Basic Business Math - Study Notes

Basic Business Math

Study Notes

Entry Level

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2 | P a 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. Understanding Decimals ................................................................................................. 21

15. Adding and Subtracting Decimals .................................................................................... 22

16. Multiplying Decimals ....................................................................................................... 23

17. Dividing Decimals ............................................................................................................ 24

18. Understanding Percentages ............................................................................................ 25

19. Solving for a Number (Portion) ....................................................................................... 26

20. Solving for a Percent (Rate) ............................................................................................. 27

21. Solving for the Whole (Base) ........................................................................................... 28

22. Place Values .................................................................................................................... 29

23. Converting Numbers ....................................................................................................... 29

24. Elements of a Percentage Problem ................................................................................. 31

25. Financial Formulas .......................................................................................................... 33

26. Profit Margins Expressed as Percentages ........................................................................ 33

27. Evaluate Telephone Costs ............................................................................................... 34

28. Understanding Ratios ...................................................................................................... 35

29. Understanding Proportions ............................................................................................. 37

30. Comparing Different Kinds of Numbers .......................................................................... 38

31. Simple Averages .............................................................................................................. 39




32. Productivity Ratios .......................................................................................................... 44

33. Proportions and Ratios .................................................................................................... 46

34. Calculating Simple, Weighted, and Moving Averages ..................................................... 47

35. Productivity Ratios .......................................................................................................... 49

36. Moving Averages ............................................................................................................. 50

37. Glossary ........................................................................................................................... 51






1. Importance of Basic Math in Business

a. 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.

b. 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.

c. 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




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.




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.




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.

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).

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 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 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:




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




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




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.




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




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: 52 = 5 x 5 = 25. In this equation, 5 is called the base, and 2 is called the exponent.




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.




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.




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




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"




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.

14. Understanding Decimals

A decimal is basically a fraction with a denominator that is a power of ten. The period, or decimal point (.), marks the place where the whole numbers end, and the decimal fractions begin.

In other words, digits to the left of a decimal point indicate a number greater than or equal to one, digits to the right of a decimal point indicate a fraction (a value less than one).

Decimals are a way of writing fractions without cumbersome numerators and denominators. For example, 3/10 is written as the decimal 0.3. The decimal point between the 0 and the 3 indicates that this number is a decimal fraction. Any fraction can be expressed as a decimal simply by dividing the numerator by the denominator.

There are three parts to a decimal number:

A digit or digits representing the whole number—Like mixed numbers, decimals include both whole numbers (to the left of the decimal point) and fractions (to the right of the decimal point). For example, the decimal 1.1, includes the whole number one and the fraction one-tenth.

A decimal point—Decimal numbers have digits before and after the decimal point—even if that digit is just zero. Whole numbers have no fractional part, so that part is expressed as point zero. For example, the whole number eight would be 8.0, expressed as a decimal.

A digit or digits representing the fraction—Some decimal numbers have only fractions, and no whole numbers. The whole number part is expressed as 0. For example, the fraction five-tenths would be written as zero point five (0.5).

Place value Placement of the decimal point in a decimal number is based on the concept of place value as follows:




Moving from right to left in a decimal number, each place value increases by a power of ten. In other words, if the first place value is ones, the next place value to the left is tens (10 x ones), the next to the left is hundreds (10 x tens), and so on.

The first place value to the left of the decimal point represents ones. Whatever digit is in that place (0, 1, 2, 3, 4, 5, 6, 7, 8, or 9) tells you how many ones are in the number.

The first place value to the right of the decimal point represents tenths. So whatever digit is in that place tells you how many tenths there are in the number.

Moving from left to right in a decimal number, each place value decreases by a power of one-tenth. So if the first place value is tenths, the next place value to the right is hundredths (1/10 x tenths), the next to the right is thousandths (1/10 x hundredths), and so on.

When working with decimal numbers, think of zeroes as placeholders. For example, in the number 100, there are no tens, and no ones. But those places are filled with zeroes, so you know that the 1 is in the hundreds place.

In decimal numbers, the value of each digit depends on its place, or position, in the number. Values to the left of the decimal point are whole numbers. Values to the right of the decimal point are fractions.

15. Adding and Subtracting Decimals

Adding and subtracting decimals is basically the same as adding and subtracting whole numbers. You work from right to left, adding or subtracting numbers of the same place value.

The only tricky part is placement of the decimal. You must be sure to line up the numbers being added or subtracted, so that all the decimal points are in a vertical line. Then add or subtract each column of digits, starting on the right and working left. You will need to borrow and carry, the same as you do when adding and subtracting whole numbers.

The steps to add decimal numbers are explained below:

Align the decimal points—The first step is to align the decimal points vertically. Add trailing zeroes—Sometimes zeros have to be added, so all the numbers being

added have the same number of digits to the right of the decimal point. Add each column—Remember to carry numbers if it is necessary. Place decimal point—The last step is to place the decimal point—so there are the

same number of digits to the right of each decimal number.

The steps to subtract decimal numbers are explained below:

Align the decimal points—The first step is to align the decimal points vertically. Add trailing zeroes—Sometimes zeros have to be added to make sure all the

numbers line up.




Perform the subtraction—Now subtract each digit in order from right to left. Remember that it may be necessary to borrow from the next place value in order to subtract the digits.

Place decimal point—The last step is to place the decimal point—so there are the same number of digits to the right of each decimal number.

The rule in addition and subtraction is that the decimal points must always be lined up. Sometimes you might have to add zeros at the beginning or end of a decimal number to get them to line up correctly. That doesn't change the value of the number. Never put a 0 between the decimal point and any other digit, however. That would change the value!

Once you have mastered the addition and subtraction of decimals, you will be able to handle many new job tasks, and make existing tasks easier. Just remember to put the numbers in a vertical arrangement with the decimal points lining up, add zeroes as necessary, and add or subtract as indicated.

16. Multiplying Decimals

Multiplying decimals can be an important skill. If you can multiply whole numbers, you can multiply decimals.

The trickiest part of multiplying decimals may be counting the total number of decimal places in the problem, so you can put the decimal point in the right place in the answer. Once you have a good understanding of where to place the decimal point in your product, multiplying decimals is a straightforward process.

The steps in the process are as follows. Note that the answer in a multiplication problem is called the product.

Delete trailing zeroes—To simplify the multiplication process, you should remove trailing zeroes from the decimal fractions (the numbers to the right of the decimal points).

Multiply the numbers—Then multiply, as if the operation involved whole numbers. If the last digit to the right of the decimal point is zero, that digit is left out of the equation.

Count the total number of decimal places in both numbers—After multiplying, the next step is to look at the two numbers being multiplied. Count how many digits are to the right of the decimal points.

Insert the decimal point in the product—The last step is to place the decimal point. When you multiply decimal numbers, temporarily disregard the decimal points and multiply the numbers like multiplying any whole numbers. Then just count up the decimal places and move the decimal point to its proper location.

Putting a decimal in the wrong place in a multiplication problem makes a big difference. Moving a decimal one place to the left reduces the product by one-tenth. Moving it one place to the right multiplies the answer by ten.




Multiplying decimals is used on the job to calculate money values, dimensions, volume, and trends. Next time you use your calculator to figure costs or expenses, think about the math, and how the process works.

17. Dividing Decimals

The process of dividing decimals is very similar to the process of dividing whole numbers. It can be a useful skill in many situations.

Dividing a decimal number by a whole number If you can divide whole numbers, you can divide decimals. The steps in the process are as follows:

Place the divisor before the division bracket and place the dividend under it. Then proceed with the division, as usual. Place the decimal point in the answer (quotient) directly over the decimal point in

the dividend.

Sometimes, when you divide numbers (1.0 divided by 3.0, for example), the quotients have infinite decimal places. That means you have to decide how precise your answer needs to be, and use rounded numbers.

Rounding decimal numbers To round a decimal number, find the digit occupying the place value you need (the rounding digit). Then look at the digit to the right of that digit. If the digit to the right of the rounding digit is less than 5, leave the rounding digit unchanged. If it is five or more, then add one to the rounding digit. Remove all digits following the rounding digit.

Rounding decimal numbers really isn't difficult if you just remember that the key to whether you round up or down is the number to the immediate right of the rounding digit—five or more, round up; less than five, round down.

Dividing a decimal number by another decimal number Dividing a decimal number by another decimal number is basically the same process as dividing a decimal number by a whole number, except there is one more step. You must convert the divisor (the decimal you're dividing into the other decimal) to a whole number.

Convert the divisor to a whole number by moving the decimal point all the way to the right. Then, just move the decimal point the same number of places to the right in the dividend (the number you are dividing the divisor into).

Division is often required in business situations where you know two numbers (consumption per hour versus total consumption, unit size versus total area, price versus quantity, for example) and have to calculate a third. In fact, division is a critical skill for business analysis purposes.




Being able to divide decimal numbers is an important skill in dealing with business finance, productivity, and other areas. When you can do the math, you can perform analyses and solve problems that improve performance and profits—and make yourself a more valuable member of the workforce.

18. Understanding Percentages

Percent simply means parts per hundred. Anything that can have a quantity associated with it can be defined as a percentage. If you divide a whole quantity into 100 equal units, one of those units equals one percent.

Since 100% is a multiple of ten, percentages bear a close relationship to decimal numbers. One hundred percent is one whole quantity, so it would be written as the decimal number one: 1.0. Percentages can easily be converted into decimal numbers simply by moving the decimal point two places to the left. So, for example, 1% = 0.01, 15% = 0.15, 90% = 0.90. You need to know the following to solve many problems involving percentages:

One percent is 1/100 of the whole amount (100%). Ten percent is ten times that, or 1/10 of the whole amount. In decimal terms, 10% is 0.1. To find 10% of any decimal number, just move the decimal point one place to the left. For example, 10% of 25 is 2.5.

Seventy-five percent represents 75 of the 100 units that the whole was divided into—or 3/4. Again, divide 4 (the denominator) into 3 (the numerator), or move the decimal point two places in the percentage, to find the decimal fraction 0.75.

Percentages are parts of the whole. For example, 25% is the same as 25/100. The fraction can be converted to a decimal (0.25) by dividing the numerator by the denominator. Moving the decimal point two places to the left converts the decimal to a percentage (25%).

Percentages can even be more than 100%. It just means there are more units than were designated as one whole. For example, if your whole is a dozen eggs, and you have 15 eggs, you have 125% of the whole, or 1.25 (15 divided by 12).

Whatever your total quantity is, whether 333.3 cabbages or 10,000 paper clips, they get divided by 100, and each 1/100 represents 1% of the total. To divide a number by 100, just move the decimal point two places to the left.

The quantity 1% does not have to correspond to a single unit—or even to whole units. For example, if a company sold 20 truck tires, 1%, or one hundredth, of that number is 0.2, or two-tenths of a truck tire. Therefore, each complete tire is 5% of the total.

Using percentages allows comparisons between different parts of a whole amount, by expressing those parts relative to the defined whole.




19. Solving for a Number (Portion)

Generally, in percentage problems, two values are given, and the third must be calculated from those given values.

These three key elements in a typical percent problem are:

Base—the whole quantity or value Rate—a percentage, decimal, or fraction Portion—the result when the base is multiplied by the rate

The mathematical process, or formula, to determine Portion is: Portion = Rate x Base.

Solving for a portion In a business environment, there are three steps to solve for a portion (not a percent and not a whole):

Identify the key elements Set up the formula Calculate portion

Keep in mind that Rate is usually expressed as a percentage, or as a fraction or decimal; the Base is usually preceded by the word "of" when you have to solve for a portion, and the Portion is a part that is neither a percentage nor the whole quantity or value.

When inserting Rate into the formula to determine Portion (Portion = Rate x Base), you need to convert percentages into decimals before multiplying. To do this, just move the decimal point two places to the left, adding zeroes, if necessary.

Pie Chart A simple pie chart is often used to help remember what operation (multiplication or division) to use when solving for the different elements (Portion, Rate, or Base) in a percentage problem.

The top half of the chart represents the Portion. The bottom half is divided into two parts, representing Base and Rate. The line separating Portion from Base and Rate means divide. The vertical line separating Base from Rate means multiply.

To use the chart, just cover the element you're solving for. The two elements, and the dividing line that is left show what you have to do to solve for that element. For example:

If you are solving for Portion, you are left with Base x Rate. So, Portion = Base x Rate. If you are solving for Base, you are left with Portion/Rate. So, Base = Portion/Rate.

Keep in mind that Rate is usually expressed as a percentage, or as a fraction or decimal; the Base is usually preceded by the word "of" when you have to solve for a Portion, and the Portion is a part that is neither a percentage nor the whole quantity or value. If the Rate is less than 100%, the Portion is always less than the Base. The Rate, if expressed as a




percentage or a fraction, must be converted to a decimal number before you can perform the multiplication.

The key to solving workplace problems involving percentages is to identify the three elements. Then you can plug the numbers into the formula Portion = Rate x Base, and perform calculations to determine the missing element.

20. Solving for a Percent (Rate)

Percentages are useful because they make it very easy to compare things. For example, profits might be expressed as a percentage of revenues, marketing costs as a percentage of sales, or statistics in terms of a percentage of change.

Pie Chart A simple pie chart is often used to help remember what operation (multiplication or division) to use when solving for the different elements (Portion, Rate, or Base) in a percentage problem.

The top half of the chart represents the Portion. The bottom half is divided into two parts, representing Base and Rate. The line separating Portion from Base and Rate means divide. The vertical line separating Base from Rate means multiply.

When you're trying to determine a percentage (Rate), the pie chart tells you that the formula is Rate = Portion divided by Base. Cover up the element you're solving for, and look at the relationship between the remaining two elements.

Remember, Rate is the percentage you're solving for. Portion represents a part of the whole, and Base is the whole object or number that you're taking a percentage of.

Identifying Portion can be somewhat tricky when you are determining Rate of increase or decrease. You need to identify the original and the new amounts of the Portion, and find the difference between them.

Solving for Rate In solving for Rate, the process is to compare Portion to Base, and express the results using the % symbol. A decimal is easily changed to a percent by moving the decimal point two places to the right and adding a percentage sign.

Businesses frequently use percentages to clearly communicate comparative data in a wide range of areas. If you don't know how percentages work, you may have difficulty understanding important information. When solving for a percentage (Rate) in your business environment, there are three steps you should use:

1. Identify the key elements. 2. Set up the formula. 3. Calculate the rate.




Once you know you're looking for Rate, it's easy to use the pie chart to come up with the formula: Rate = Portion/Base. Then it's just a matter of plugging in the appropriate numbers, and doing the math operation.

People are data-driven these days, and figuring out percentages is a necessary skill for many employees. Percentages are very helpful in the presentation of business data. Just remember that they are considered in relation to the whole (Base) and are always calculated on the basis of 100.

21. Solving for the Whole (Base)

When the Portion (part of the whole quantity) is known, and you also know the percentage (Rate) of the whole the portion represents, it is possible to determine the Base (the whole quantity).

The Portion over Base x Rate pie chart Solving for Base is a variation of the formula to solve any percentage problem. Keeping the pie chart in mind will help you remember all the appropriate formulas. The pie chart is divided into three parts:

The top half is labeled Portion. The line separating the top half of the pie from the bottom indicates division. The lower half of the pie is divided into Base and Rate. The line separating these two

elements indicates multiplication.

Use the Portion over Base x Rate pie chart to set up the formula. Covering Base, you are left with Portion over Rate. So the formula to solve for Base is Portion divided by Rate.

When solving for the whole (Base) in your business environment, there are three steps you should follow:

Identify the key elements—As in other percentage problems, identifying the key elements is the first step. For example, if it is known that 100 is 20% of the whole quantity, 100 is the Portion (a part that is not the whole quantity and is not a percentage) and 20% is the Rate (a fraction of the Base expressed as a percentage).

Set up the formula—Use the Portion over Base x Rate pie chart to set up the formula. Covering Base, you are left with Portion over Rate. So the formula to solve for Base is Portion divided by Rate.

Perform the calculation—Given the formula, Base = Portion/Rate, you know that Base = 100/20%. To make the math easier, change the percent to a decimal by moving the decimal point two places to the left. So 20% becomes .20. So now the problem is Base = 100/.20 = 500.

Performing these calculations and recognizing the significance of these numbers takes time and patience to master, but the results will surprise you—improved understanding and hence, improved job performance.




22. Place Values

help yourself 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 thousandths column, the 7 is entered in the hundredths column, the 0 is entered in the tenths column, the 5 is entered in the ones column, and so on. Each column corresponds to that digit's place value.

Numbers Thousandths Hundreds Tens Ones Decimal Point

Tenths Hundredths Thousandths

1,345.076 1 3 4 5 . 0 7 6

28.690 0 0 2 8 . 6 9 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 5 in the ones column in the first example represents five ones, or 5. If it was in the next column to the left (the tens place), it would represent five tens, or 50. If it was in the next column to the right (the tenths place), it would represent five tenths, or 0.5.

23. Converting Numbers

help yourself convert between fractions, percentages, and decimal numbers.




Sometimes to solve a problem, you need to convert all of the values into the same type of expression. Or you might need the solutions expressed as decimals, fractions, or percentages.

a. Converting Decimals to Percentages

Turning decimals into percentages is relatively simple. All you need to do is multiply the number by 100 to find the percentage, or move the decimal point two places to the right.

For example:

0.948 = 94.8% 0.3 = 30% (Add a zero, so you can move the decimal point two places.) 4.75 = 475% (Note that a whole number represents 100%.)

b. Converting Decimals to Fractions

The digits to the right of the decimal point in a decimal number are already a fraction, since those place values represent tenths, hundredths, thousandths, and so on. All you really need to do is cancel the fraction down to simplify it:

For example:

12 . 5 is 12 5/10, which can be canceled to 12 1/2 3.125 is 3 125/1000, which can be canceled 3 1/8

c. Converting Percentages to Decimals

Percentages are always expressed in hundredths. So you can write them in terms of place value. For example, 75% is 75/100, or 0.75. As a practical matter, all you have to remember is that the decimal point gets moved two places to the left to convert a percentage to a decimal number. For example:

25% = 0.25 5% = 0.05 (Add a zero, so you can move the decimal point two places.) 178% = 1.78 (Note that when a number is above 100%, there will be whole numbers in the

decimal number.)

a. Converting Percentages to Fractions

Percentages are fractions of a whole, just like decimal numbers. You can easily make a percentage into a fraction just by putting the number over 100. For example 50% is just 50 out of 100, or 50/100. Then the fraction can be simplified by canceling.

For example:

50% = 50/100 = 1/2 2.5% = 25/1000 (Note that the numerator and denominator were both multiplied by 10 to

get rid of the decimal point in the numerator.) = 1/40




435% = 425/100 = 17/4 = 4 1/4 (Note that when a percentage is above 100, the solution will be an improper fraction or mixed number.)

b. Converting Fractions to Decimals

Fractions can be converted into decimal numbers just by dividing the numerator by the denominator. Using a calculator makes this easy!

For example:

1/2 = 1 divided by 2 = 0.5 3/4 = 3 divided by 4 = 0.75 3 7/8 = 3.875 (Ignore the whole number and divide the numerator by the denominator: 7

divided by 8 = 0.875, + 3 = 3.875.)

Note that some numerators cannot be evenly divided by the denominator; 2/3, for example. If you divide three into two, you get 0.0.66666666666 … going on forever. These "recurring decimals" are usually rounded off to the nearest tenth (0.7), hundredth (0.67), or thousandth (0.667).

c. Converting Fractions to Percentages

To convert a fraction to a percentage, the denominator must be changed to 100. For example, to convert the fraction 1/2 to a percentage, the numerator and the denominator would both have to be multiplied by 50: 1/2 = 50/100, which is 50%. This can be done more easily by dividing the numerator by the denominator (converting the fraction to a decimal), and then converting the quotient into a percentage. For example: 2/5 = 2 divided by 5 = 0.4 = 40% (See Converting Decimals to Percentages.)

3/8 = 0.375 = 37.5% 5/16 = 0.3125 = 31.25% 7/3 = 2.333 (rounded to the nearest thousandth) = 233.3% 1 3/4 = 1.75 = 175%

24. Elements of a Percentage Problem

Use this SkillGuide to determine what formula to use to solve for Portion, Rate, or Base.

The pie chart below will help you remember the various formulas you need to solve a percentage problem for Portion, Base, or Rate. The horizontal line dividing the top half of the circle from the bottom means divide. The vertical line dividing the lower half of the circle means multiply.

To use this chart, simply put your hand over the element you're solving for, and look at the relationship between the remaining two elements. For example, if you are solving for Portion, cover the top half of the circle, which leaves you with Base multiplied by Rate. That is the formula to solve for Portion. If you want to find Rate, cover the lower right quarter of the circle, so you are left with Portion over (divided by) Base. That is the formula to solve for




Rate. If you cover the lower left hand quarter of the circle, you get the formula to solve for BASE, which is Portion over (divided by) Rate. To summarize:

Portion = Base x Rate Base = Portion ÷ Rate Rate = Portion ÷ Base

The Base is the whole quantity or value to which the rate, or percent, is applied. In other words, the Base is 100% of the quantity being considered. Usually, the Base is the number that follows the word "of."

The Portion is a part that is not the whole quantity and is not a percentage. When determining rate of increase or decrease, you need to identify the original and the new amounts, and find the sum, or the difference between them.

The Rate or percent is the part of the base that you must calculate—a percentage, decimal, or fraction. The Rate is the number with the "%" symbol--the parts out of 100 that you are dealing with.




25. Financial Formulas

Use this Learning Aid to calculate financial figures, in the test for the lesson "Using Decimals."

Financial Figure Formula

Gross Profit Margin gross profit margin ratio = gross profit margin ÷ sales

Markup markup = profit margin ÷ selling price

Current Ratio current assets ÷ current liabilities

26. Profit Margins Expressed as Percentages

Instructions: Use this Follow-on Activity to practice your new math skills by calculating your company's profit margin--and how that margin is affected when prices are discounted to promote sales. Most businesses use percentages to calculate discounts and commissions. That means anyone working as a salesperson has to be able to work out percentages. To get a customer to buy a product, your company may have to offer discount terms--which is just a percentage off the regular price. The problem is, the bigger the discount, the smaller the profit. See if you can get the following information about a product your company manufactures or sells.

Selling Price

Cost to Your Company

Once you have these numbers, try to figure out the following percentages.

Company’s Profit Margin (Portion = Selling Price – Cost to Your Company; Base = Selling Price)

Now calculate what your company’s profit margin would be if the following discounts were offered. To do this, reduce the Selling Price by these percentages, and then perform the operation to calculate the Company’s Profit Margin (Portion = Selling Price – Cost to Your Company; Base = Selling Price).

10%

15%




30%

27. Evaluate Telephone Costs

Instructions: Use this Follow-on Activity to practice math operations using decimal numbers--and find out how much your organization could save by finding a cheaper long-distance telephone services provider.

For many companies, utility costs are a significant expense of doing business. Telephone bills, for example, can take a big bite out of your organization's profits. Could your company save money by using carriers that offer reduced calling charges, reduced minimum call charges, or per second billing? Use your new math skills to evaluate your company's telephone costs. First, get your organization's telephone expenses from its income statement and balance sheet data.

Then find out from someone in telecommunications how much you pay per minute for telephone service. Then search online for a provider that offers a lower corporate rate. After you've found a cheaper rate, complete the following calculations. Round all numbers to the nearest thousandth, except as otherwise noted.

Item #

Facts

Example

Your Figures

1 Annual Telephone expense $548,310.00

2 Cost per minute (round to the nearest ten-thousandth)

$0.0390

3 Divide total telephone expense by cost per minute

4 = Total minutes billed 14,059,230.769

5 Competitive Rate (round to the nearest ten-thousandth)

$0.0345

6 Multiply by total minutes billed

7 = Total cost at competitive rate 485,043.462

8 Total cost at competitive rate - Annual telephone expense

9 = Savings $63,266.538

10 Now multiply your cost per minute by




0.017 (1/60 rounded to the nearest thousandth)

11 = Cost per second (round to the nearest millionth)

$0.000663

12 Time a couple of your next calls, and record them in seconds. For example, a 2 minute, 30 second call would be 150 seconds.

150 second

13 Multiply the seconds of your call by the cost per second

14 Cost of the call $0.099

15 Round the time of the call up to the next minute

3 minutes

Multiply times the per minute charge

16 Cost billed per minute $0.117

17 Subtract the per second charge from the per minute charge

18 Savings $0.018

19 Divide the Savings (18) by the Cost billed per minute (16)

0.154

20 Multiply the Quotient (19) by Annual Telephone Expense (1)

$84,439.74

The last figure you calculate (20) is the possible savings from billing per second, if your call represents an average call at your organization.

28. Understanding Ratios

Ratios are used widely in business because they make statistics easier to analyze and compare.

A ratio is a comparison of two numbers. Ratios are expressed as x to y; x:y, or x/y. When expressing ratios in words, use the word to—the ratio of something to something else. The ratio x:y means that for every x number of something, there are y number of something else.




For example, if the number of cats and dogs are in a 2:5 ratio, it means that for every 2 cats, there are 5 dogs. So, if the cat to dog ratio is 2 to 5 (2:5, 2/5), and there are 6 cats, that means there are 15 dogs.

Forming a ratio Always list the amounts in the same order as they are stated. For example, 3 doctors to 7 lawyers must be stated as 3:7. The numbers or measurements being compared are called the terms of the ratio.

Simplifying ratios Ratios are generally reduced to their lowest terms. The terms of a ratio are reduced by dividing both by as many common factors as possible. For example, the ratio 6:15 can be reduced to 2:5 by dividing both numbers by 3 (note the similarity with reducing fractions to their lowest terms). Keep dividing until there are no more common factors, except 1.

The ratio 45:30 can be simplified by dividing both terms by the same divisors. Both terms are divisible by five, for example, which reduces the ratio to 9:6. Both of those terms can then be divided by 3 to reduce the ratio to its simplest form: 3:2.

Simplifying ratios in this manner makes them easier to understand because reduced ratios express relationships between the numbers more simply.

Ratios as fractions Ratio may be expressed as fractions. For example, the ratio 3 to 4 can be written as the fraction 3/4.

Financial ratios Financial ratios express the relationships between two or various financial figures in the form of percentages or fractions. Using balance sheet data for a company, an analyst can compute the company's debt-to-worth ratio, for example. This is helpful because a low ratio often indicates greater long term financial safety.

Businesspeople use financial ratios to help them manage their organizations. This information is generally expressed as percentages or decimal numbers, because single numbers are easy to compile and compare.

Ratios as decimal numbers To convert ratios into decimal numbers, just divide the first term by the second term. Use a calculator. For example, the ratio 3:7, or 3/7, would be expressed as 0.43, rounded to the nearest hundredth.

Ratios as percentages Ratios can also be expressed as percentages. Percentages are calculated using the equation (x/y) x 100, where x and y are the terms of the ratio (x to y). Just divide x by y, then move the decimal point two places to the left and add a percent sign (%).

That means the ratio 3:7 could be expressed as the decimal number 0.43, or it could be expressed as a percentage—43%.




To assess how a business is doing, you need more than isolated numbers. Each number has to be viewed in the context of the whole picture. Simple ratios can be a powerful tool because they allow you to immediately grasp the relationship expressed.

29. Understanding Proportions

If one ratio can be reduced to or is a factor of another ratio, then the ratios are equal. For example, the ratio 2:1 is equal to the ratio 4:2, which is equal to the ratio 12:6.

When working with proportion equations, if you know one of the ratios, and only one of the numbers of the other ratio, you can figure out the missing number. For example, if the ratio of total liabilities to total assets is 1:2, and the total liabilities are $35,000, you can solve for the total assets figure.

You're basically comparing two equal ratios: 1:2 = $35,000:x, where x is the unknown amount of the total assets. Solving this equation, you can find the value of x ($70,000). In solving for the unknown in a proportion equation, it's important to recall that a ratio can also be expressed as a fraction. For example, the ratio of 3 to 4 can also be written as 3/4.

An equation is just two algebraic expressions separated by an equal sign. An equation stating that two ratios are equal is a proportion. When one of the four numbers in a proportion equation is unknown, you can find the unknown number by following these five steps:

1. Set up the proportion: 3/4 = x/16 When solving a proportion to find an unknown number, the first step is to set up the two ratios that are in the proportion as an equation. Use a letter in place of the unknown number.

For example: If you know that 3 to 4 is the same as some unknown number to 16, you would write the equation as 3/4 = x/16. In this equation, you are solving for x. Every proportion has two cross products. For any proportion, if a/b = c/d, then ad = bc. In this example, the cross products are (3)(16) and (4)(x). So 3/4 = x/16 can be expressed as 3 x 16 = 4x.

2. Use cross products If any three terms in a proportion are given, the fourth may be found by using cross products. An easy way to remember this is to say that in a proportion, the product of the means is equal to the product of the extremes.

In the equation x/y = a/b, the values in the y and a positions are called the means, and the values in the x and b positions are called the extremes. A basic defining property of a proportion is that ya = xb.

3. Solve the side of the equation with no unknown number Knowing that 3 x 16 = 4x, you can do the math on the left side of this equation: 3 x 16 = 48. The equation then becomes 48 = 4x.




4. Isolate the unknown Divide both sides by 4 to isolate the unknown number. The fours on the right side of the equation cancel each other out, so 48 = 4x becomes 48/4 = x.

5. Solve for the unknown Finally, just do the division on the left side of the equation 48/4 = x. Dividing 48 by 4 you get 12, so the answer to this problem is 12 = x.

After you solve a proportion, you can use cross multiplication to see if you have done your math correctly. If the products of cross multiplication are equal, then the ratios are a true proportion.

Understanding proportions is very helpful when you need to make projections or calculate quantities. Proportions empower you to estimate and calculate unknown numbers from known figures.

30. Comparing Different Kinds of Numbers

A rate is a ratio that compares two different kinds of numbers, such as miles per hour or dollars per gallon. Notice that the units—miles, hours, dollars, and gallons—are all different.

The word per always indicates a rate. For example, gasoline filters might be on sale for $6.24 per dozen. The word per can be replaced by the / symbol. So in a problem, $6.24 per dozen could also be written as 6.24/12, or .52/1. When rates are expressed as a quantity of 1, such as 25 feet per second, 65 miles per hour, or $1.90 per gallon, they are called unit rates. You can find the unit rate by dividing the first term of the ratio by the second term.

If you have a multiple-unit price, such as $1,500 for 50 hours of work, and want to find the single-unit rate, divide the multiple-unit price by the number of units ($1,500/50 hours = $30/hour).

Of course, in rate problems, you often have to do more than just solve for the unit rate. You may need to calculate speed, interest, distance, or any number of other factors that depend upon rate calculations.

Any rate problem can be solved using a proportion. If you have a multiple-unit rate, and want to find the single-unit rate, write a ratio equal to the multiple-unit rate with 1 as the second term.

For example, say, you had a ratio of 65:36 and another ratio in which the first term is 195, but you need to find the second term. To find the second term, do the following:

Set up the proportion—A proportion is two equal ratios. In this example, the ratios are 65:36 and 195:x, so the proportion would be: 65:36 = 195:x.

Use cross products—Next, change the proportion, 65:36 = 195:x, to an equation using fractions, like this: 65/36 = 195/x. In a proportion, the product of the means




(36 x 195, in this example) always equals the product of the extremes (65x), so 36 x 195 = 65x.

Isolate the variable—After using cross products to make the equation 36 x 195 = 65x, do the math operation on the side with no variable (x): 36 x 195 = 7,020. So the equation becomes 7,020 = 65x. Then isolate the variable by dividing both sides by 65: 7,020/65 = x.

Solve the equation—Finally, solve the equation 7,020/65 = x by dividing 7,020 by 65. The quotient, 108, is the value of x: 108 = x.

Although all rate problems can be solved using the proportion, it's simpler to use a formula: Rate = Distance/Time. This formula is derived from the proportion calculation, but it's a shortcut that eliminates one multiplication step. This is a formula used for specific types of rate problems involving distance and time. It can be used to solve for Rate (Rate = Distance/Time), Distance (Distance = Rate x Time), or Time (Time = Distance/Rate).

When using these equations, it's important to make your units match. If the problem gives a rate in miles per hour (mph), the time needs to be in hours, and the distance in miles. If the units do not match, you will need to convert them so they are all the same units. For example, if the time is given in minutes, you will need to divide by 60 to convert it to hours before you can use the equation to find the distance in miles.

In rate problems, setting up the equations is the hardest part. Once that's done, calculating the unknown number is relatively easy.

31. Simple Averages

A simple average is a single number that is the result of a calculation performed on a group of numbers. This average typifies the value of all the numbers in the group, taking their individual differences into account.

When a series is made up of different numbers, the simple average is determined by adding up all the different values and then dividing the result by the number of values.

Say you need to find the average of the following list of numbers (data set): 44, 20, 71, 12, 18, 9. There are three steps to determine the simple average, or arithmetic mean. Note that a data point is a single value, or number, from the data set being averaged. Select each numbered step below for more information.

Find the sum of (add) all the data points—The first step is to add up all the individual numbers (data points) in the entire list of numbers (the data set). In this example: 44 + 20 + 71 + 12 + 18 + 9 = 174. The total of all data points in this data set is 174.

Count the number of data points—The next step is to count the number of data points. Count each data point once, regardless of value. In this example, there are six data points—44, 20, 71, 12, 18, and 9.




Divide the sum of the data points by the number of data points—The final step to calculate the simple average is to divide the total of all the data points, 174, by the number of data points, 6. One hundred seventy-two divided by 6 is 29, so 29 is the simple average of all the data points.

Averaging numbers in this way is the simplest way to summarize what all the data has in common. The average tells something about the larger pattern of data that no single number reveals on its own.

The simple average, or arithmetic mean:

takes into account all the data collected is particularly useful if the range of the data is fairly narrow can be influenced by very large or small values in the data set.

The simple average is most useful under the condition that the data points have nearly the same values, with some higher and some lower. In other words, the simple average is most useful when there are no extreme values in the data set.

Using averages can help you find important information from cumulative data. You get an overall picture instead of numerous individual numbers.

a. Sales Data for a Retail Operation

Instructions: Use this Learning Aid to calculate moving averages in the lesson "Using Averages."

Month Sales

January $1,680.00

February $1,410.00

March $1,600.00

April $1,540.00

May $1,610.00

June $1,070.00

July $920.00




August $730.00

September $1,870.00

October $1,880.00

November $1,550.00

December $2,360.00

b. Prices of Common Stock


Date 2/1 2/2 2/3 2/4 2/5 2/6 2/7 2/8 2/9

Price per share

$80.10 $78.20 $75.10 $75.00 $74.80 $76.20 $78.40 $77.20 $76.50

c. Demand for Kerosene


The table below shows the demand for kerosene for each of the last 12 months.

Month Demand (gallons)

January 1,090

February 1,095

March 1,100

April 1,105

May 1,110

June 1,115




July 1,108

August 1,110

September 1,090

October 1,080

November 1,060

December 1,050

d. Sales Data for a Retail Operation


Month Sales

January $1,680.00

February $1,410.00

March $1,600.00

April $1,540.00

May $1,610.00

June $1,070.00

July $920.00

August $730.00

September $1,870.00

October $1,880.00




November $1,550.00

December $2,360.00

e. Prices of Common Stock


Date 2/1 2/2 2/3 2/4 2/5 2/6 2/7 2/8 2/9

Price per share

$80.10 $78.20 $75.10 $75.00 $74.80 $76.20 $78.40 $77.20 $76.50

f. Demand for Kerosene


The table below shows the demand for kerosene for each of the last 12 months.

Month Demand (gallons)

January 1,090

February 1,095

March 1,100

April 1,105

May 1,110

June 1,115

July 1,108

August 1,110

September 1,090

October 1,080

November 1,060




December 1,050

32. Productivity Ratios

Instructions: Use this Follow-on Activity to practice your new math skills by calculating these ratios to evaluate the productivity of your workforce and company in terms of producing core products or services.

a. Sales per employee

Find out what your company's total sales were for the year, and then find out how many employees

your company has. Reduce that ratio to a unit rate--sales per single employee. The ratio provides a

useful productivity measure, which is also useful to determine the level of sales required to support

increased staffing levels.

Example: If a company has 54 employees and annual sales volume is $5,953,500.00, the ratio of sales to employees is $5,953,500.00/54. Reduce that to a unit rate by dividing $5,953,500.00 by 54. The result, 110,250.00, is the sales per employee.

b. Gross profit dollars per employee

This measure combines an item from your company's income statement--gross profit--with employees. Profits are divided by employees. It provides a measure of personnel productivity.

Example: Say the company's gross profit was $1,786,050.00, and it has 54 employees. That's a ratio of 1,786,050.00/54. To get the unit rate, divide 1,786,050.00 by 54. The result, $33,075.00, indicates profit per employee.

c. Payroll per employee

This ratio uses the wages and salaries figure from your company's income statement and the number of employees in your business. Divide the number of employees into wages and salaries. Payroll per employee indicates the expected level of pay for an average employee.

Example: If the company's wages and salaries are $2,994,408.00, and it has 54 employees, that's a ratio of $2,994,408.00/54. To get the unit rate, divide $2,994,408.00 by 54. The result, $55,452.00, indicates payroll per employee.

d. Weighted Averages

A weighted average is an average that takes into account the relative precision or importance of the data used in the calculation. Use weighted averages to address situations where the elements being averaged are not equivalent in some respect. To compensate for this inequality, weights are attached to each element.

The weighted average is a useful calculation. For example, say you have values of 25, 50, 75, and 100. You want to assign weight of 75% (0.75) to the values 25 and 50, and a weight of




25% (0.25) to the values of 75 and 100. The simple average of the numbers 25, 50, 75, and 100 is 62.5.

The steps below show how to calculate the weighted average:

Multiply the value with the weight—The first step is to multiply the value with the weight. For example, you need to multiply 25 and 50 by 75% (0.75). That gives weighted values of 18.75 and 37.5. Multiplying 75 and 100 by the weight assigned to them—25% (0.25)—gives weighted values of 18.75 and 25.

Total the results—The second step is to total the results (the weighted values). For example, the weighted values are 18.75, 37.5, 18.75, and 25. 18.75 + 37.5 + 18.75 + 25 = 100. So the total of the products from the first step—the weighted values—is 100.

Divide the total by the sum of the weights—The last step to find the weighted average is to divide the total (100) by the sum of the weights. The weights are .75, .75, .25, and .25. Each value has a weight. The sum of those weights is 2.0. Dividing 100 by 2.0 gives the final answer—the weighted average, which is 50.

The weighted average method allows you to adjust to experience, trends, and facts, but the weights you choose will affect the results! That's because, to calculate weighted averages, you divide the total of the weighted values by the sum of the weights.

Often, weights are assigned such that all the weights sum to 1.0, or 100%. Weights for class grades may be as follows: homework 20%, quizzes 20%, exams 40%, final exam 20%. Weights may be determined by confidence in the data, by the data's contribution to overall results, or by quantities.

Depending upon the data, any fraction, percentage, or decimal number may be appropriate to adjust the value of a data point. If you are using a percentage as the weight, you'll need to convert the percentage value to a decimal (simply by moving the decimal point two places to the left) before multiplying the data point by that weight.

Weighted averages are calculated in such a way that some data points affect the result more or less than others. Of course, weighting components properly helps produce more meaningful results.

e. Moving Averages

Moving averages smooth temporary fluctuations in a series of data measurements and make it easier to spot trends. This makes the technique especially helpful in volatile business environments.

A moving average is an average of a fixed number of consecutive values or measurements, updated periodically at regular intervals.

Calculating a moving average is like taking a sample of a constantly changing stream of information. As more information is added to the data set, the average moves to accommodate that new data. Here are the steps to determine a moving average:




Gather the data points within a selected period—Consider the following progression, from day one to day eight:25.75, 27.50, 27.95, 27.85, 28.20, 28.50, 28.75, 28.80.

Add selected data points—To take a five-day moving average on day five, add up the figures for days 1-5: 25.75 + 27.50 + 27.95 + 27.85 + 28.20 = 137.25.

Divide by the number of data points—Since there are five data points, divide 137.25 by 5. The answer, 27.45, is the five day moving average.

Continue the process by adding the latest period data while dropping the first period of the calculation—To move the average to the next 5-day period, simply drop the oldest (day 1) figure, and add the latest (day 6) figure: 27.50 + 27.95 + 27.85 + 28.20 + 28.50 = 140. Again, divide by 5, since there are 5 data points. The five-day moving average is 28.

The next step is the same, except you drop day 2 and add day 7: 27.95 + 27.85 + 28.20 + 28.50 +28.75 = 141.25. There are always going to be 5 data points, since this is a 5-day moving average. So divide 141.25 by 5. The moving average is 28.25.

This is a continuous process. Now, you drop day 3 and add day 8: 27.85 + 28.20 + 28.50 + 28.75 + 28.80 = 142.1. Dividing 142.1 by 5, you get the 5-day moving average: 28.42.

A moving average is the sum of the measurements or values over a certain number of time periods divided by the number of time periods to get an average for that period, and that average moves with the addition of new data. This process reduces the effect of fluctuations in the data and produces a stronger indication of the trend over the period being analyzed.

33. Proportions and Ratios

Instructions: Use this SkillGuide to help understand and solve proportions.

A ratio is a comparison of two numbers. The two terms in a ratio are generally separated by a colon (:), but ratios can also be written as fractions. For example, if you want to express the ratio of 2 and 3, you can write it as 2:3 or as a fraction 2/3.

A proportion is an equation with a ratio on each side. It is a statement that two ratios are equal. For example, 12/16 = 3/4 is an example of a proportion. Ratios are equal if their cross products are equal, in this case, if 12× 4 = 16× 3. Since both of these products equal 48, the ratios are equal.

When one of the four numbers in a proportion is unknown, cross products may be used to solve the proportion by finding the unknown number. For example, if you have the equation 2/3 = x/6, you need to solve for x.

Using cross products makes the equation 3x = 2 x 6, so 3x = 12. Dividing both sides by 3, x = 12 ÷ 3, so that x = 4.

Use the table below to help solve a proportion to find an unknown number:




To use this form, just write your proportion as fractions. So 3:4 = 15:20 would be 3/4 = 15/20. Then just plug the numbers into the appropriate boxes above. Write a letter--"x" will do--in the box to represent the unknown number. In the bottom set of boxes, on the side where you have two numbers, do the multiplication. Then divide the product of that multiplication by the known number on the other side of the equation. That will give you the value of "x."

34. Calculating Simple, Weighted, and Moving Averages

Instructions: Use this SkillGuide to help understand and calculate simple averages, weighted averages, and moving averages in the workplace.

Problems involving averages are very common in the workplace. They can be classified into three major categories: simple averages, weighted averages, and moving averages. Here are the steps to calculate each category:

a. Simple Average

A simple average of x number of data points is their sum divided by x: simple average = sum/x.

Example: The average of 35, 70, and 39 (three data points) is (35 + 70 + 39)/3 = 144/3 = 48

b. Weighted Average

A weighted average is the sum of the weighted values divided by the sum of the weights themselves. You can assign any weights you want to the data points.

Example: In a workplace, employees' performance is graded on the basis of Sales (50% of the grade), customer service (20% of the grade), internal customer service (20% of the




grade), and attitude (10% of the grade). The percentages are the weights to be assigned to the score (data point) in each of the four categories. An employee gets the following scores: a 70 in Sales, an 80 in Customer Service, a 90 in Internal Customer Service, and a 100 in Attitude. Then, calculate this way:

Score Weight Value 70 x 50% (0.5) = 35 80 x 20% (0.2) = 16 90 x 20% (0.2) = 18 100 x 10% (0.1) = 10

Weighted average = (35 + 16 + 18 + 10)/1.0 = 79/1.0 = 79

Here, 70 has a weight of 0.5--it makes up 50% of the overall evaluation grade, whereas 100 has a weight of just 0.1, making up just 10% of the overall grade. So the weighted average is closer to 70 than to 100.

c. Moving Average

A simple moving average is calculated by adding the data points in a specified period, then dividing the result by the number of data points. The average "moves" and changes as the oldest value is dropped out of the calculation and the newest value is added in. In a simple moving average, every data point is given equal weight.

Example: Prices for a commodity vary as follows:

January February March April May June July

208.5 209.9 227.1 208.5 194.5 166.0 129.4

A 5-month simple moving average is calculated by adding the prices for the 5 consecutive months and dividing the total by 5. So, for the first five months, the moving average would be:

Moving average = (208.5 + 209.9 + 227.1 + 208.5 +194.5)/5 = 1048.5/5 = 209.7

Continuing the process, the next price in the average is June, which is 166.0, so this new data point would be added and the oldest data point, which is January (208.5), would be dropped. The second 5-month moving average would be calculated as follows:

Moving average = (209.9 + 227.1 + 208.5 +194.5 + 166.0)/5 = 1006.0/5 = 201.2




The calculation is repeated as new prices appear on the chart--each new data point is added and the oldest data point is dropped.

The next price in the average is July, which is 129.4, so this new data point would be added and the oldest data point, which is February (209.9), would be dropped. The third 5-month moving average would be calculated as follows:

Moving average = (227.1 + 208.5 +194.5 + 166.0 + 129.4)/5 = 925.5/5 = 185.1

35. Productivity Ratios

Instructions: Use this Follow-on Activity to practice your new math skills by calculating these ratios to evaluate the productivity of your workforce and company in terms of producing core products or services.

a. Sales per employee

Find out what your company's total sales were for the year, and then find out how many employees

your company has. Reduce that ratio to a unit rate--sales per single employee. The ratio provides a

useful productivity measure, which is also useful to determine the level of sales required to support

increased staffing levels.

Example: If a company has 54 employees and annual sales volume is $5,953,500.00, the ratio of sales to employees is $5,953,500.00/54. Reduce that to a unit rate by dividing $5,953,500.00 by 54. The result, 110,250.00, is the sales per employee.

b. Gross profit dollars per employee

This measure combines an item from your company's income statement--gross profit--with employees. Profits are divided by employees. It provides a measure of personnel productivity.

Example: Say the company's gross profit was $1,786,050.00, and it has 54 employees. That's a ratio of 1,786,050.00/54. To get the unit rate, divide 1,786,050.00 by 54. The result, $33,075.00, indicates profit per employee.

c. Payroll per employee

This ratio uses the wages and salaries figure from your company's income statement and the number of employees in your business. Divide the number of employees into wages and salaries. Payroll per employee indicates the expected level of pay for an average employee.

Example: If the company's wages and salaries are $2,994,408.00, and it has 54 employees, that's a ratio of $2,994,408.00/54. To get the unit rate, divide $2,994,408.00 by 54. The result, $55,452.00, indicates payroll per employee.




36. Moving Averages

Instructions: Use this Follow-on Activity to practice your new math skills by calculating a moving average that shows the average value of your company's stock over a period of time.

A moving average is an indicator that shows the average value of any data series over a period of time. As the data points change, their average price moves up or down. For example, consider the following table, which shows the daily stock prices of a company:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Date 4/5 4/6 4/7 4/8 4/9 4/12 4/13 4/14 4/15 4/16 4/19 4/20 4/21 4/22 4/23

Value

20.96

21.20

19.83

20.0

19.82

20.04

19.76

19.92

20.02

19.87

19.54

19.63

19.45

19.10

19.26

In the following table, print the date and the price per share of your company, or of any other security you want to evaluate. Then fill in the blanks to calculate a series of six 5-day moving averages.

1 2 3 4 5 6 7 8 9 10

Date

Value

Add the values in columns 1-5: _________ Divide by 5 to get the simple moving average for the first 5-day period: _______

Add the values in columns 2-6: _________ Divide by 5 to get the simple moving average for the second 5-day period: _______

Add the values in columns 3-7: _________ Divide by 5 to get the simple moving average for the third 5-day period: _______

Add the values in columns 4-8: _________ Divide by 5 to get the simple moving average for the fourth 5-day period: _______

Add the values in columns 5-9: _________ Divide by 5 to get the simple moving average for the fifth 5-day period: _______

Add the values in columns 6-10: _________ Divide by 5 to get the simple moving average for the sixth 5-day period: _______

Now look at the six moving averages you came up with. The general trend is: _____ Up _____ Down




37. 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.

Base The Base is the whole quantity or value to which the rate, or percent, is applied. In other words, the Base is 100% of the quantity being considered. Usually, the Base is the number that follows the word "of." Current ratio The current ratio is determined by dividing the current assets by the current liabilities. This figure indicates how easily business assets can be turned into cash. Decimal fraction A decimal fraction is a fraction expressed in decimal notation. For example, three-quarters expressed as a decimal fraction is 0.75. It is the part of a decimal number that is to the right of the decimal point--representing a value less than one. Dividend A dividend is a number to be divided by another number. Divisor A divisor is a number that another is being divided by. Operations Operations include addition, subtraction, multiplication, and division, which are used to combine numbers and variables in a math problem. Percentage A percentage is a part of a whole expressed in hundredths. Generally, the word "percent" is used as part of a numerical expression--profits were down by two percent, for example. The term percentage is used to suggest a portion--a large percentage of our inventory was spoiled, for example. Pie chart The pie chart is a visual tool that can help determine what formula to use to solve for Base, Portion, or Rate. Simply cover up the element being solved for, and the pie chart shows whether the remaining elements should be multiplied or divided. Portion The Portion is a part that is not the whole quantity and is not a percentage. When determining rate of increase or decrease, you need to identify the original and the new amounts, and find the sum, or the difference between them. Quotient The quotient is the answer to a division problem--the result from dividing one number by another number. Rate The Rate or percent is the part of the base that you must calculate--a percentage, decimal, or fraction. The Rate is the number with the "%" symbol--the parts out of 100 that you are dealing with. Variable A variable is simply a letter that stands for an unknown number.




Current ratio The current ratio is determined by dividing the current assets by the current liabilities. This figure indicates how easily business assets can be turned into cash. Dividend A dividend is a number to be divided by another number. Divisor A divisor is a number that another is being divided by. Equation An equation is a mathematical sentence stating that two expressions (in this case, ratios) are equal. Operations Operations include addition, subtraction, multiplication, and division, which are used to combine numbers and variables in a math problem. Percentage A percentage is a part of a whole expressed in hundredths. Generally, the word "percent" is used as part of a numerical expression--profits were down by 2 percent, for example. The term percentage is used to suggest a portion--a large percentage of our inventory was spoiled, for example. Quotient The quotient is the answer to a division problem--the result from dividing one number by another number. Variable A variable is simply a letter that stands for an unknown number.


Basic Business Math - Study Notes

Business

Transcript of Basic Business Math - Study Notes