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# Basic Business Math - Study Notes

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A basic understanding of decimals and percentages is key to any businessperson, whether tallying costs for warehouse supplies or estimating resource allocation. …

A basic understanding of decimals and percentages is key to any businessperson, whether tallying costs for warehouse supplies or estimating resource allocation.

Therefore learn to use decimals, addition, subtraction, multiplication, and division; and to solve problems involving percentages.

Also, knowledge of ratios and averages is indispensable in the business world. Using real-world scenarios, this course explains the concepts of ratio, proportion, and how to compare different kinds of numbers; and discusses simple, weighted, and moving averages.

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• 1. Basic Business Math Study Notes Entry Level http://SlideShare.net/OxfordCambridge
• 2. http://SlideShare.net/OxfordCambridge 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 2|P a g e Basic Business Math
• 3. http://SlideShare.net/OxfordCambridge 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 http://SlideShare.net/OxfordCambridge 3|P a g e Basic Business Math
• 5. http://SlideShare.net/OxfordCambridge 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 sixpart whole. 7/6 is six parts, plus one extra one, written as 1/6. So the improper fraction 7/6 could be written as the mixed number 1 1/6. Mixed numbers consist of a whole number followed by a proper fraction. 3. Mixed numbers You use mixed numbers when you need to count whole units and parts at the same time. For example: If you have three full cans of soda and one half-full can of soda, you write it like this: 3 1/2, and say it like this three and one half. It's really 3 + 1/2. That's why we say the and. 5|P a g e Basic Business Math
• 7. http://SlideShare.net/OxfordCambridge 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. 7|P a g e Basic Business Math
• 10. http://SlideShare.net/OxfordCambridge    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 10 | P a g e Basic Business Math
• 17. http://SlideShare.net/OxfordCambridge 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÷2x7-3=x There are no exponents, so the next step is to do the division and multiplication. Working from left to right, 8 ÷ 2 = 4; 4 x 7 = 28. So now the equation is: 4 + 28 - 3 = x Now, perform all additions and subtractions, working from left to right: 4 + 28 = 32; 32 - 3 = 29. So the solution is: x =29 Hint: To solve equations correctly, it might help to remember the phrase: "Please excuse my dear Aunt Sally" (P E M D A S): Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. 17 | P a g e Basic Business Math
• 18. http://SlideShare.net/OxfordCambridge 11. Place Values Help yourself to understand and identify place values in numbers that you want to round. To use this tool, write the number you want to round in the box in the left column. Then, starting with the last digit on the right, enter each digit of your number into the boxes from right to left under the place values. For example, the number 1,345,076 is entered on the left. Then the 6 is entered in the ones column, the 7 is entered in the tens column, the 0 is entered in the hundreds column, the 5 is entered in the thousands column, and so on. Each column corresponds to that digit's place value. Numbers Millions Hundred Thousands Ten Thousands Thousands Hundreds Tens Ones Example 1,345,076 1 3 4 5 0 7 6 Example 28,670 0 0 2 8 6 7 0 Note that the place values vary by a factor of 10. The column to the left of a place value is ten times that place value, the column to the right of a place value is one-tenth of that place value. For example, the 7 in the tens column in the first example represents seven tens, or 70. If it was in next column to the left (the hundreds place), it would represent seven hundreds, or 700. If it was in next column to the right (the ones place), it would represent seven ones, or 7. 18 | P a g e Basic Business Math
• 21. http://SlideShare.net/OxfordCambridge 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: 21 | P a g e Basic Business Math
• 24. http://SlideShare.net/OxfordCambridge 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. 24 | P a g e Basic Business Math
• 25. http://SlideShare.net/OxfordCambridge 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. 25 | P a g e Basic Business Math
• 26. http://SlideShare.net/OxfordCambridge 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 26 | P a g e Basic Business Math
• 27. http://SlideShare.net/OxfordCambridge 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. 27 | P a g e Basic Business Math
• 28. http://SlideShare.net/OxfordCambridge 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. 28 | P a g e Basic Business Math
• 29. http://SlideShare.net/OxfordCambridge 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 1,345.076 1 3 4 5 28.690 0 0 2 8 Decimal Point Tenths Hundredths Thousandths . 0 7 6 . 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. 29 | P a g e Basic Business Math
• 30. http://SlideShare.net/OxfordCambridge 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 30 | P a g e Basic Business Math
• 31. http://SlideShare.net/OxfordCambridge  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 31 | P a g e Basic Business Math
• 32. http://SlideShare.net/OxfordCambridge 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.   32 | P a g e Basic Business Math
• 34. http://SlideShare.net/OxfordCambridge 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 8 Total cost at competitive rate - Annual telephone expense 9 = Savings 10 Now multiply your cost per minute by 34 | P a g e 485,043.462 \$63,266.538 Basic Business Math
• 35. http://SlideShare.net/OxfordCambridge 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. 35 | P a g e Basic Business Math
• 36. http://SlideShare.net/OxfordCambridge 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%. 36 | P a g e Basic Business Math
• 37. http://SlideShare.net/OxfordCambridge 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. 37 | P a g e Basic Business Math
• 38. http://SlideShare.net/OxfordCambridge 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 38 | P a g e Basic Business Math
• 39. http://SlideShare.net/OxfordCambridge   (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. 39 | P a g e Basic Business Math
• 40. http://SlideShare.net/OxfordCambridge  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 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 40 | P a g e Sales \$920.00 Basic Business Math
• 41. http://SlideShare.net/OxfordCambridge August \$730.00 September \$1,870.00 October \$1,880.00 November \$1,550.00 December \$2,360.00 b. Prices of Common Stock Instructions: Use this Learning Aid to calculate moving averages in the lesson "Using Averages." 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 Instructions: Use this Learning Aid to calculate moving averages in the lesson "Using Averages." The table below shows the demand for kerosene for each of the last 12 months. Month January 1,090 February 1,095 March 1,100 April 1,105 May 1,110 June 41 | P a g e Demand (gallons) 1,115 Basic Business Math
• 42. http://SlideShare.net/OxfordCambridge 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 Instructions: Use this Learning Aid to calculate moving averages in the lesson "Using Averages." Month 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 42 | P a g e Sales \$1,880.00 Basic Business Math
• 43. http://SlideShare.net/OxfordCambridge November \$1,550.00 December \$2,360.00 e. Prices of Common Stock Instructions: Use this Learning Aid to calculate moving averages in the lesson "Using Averages." 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 Instructions: Use this Learning Aid to calculate moving averages in the lesson "Using Averages." The table below shows the demand for kerosene for each of the last 12 months. Month 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 43 | P a g e Demand (gallons) 1,060 Basic Business Math
• 44. http://SlideShare.net/OxfordCambridge 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 44 | P a g e Basic Business Math
• 45. http://SlideShare.net/OxfordCambridge 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: 45 | P a g e Basic Business Math
• 46. http://SlideShare.net/OxfordCambridge     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: 46 | P a g e Basic Business Math
• 47. http://SlideShare.net/OxfordCambridge 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 47 | P a g e Basic Business Math
• 48. http://SlideShare.net/OxfordCambridge 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 48 | P a g e Basic Business Math