply to determine how much you have in total, the
equation would look like this: x = 6 x 50.
To solve problems efficiently and accurately, you need to understand the relationship
between the information you have and the information you need. An equation expresses
this relationship in mathematical terms.
An equation is like a scale, with two sides separated by the equal (=) sign. Rather than
balancing objects and weights, an equation balances constants, variables, and mathematical
symbols.
Terminology
There are some terms you will need to know to understand equations:
Variable—A variable is simply a letter that stands for an unknown number.
Exponent—An exponent is a number that tells how many times another number is
to be multiplied. For example, to show 8 to the third power—the exponent is 3,
indicating that 8 is used as a factor 3 times = 8 x 8 x 8.
Coefficient—A coefficient is the numeric part of a term that contains a variable. It's a
number that the variable is multiplied by. For example, if the variable x is multiplied
by 3 in your equation, that can be simply expressed as 3x. The coefficient is 3.
Constant—A constant is a number value that never changes. For example, the
constant in 2x + 10 is 10.
Operations—Operations include addition, subtraction, multiplication, and division,
which are used to combine numbers and variables in an equation.
Isolating the variable
An equation generally has a single unknown quantity, called a variable, represented by a
letter—x, y, or z, for example. The main idea in solving the equation is to isolate the
variable.
In other words, you want to get terms containing the variable on one side of the equation.
Move all other variables and constants to the other side of the equation. For example, your
equation might be x + 11 = 25. The key to solving this problem is to isolate x, like this: x = 25-
11.
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items to buy? How much resin to add to an epoxy mixture? How many hours it will
take to drive to a client's office?
2. Identify the information you know (the constants)—In order to calculate an answer,
you need information to relate it to. For example, if you're trying to calculate your
profit margin, you will need to know how much you bought goods for and how much
you sold them for. Those are the constants you need to calculate the variable.
3. Decide which operation(s) you can use to find the answer—Decide which operation
or operations you will apply to your known numbers (constants) to find the solution
(variable). For example, if you have six 50 gallon drums of waste material, you will
have to multiply to determine how much you have in total.
4. Write an equation to express the problem—Your last step is to write the problem as
an equation, so you can solve it. For example, if you have six 50 gallon drums of
waste material, and you will multiply to determine how much you have in total, the
equation would look like this: x = 6 x 50.
To solve problems efficiently and accurately, you need to understand the relationship
between the information you have and the information you need. An equation expresses
this relationship in mathematical terms.
An equation is like a scale, with two sides separated by the equal (=) sign. Rather than
balancing objects and weights, an equation balances constants, variables, and mathematical
symbols.
Terminology
There are some terms you will need to know to understand equations:
Variable—A variable is simply a letter that stands for an unknown number.
Exponent—An exponent is a number that tells how many times another number is
to be multiplied. For example, to show 8 to the third power—the exponent is 3,
indicating that 8 is used as a factor 3 times = 8 x 8 x 8.
Coefficient—A coefficient is the numeric part of a term that contains a variable. It's a
number that the variable is multiplied by. For example, if the variable x is multiplied
by 3 in your equation, that can be simply expressed as 3x. The coefficient is 3.
Constant—A constant is a number value that never changes. For example, the
constant in 2x + 10 is 10.
Operations—Operations include addition, subtraction, multiplication, and division,
which are used to combine numbers and variables in an equation.
Isolating the variable
An equation generally has a single unknown quantity, called a variable, represented by a
letter—x, y, or z, for example. The main idea in solving the equation is to isolate the
variable.
In other words, you want to get terms containing the variable on one side of the equation.
Move all other variables and constants to the other side of the equation. For example, your
equation might be x + 11 = 25. The key to solving this problem is to isolate x, like this: x = 25-
11.
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You can add, subtract, multiply, or divide both sides of an equation by any number, and they
will still be equal. In this case, you subtract 11. That gets rid of the 11 on the left (11 - 11 = 0)
and adds -11 on the right side.
Solving an equation
There are four steps to solve an equation as follows:
simplify both sides
move all terms with the variable to one side
simplify both sides again
if x has a coefficient, divide both sides by the coefficient.
Simplifying equations makes them easier to understand and work with. This may be done by
using the same denominator for terms or by adding, subtracting, multiplying, or dividing
both sides by the same number. Remember, what you do to one side of the equation, you
must do to the other.
Although equations might seem like a cumbersome way to solve problems, it is foolproof. If
you have accurate numbers, come up with the right equation, and follow the steps set forth
in this topic, you will get the correct answer.
9. The Order of Operations
The order of operations is very important when simplifying equations. The order of
operations defines the order in which you should perform each operation in an equation,
such as addition, subtraction, multiplication and division.
Understanding and following the order of operations is critical to simplifying and solving
equations. Without it, you would never know if you were interpreting an equation in the
right way to come up with the correct answer.
Consider how failure to use the order of operations can result in a wrong answer to a
problem. Take a simple equation, like 5 + 7 x 3. You could simplify this problem working left
to right. You'd add five plus seven to get 12, then multiply 12 by 3 to get a final answer of
36. Or you might decide to simplify the problem by multiplying 7 x 3 first to get 21. But if
you add 5 to that result, you get only 26. You can see why the order of operations is
important.
When equations have more than one operation, it's important to follow rules for the order
of operations. This is simply a convention that establishes which operations should be
performed in what order.
The order of operations is as follows:
1. parenthesis and brackets—Simplify the inside of parentheses and brackets first. To
solve equations correctly, it might help to remember the phrase: Please excuse my
15 | Pa g e B a s i c B u s i n e s s M a t h
You can add, subtract, multiply, or divide both sides of an equation by any number, and they
will still be equal. In this case, you subtract 11. That gets rid of the 11 on the left (11 - 11 = 0)
and adds -11 on the right side.
Solving an equation
There are four steps to solve an equation as follows:
simplify both sides
move all terms with the variable to one side
simplify both sides again
if x has a coefficient, divide both sides by the coefficient.
Simplifying equations makes them easier to understand and work with. This may be done by
using the same denominator for terms or by adding, subtracting, multiplying, or dividing
both sides by the same number. Remember, what you do to one side of the equation, you
must do to the other.
Although equations might seem like a cumbersome way to solve problems, it is foolproof. If
you have accurate numbers, come up with the right equation, and follow the steps set forth
in this topic, you will get the correct answer.
9. The Order of Operations
The order of operations is very important when simplifying equations. The order of
operations defines the order in which you should perform each operation in an equation,
such as addition, subtraction, multiplication and division.
Understanding and following the order of operations is critical to simplifying and solving
equations. Without it, you would never know if you were interpreting an equation in the
right way to come up with the correct answer.
Consider how failure to use the order of operations can result in a wrong answer to a
problem. Take a simple equation, like 5 + 7 x 3. You could simplify this problem working left
to right. You'd add five plus seven to get 12, then multiply 12 by 3 to get a final answer of
36. Or you might decide to simplify the problem by multiplying 7 x 3 first to get 21. But if
you add 5 to that result, you get only 26. You can see why the order of operations is
important.
When equations have more than one operation, it's important to follow rules for the order
of operations. This is simply a convention that establishes which operations should be
performed in what order.
The order of operations is as follows:
1. parenthesis and brackets—Simplify the inside of parentheses and brackets first. To
solve equations correctly, it might help to remember the phrase: Please excuse my
http://SlideShare.net/OxfordCambridge
16 | Pa g e B a s i c B u s i n e s s M a t h
dear Aunt Sally (P E M D A S): Parentheses and brackets, Exponents, Multiplication
and Division, Addition and Subtraction. To solve an equation, start inside the
parentheses and brackets. That must be done before you deal with any exponent of
the parenthesis or remove the parenthesis.
In the equation 10 = y + 54 - 8, you can't do the addition, since the variable y is an
unknown number. But you can subtract 8 from 54, so the equation becomes 10 = y +
46. Now, just subtract 46 from both sides to solve the equation: -36 = y.
2. Exponents—Simplify the exponent of a number before you multiply, divide, add, or
subtract it. An exponent is written above the number to be exponentiated to
indicate that the number will be multiplied by itself a number of times. For example,
the number 5 exponent 3 means 5 x 5 x 5.
3. Multiplication and division—Multiply and divide in the order those operations
appear from left to right. Since there are no exponents in the equation 10 = y + 9 x 6
- 8, the next step is to multiply and divide in the order that they appear—from left to
right. In this equation, there is no division, only multiplication, so your next step is to
multiply 9 and 6. The result is 54. So the equation becomes 10 = y + 54 - 8.
4. Addition and subtraction—Simplify addition and subtraction in the order they
appear from left to right.
In the equation 10 = y + 54 - 8, you can't do the addition, since the variable y is an
unknown number. But you can subtract 8 from 54, so the equation becomes 10 = y +
46. Now, just subtract 46 from both sides to solve the equation: -36 = y.
To solve equations correctly, remember the phrase, Please excuse my dear Aunt Sally. This
means that you should do what is possible within parentheses first, then exponents, then
multiplication and division (from left to right), and then addition and subtraction (from left
to right).
10. Applying the Rules of Order
Make sure you solve the operations within an equation--parentheses, exponentiation, multiplication, division,
addition, subtraction--in the correct order, so you will get the correct answer.
When solving mathematical equations, there can be only one correct answer.
Mathematicians have devised a standard order of operations for calculations involving more
than one arithmetic operation. The order of operations, from first to last, is given below.
Operations with the same precedence (addition and subtraction, or multiplication and
division) are performed from left to right.
1. Perform any calculations inside parentheses. Expressions within nested
parentheses are evaluated from inner to outer.
2. Apply all exponents. (Exponents are a way to show a number multiplied by itself a
certain number of times. Example: 5
2
= 5 x 5 = 25. In this equation, 5 is called the
base, and 2 is called the exponent.
16 | Pa g e B a s i c B u s i n e s s M a t h
dear Aunt Sally (P E M D A S): Parentheses and brackets, Exponents, Multiplication
and Division, Addition and Subtraction. To solve an equation, start inside the
parentheses and brackets. That must be done before you deal with any exponent of
the parenthesis or remove the parenthesis.
In the equation 10 = y + 54 - 8, you can't do the addition, since the variable y is an
unknown number. But you can subtract 8 from 54, so the equation becomes 10 = y +
46. Now, just subtract 46 from both sides to solve the equation: -36 = y.
2. Exponents—Simplify the exponent of a number before you multiply, divide, add, or
subtract it. An exponent is written above the number to be exponentiated to
indicate that the number will be multiplied by itself a number of times. For example,
the number 5 exponent 3 means 5 x 5 x 5.
3. Multiplication and division—Multiply and divide in the order those operations
appear from left to right. Since there are no exponents in the equation 10 = y + 9 x 6
- 8, the next step is to multiply and divide in the order that they appear—from left to
right. In this equation, there is no division, only multiplication, so your next step is to
multiply 9 and 6. The result is 54. So the equation becomes 10 = y + 54 - 8.
4. Addition and subtraction—Simplify addition and subtraction in the order they
appear from left to right.
In the equation 10 = y + 54 - 8, you can't do the addition, since the variable y is an
unknown number. But you can subtract 8 from 54, so the equation becomes 10 = y +
46. Now, just subtract 46 from both sides to solve the equation: -36 = y.
To solve equations correctly, remember the phrase, Please excuse my dear Aunt Sally. This
means that you should do what is possible within parentheses first, then exponents, then
multiplication and division (from left to right), and then addition and subtraction (from left
to right).
10. Applying the Rules of Order
Make sure you solve the operations within an equation--parentheses, exponentiation, multiplication, division,
addition, subtraction--in the correct order, so you will get the correct answer.
When solving mathematical equations, there can be only one correct answer.
Mathematicians have devised a standard order of operations for calculations involving more
than one arithmetic operation. The order of operations, from first to last, is given below.
Operations with the same precedence (addition and subtraction, or multiplication and
division) are performed from left to right.
1. Perform any calculations inside parentheses. Expressions within nested
parentheses are evaluated from inner to outer.
2. Apply all exponents. (Exponents are a way to show a number multiplied by itself a
certain number of times. Example: 5
2
= 5 x 5 = 25. In this equation, 5 is called the
base, and 2 is called the exponent.
http://SlideShare.net/OxfordCambridge
17 | Pa g e B a s i c B u s i n e s s M a t h
3. Perform all multiplications and divisions, working from left to right.
4. Perform all additions and subtractions, working from left to right.
For example, consider the following equation:
4 + 8 ÷ 2 x 7 - (9÷ 3) = x
First perform the calculation with the parentheses (9 ÷ 3 = 3). So now the equation is:
4 + 8 ÷ 2 x 7 - 3 = x
There are no exponents, so the next step is to do the division and multiplication. Working
from left to right, 8 ÷ 2 = 4; 4 x 7 = 28. So now the equation is:
4 + 28 - 3 = x
Now, perform all additions and subtractions, working from left to right: 4 + 28 = 32; 32 - 3 =
29. So the solution is:
x =29
Hint: To solve equations correctly, it might help to remember the phrase: "Please excuse my
dear Aunt Sally" (P E M D A S): Parentheses, Exponents, Multiplication and Division, Addition
and Subtraction.
17 | Pa g e B a s i c B u s i n e s s M a t h
3. Perform all multiplications and divisions, working from left to right.
4. Perform all additions and subtractions, working from left to right.
For example, consider the following equation:
4 + 8 ÷ 2 x 7 - (9÷ 3) = x
First perform the calculation with the parentheses (9 ÷ 3 = 3). So now the equation is:
4 + 8 ÷ 2 x 7 - 3 = x
There are no exponents, so the next step is to do the division and multiplication. Working
from left to right, 8 ÷ 2 = 4; 4 x 7 = 28. So now the equation is:
4 + 28 - 3 = x
Now, perform all additions and subtractions, working from left to right: 4 + 28 = 32; 32 - 3 =
29. So the solution is:
x =29
Hint: To solve equations correctly, it might help to remember the phrase: "Please excuse my
dear Aunt Sally" (P E M D A S): Parentheses, Exponents, Multiplication and Division, Addition
and Subtraction.
http://SlideShare.net/OxfordCambridge
18 | Pa g e B a s i c B u s i n e s s M a t h
11. Place Values
Help yourself to understand and identify place values in numbers that you want to round. To use this tool, write
the number you want to round in the box in the left column. Then, starting with the last digit on the right, enter
each digit of your number into the boxes from right to left under the place values. For example, the number
1,345,076 is entered on the left. Then the 6 is entered in the ones column, the 7 is entered in the tens column,
the 0 is entered in the hundreds column, the 5 is entered in the thousands column, and so on. Each column
corresponds to that digit's place value.
Numbers Millions Hundred
Thousands
Ten
Thousands
Thousands Hundreds Tens Ones
Example 1,345,076 1 3 4 5 0 7 6
Example 28,670 0 0 2 8 6 7 0
Note that the place values vary by a factor of 10. The column to the left of a place value is
ten times that place value, the column to the right of a place value is one-tenth of that place
value. For example, the 7 in the tens column in the first example represents seven tens, or
70. If it was in next column to the left (the hundreds place), it would represent seven
hundreds, or 700. If it was in next column to the right (the ones place), it would represent
seven ones, or 7.
18 | Pa g e B a s i c B u s i n e s s M a t h
11. Place Values
Help yourself to understand and identify place values in numbers that you want to round. To use this tool, write
the number you want to round in the box in the left column. Then, starting with the last digit on the right, enter
each digit of your number into the boxes from right to left under the place values. For example, the number
1,345,076 is entered on the left. Then the 6 is entered in the ones column, the 7 is entered in the tens column,
the 0 is entered in the hundreds column, the 5 is entered in the thousands column, and so on. Each column
corresponds to that digit's place value.
Numbers Millions Hundred
Thousands
Ten
Thousands
Thousands Hundreds Tens Ones
Example 1,345,076 1 3 4 5 0 7 6
Example 28,670 0 0 2 8 6 7 0
Note that the place values vary by a factor of 10. The column to the left of a place value is
ten times that place value, the column to the right of a place value is one-tenth of that place
value. For example, the 7 in the tens column in the first example represents seven tens, or
70. If it was in next column to the left (the hundreds place), it would represent seven
hundreds, or 700. If it was in next column to the right (the ones place), it would represent
seven ones, or 7.
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19 | Pa g e B a s i c B u s i n e s s M a t h
12. Project Cost Estimate
Use this Follow-on Activity to use your new math skills to understand the costs involved in the work you are
currently doing for your employer.
It's always useful to try and determine whether projects are going to be accomplished
within the allocated budget. Consider a project or job that you are currently working on, and
see if you can create an accurate project budget. Focus on one project or individual task
First, list the information below. Write a description of the work and your numbers for items
1, 2, 4, and 6 in the column labeled "Your estimate." If possible, work with mentors or
coworkers, and have them comment on your figures. This will help you come up with more
accurate estimates of actual costs and requirements.
Item
#
Facts
Your estimate
Co-worker's or
mentor's
comments
Description of Work:
1 Number of days required for
completion of the job or project
2 x Hours required (estimated hours per
day)
3 = Total hours
4 x Labor cost per hour (based upon your
hourly rate)
5
= Total labor cost
6 + Materials Cost (including parts and
raw materials required to complete the
project)
7 = Estimated cost for the project
Now take the number of days (Item #1), and multiply that figure by the hours per day (Item
#2). This gives you the total hours that will be spent on this project (Item #3). Multiply the
total hours by the labor cost per hour (Item #4). This gives you the total labor cost of your
project (Item #5). Now, add together the costs of all the materials required to complete this
19 | Pa g e B a s i c B u s i n e s s M a t h
12. Project Cost Estimate
Use this Follow-on Activity to use your new math skills to understand the costs involved in the work you are
currently doing for your employer.
It's always useful to try and determine whether projects are going to be accomplished
within the allocated budget. Consider a project or job that you are currently working on, and
see if you can create an accurate project budget. Focus on one project or individual task
First, list the information below. Write a description of the work and your numbers for items
1, 2, 4, and 6 in the column labeled "Your estimate." If possible, work with mentors or
coworkers, and have them comment on your figures. This will help you come up with more
accurate estimates of actual costs and requirements.
Item
#
Facts
Your estimate
Co-worker's or
mentor's
comments
Description of Work:
1 Number of days required for
completion of the job or project
2 x Hours required (estimated hours per
day)
3 = Total hours
4 x Labor cost per hour (based upon your
hourly rate)
5
= Total labor cost
6 + Materials Cost (including parts and
raw materials required to complete the
project)
7 = Estimated cost for the project
Now take the number of days (Item #1), and multiply that figure by the hours per day (Item
#2). This gives you the total hours that will be spent on this project (Item #3). Multiply the
total hours by the labor cost per hour (Item #4). This gives you the total labor cost of your
project (Item #5). Now, add together the costs of all the materials required to complete this
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20 | Pa g e B a s i c B u s i n e s s M a t h
work (Item #6). Add that to the total labor cost (Item #5), and you get an estimated cost for
the entire project (Item #7). Note that this does not take into account the costs of
equipment and physical facilities.
13. Business Finance
Use this Follow-on Activity to practice your new math skills. This will help you understand some of the financial
issues that are looked at by managers and investors.
Businesses are always concerned about their financial health--and about how their financial
status looks on paper. In this exercise, you will use information about your company's
financial position to calculate some key financial ratios. See if you can get a copy of your
company's annual report or financial statement. Here are the figures you need. Copy them
from your source material into the boxes below next to the appropriate description.
Item #
Description
Number
1 Cash
2 Accounts receivable
3 Stocks & Bonds
held for investment
4 Accounts payable
5 Other current
liabilities
6 Net income
7 Sales
8 Total debt
9 Total assets
Once you've gotten as many of these financial figures as you can (estimates are OK, too, for
practice purposes), plug them into the following equations. Below each equation is a brief
explanation of what the figure means.
Cash (Item #1) + Accounts Receivable (Item #2) + Stocks & Bonds (Item #3) / Current
Liabilities (Accounts payable (Item #4) + Other current liabilities (Item #5) = the "Quick Ratio"
20 | Pa g e B a s i c B u s i n e s s M a t h
work (Item #6). Add that to the total labor cost (Item #5), and you get an estimated cost for
the entire project (Item #7). Note that this does not take into account the costs of
equipment and physical facilities.
13. Business Finance
Use this Follow-on Activity to practice your new math skills. This will help you understand some of the financial
issues that are looked at by managers and investors.
Businesses are always concerned about their financial health--and about how their financial
status looks on paper. In this exercise, you will use information about your company's
financial position to calculate some key financial ratios. See if you can get a copy of your
company's annual report or financial statement. Here are the figures you need. Copy them
from your source material into the boxes below next to the appropriate description.
Item #
Description
Number
1 Cash
2 Accounts receivable
3 Stocks & Bonds
held for investment
4 Accounts payable
5 Other current
liabilities
6 Net income
7 Sales
8 Total debt
9 Total assets
Once you've gotten as many of these financial figures as you can (estimates are OK, too, for
practice purposes), plug them into the following equations. Below each equation is a brief
explanation of what the figure means.
Cash (Item #1) + Accounts Receivable (Item #2) + Stocks & Bonds (Item #3) / Current
Liabilities (Accounts payable (Item #4) + Other current liabilities (Item #5) = the "Quick Ratio"
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21 | Pa g e B a s i c B u s i n e s s M a t h
This number is of particular interest to lenders are interested in this ratio because it
indicates assets that can be quickly converted into cash to meet short-term liabilities.
Net Income (Item #6) / Sales (Item #7) = the "Profit Margin"
This number basically shows your company's bottom line--how much of each sales dollar is
profit.
Total debt (Item #8) / Total assets (Item #9) = the "Debt Ratio"
This number indicates how much of a company’s assets are financed through loans.
This kind of financial information provides managers with an objective basis for comparing
the performance of your company with other businesses in the industry.
21 | Pa g e B a s i c B u s i n e s s M a t h
This number is of particular interest to lenders are interested in this ratio because it
indicates assets that can be quickly converted into cash to meet short-term liabilities.
Net Income (Item #6) / Sales (Item #7) = the "Profit Margin"
This number basically shows your company's bottom line--how much of each sales dollar is
profit.
Total debt (Item #8) / Total assets (Item #9) = the "Debt Ratio"
This number indicates how much of a company’s assets are financed through loans.
This kind of financial information provides managers with an objective basis for comparing
the performance of your company with other businesses in the industry.
http://SlideShare.net/OxfordCambridge
22 | Pa g e B a s i c B u s i n e s s M a t h
14. Glossary
amortize
To amortize means to pay off a loan through a series of equal payments. Generally, the initial payments include
more interest than principal and toward the end of the amortization period; the payment includes more principal
than interest.
coefficient
A number or letter put before a letter or quantity, known or unknown, to show how many times the latter is to be
taken; as, 6x; bx; here 6 and b are coefficients of x.
constant
A constant is a number value that never changes. For example, the constant in 2x + 10 is 10.
denominator fraction
A denominator fraction is the fraction in a division problem that is being divided into another fraction.
dividend
A dividend is a number to be divided by another number.
divisor
A divisor is a number that another is being divided by.
exponent
An exponent is a number that tells how many times another number is to be multiplied. For example, 8 to the third
power indicates that 8 is used as a factor 3 times. So 8 to the third power equals 8 x 8 x 8.
operations
Operations include addition, subtraction, multiplication, and division, which are used to combine numbers and
variables in an equation.
variable
A variable is simply a letter that stands for an unknown number.
22 | Pa g e B a s i c B u s i n e s s M a t h
14. Glossary
amortize
To amortize means to pay off a loan through a series of equal payments. Generally, the initial payments include
more interest than principal and toward the end of the amortization period; the payment includes more principal
than interest.
coefficient
A number or letter put before a letter or quantity, known or unknown, to show how many times the latter is to be
taken; as, 6x; bx; here 6 and b are coefficients of x.
constant
A constant is a number value that never changes. For example, the constant in 2x + 10 is 10.
denominator fraction
A denominator fraction is the fraction in a division problem that is being divided into another fraction.
dividend
A dividend is a number to be divided by another number.
divisor
A divisor is a number that another is being divided by.
exponent
An exponent is a number that tells how many times another number is to be multiplied. For example, 8 to the third
power indicates that 8 is used as a factor 3 times. So 8 to the third power equals 8 x 8 x 8.
operations
Operations include addition, subtraction, multiplication, and division, which are used to combine numbers and
variables in an equation.
variable
A variable is simply a letter that stands for an unknown number.
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23 | Pa g e B a s i c B u s i n e s s M a t h
23 | Pa g e B a s i c B u s i n e s s M a t h
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