Business Math Tutor. Part I SP39A-22

Transcription

1 Business Math Tutor Part I

2 Business Math Tutor Part I

3 No part of this document may be reproduced or transmitted in any form or by any means, electronic or mechanical, for any purpose, without the express written consent of U.S. Career Institute. Copyright , Weston Enterprises, Inc. All Rights Reserved. ACKNOWLEDGMENTS: Editorial Staff Trish Bowen Katy Little Design/Layout Staff Leslie Ballentine Connie Hunsader U.S. Career Institute 2001 Lowe Street Fort Collins, CO

4 Contents Introduction... 1 The Number Line... 1 Place Values... 5 Basic Math Functions... 7 Addition... 7 Subtraction... 9 Multiplication Division Order of Operation Page i

5 Page ii

6 Introduction Numbers are at the core of many business decisions and, therefore, to succeed in the business world, you need to know how to use numbers. In this supplement, you will be reviewing some of the basic math concepts you have probably learned before. In addition, you will be gaining practice using electronic calculators and 10-key adding machines, the tools most often used for math functions in business today. Finally, you will be applying math principles to situations you would encounter in the business world. In Part I of your Business Math Tutor, you will be reviewing the four basic functions in math; addition, subtraction, multiplication, and division. We will also cover some introductory items such as the number line and placement values. These make learning the four functions easier. a 2 + b 2 = c 2 Throughout this supplement, you will be presented with practical business situations in which to apply your growing mathematical knowledge. There will be exercises that will stimulate your mind and sharpen your skills. It is important that you complete each practice situation, as math is a skill that requires practice. You should use your skills as much as you can after you learn them; if you do, you will soon be able to perform each function quickly and accurately. So sharpen your pencil and open your mind. Math is fun you ll see! The Number Line A number line is a way to display numbers in a sequence. The typical number line we will use displays whole numbers ones that are not decimals or fractions. By learning the number line, you will better understand the symbols of mathematics. Let s look at the number line below: Page 1

7 We can say this number line represents all whole numbers, or more literally, the whole numbers -3, -2, -1, 0, 1, 2, 3 The three dots at each end indicate that the sequence goes on forever (infinitely). Look at that again. On the number line, the arrows also indicate the number sequence goes on forever. Anytime you have a sequence of numbers, that sequence is called a set. The above example is the set of whole numbers. If we had a different set, say 2, 3, 4, 5, then the number line would look like this: Notice the number line has only one arrow and it is pointing to the right to indicate the set of numbers is endless in that direction. The set begins at 2; therefore, it does NOT go on forever to the left. As you begin to solve mathematical equations, you will need to know what the various symbols mean. The four basic functions are represented by the following four symbols: Addition + (the plus sign) Subtraction (the minus sign) Multiplication or * (the times sign) Division or / (the divided by sign) There are other symbols you will become familiar with. The less than sign (<) and greater than sign (>) both are used to express properties of a pair of numbers. For example, we can say that 5 is less than 6. Look at the number line. Five is to the left of 6, and therefore is less than 6. By using the symbol for less than, we can shorten our equation to simply: 5 < 6. An easy way to remember the less than and greater than signs is by looking at the point. The less than sign points to the left. Its smallest point is toward the smaller number while its largest part points toward the larger number (the open part of the symbol). The greater than sign is the same way in that the larger (open) part of the symbol points toward the larger number. Therefore, we know that in the equation 7 > 4 that the 7 is greater than the 4. Less than sign Greater than sign Page 2

8 Another important symbol is the equals sign (=). This sign sets two parts of an equation apart and indicates that the two parts are equal to one another. For example, the equation: 2 2 = 4 is read as two times two equals four. In math, we also use the equals sign to show what we need to find. We might state a problem as: What does five plus two equal? By looking at the words in the sentence, we know to set the problem up like this: = And the answer is 7. The equals sign can also be written as a line under a vertical group of numbers to be added, subtracted, or multiplied. In the example below, the final group of numbers is underlined to show that you should add the numbers above the line and write the answer below the line. This symbol means the same thing as equals. So we have two ways of writing the equations for the sentence: five plus seven equals twelve. We can write it like this: = 12 Or we can write it like this: The underlining method can be used if you are performing ONLY ONE function. If you are combining two or more functions addition and subtraction, for example you must write the equation horizontally and use the equals sign. For example: = 1 Page 3

9 Try these sample problems: 1. Because 3 is to the left of 4 on the number line, we know that 3 is a. less than 4 b. greater than Write the mathematical equation for the above question using the numbers and symbols you learned. 3. Use numbers and symbols to write out this equation: Seven plus six minus four equals nine. 4. In the number sequence (1, 2, 3, 4 ) what do the three dots at the end mean? 5. Write a sentence that means the same thing as this equation 8 4 = 32. Check your answers: 1. a. less than < = 9 4. The sentence goes on forever from four (infinitely). 5. Eight times four equals thirty-two. Page 4

10 Place Values Before we get further into numbers, we need to cover place values. A good way to start is with money. Let s look at the amount: $ In this example, we say that the 4 is in the hundreds column, the 5 is in the tens column, and the 3 is in the ones column. Behind the decimal point, the 2 is in the tenths column and the 7 is in the hundredths column. What does this all mean? Look again at the example: $ Because the 4 is in the hundreds column, we can say it is worth or 400 dollars. The 5, being in the tens column, is worth 5 10 or 50 dollars. The 3 is in the ones column, so we can say it is worth 3 1 or 3 dollars. To the right of the decimal point, we now look at the cents. Cents actually are how many hundredths of one dollar there are. In this example, we can say you have 27 hundredths of one dollar. That, of course, equals 27 cents. The place values we explained above help you differentiate between, say $45.32 and $ Look at this example: 8,370, The place values break down like this: 8, 3 7 0, Hundredths Tenths Ones Tens Hundreds Thousands Ten Thousands Hundred Thousands Millions Page 5

11 Now, using what you know, label the correct place values of the underlined number in the following examples: , , , ,567, , Check your answers: 1. Tens 2. Ten Thousands 3. Ones 4. Tenths 5. Hundreds 6. Millions 7. Hundreds 8. Thousands 9. Hundredths 10. Tens Page 6

12 Basic Math Functions Addition Addition is simply adding numbers together. It s where we get the term adding machine what is sometimes called a calculator. Let s look at this example. If Bob s Sporting Goods sells $1,520 worth of baseball equipment in March and $2,950 worth of baseball equipment in April, how much money did the store collect in those two months? This is where you need to remember your place values. To set up this problem, you need to align each number so you are adding equal place values the thousands add together, the hundreds, and so on. The problem is set up like this: 1 $ 1, ,950 $ 4,470 The answer to this problem is $4,470. You find the answer by adding like place values and carrying. Start at the right and add the ones column: = 0. Then move on to the tens column: = 7. Next, add the hundreds column: = 14, but you can t write in 14, so you write the 4, then CARRY the 1 left over into the next column. Finally, add the 1 you carried to the 1 and 2 in the column. Your answer there is 4. That s how you come up with $4,470 Bob s Sporting Goods total for baseball equipment sales in March and April. Look at another example: Joan owns two bars. The first one pays $22, for liquor and the second pays $17, How much does Joan have to pay for both bars liquor? $22, , Next, do the addition. Start with the hundredths column. Add the 5 and 5 you get 10. So write down the 0 and CARRY the 1. Next, go to the tenths column and add the 1 you carried to the 7 and 2. The answer there is also 10. Write down the 0 and CARRY that 1 again. Then add the 1 you carried to the 2 and 2 in the ones column. Write that answer (5). Next, add the tens column: = 6. Then add the hundreds column: = 13. Write the 3 and CARRY the 1 to the thousands column. Add the 1 to the 2 and 7. The answer in the thousands is 10, so write the 0 and CARRY the 1 into the final column. Finally, add the 1 and the 2 and 1 in the ten thousands column to come up with 4, and your final answer. Page 7

13 $22, , $40, Joan spends $40, for liquor at her two bars. If you have more than two numbers to add together, the method is basically the same. Start by setting up your problem be careful to line up your place values. When you have to carry a number, it is the same even if that number is greater than one. For example, if you add up the tens column and come up with 24, write down the 4 and carry 2 to the next column. Here is an example of a problem that needs you to add more than two numbers together: , , , , , Now, you try some practice problems , , , , Check your answers: 1. 1, , , , Page 8

14 Now you ve learned one-fourth of the basic mathematical functions. Addition is very important to bookkeeping and in some office administrator duties. Another important function is subtraction. Subtraction is closely related to addition; therefore, it is the next section we ll cover. Subtraction Subtraction can be defined as the opposite of addition. It is the taking away of one number from another. If you have a situation where you take from one group of items apples, for example you might use subtraction. If you have 5 apples and you need to pack 3 lunches, each with one apple, how many apples would you have left? 5 3 = 2 That is an example of subtraction. You need to remember a key difference between subtraction and addition. Aside from subtraction being the taking away of an item rather than adding to it (addition), the main difference between the two is how you work through the problem. Instead of carrying, in subtraction you BORROW. Look at this example: 12 7 This problem, stated in sentence form, is: twelve minus seven equals. We know from the place value section that 12 actually means one ten and 2 ones. If we try to take 7 ones from the 2 in 12 (the 2 is in the ones column), we find it impossible. So, you borrow a pack of 10 ones from the tens column. This reduces your tens column by 1 and increases your ones column by 10. Now you still have 12, but you say it is zero tens and 12 ones. If you take 7 ones from 12 ones, you have 5 ones remaining. You probably could have figured that problem out without borrowing, but you need to learn the process by doing easier problems so you can work through more complex problems without trouble. That is the concept of borrowing. By borrowing a pack from the column to the left, you can subtract large numbers from small ones in place values. It makes more sense if you look at some numerical examples: 15 9 Page 9

15 In this example, we can state the problem as fifteen minus nine. You know that you can subtract 9 from 15. If you have 15 oranges and take away 9, you still have oranges left. To find out how many, let s look at the number line: Look at the 9. From that point to the 15, there are 6 spaces between the numbers. The 6 spaces are between the 9 and 10; the 10 and 11; the 11 and 12; the 13 and 14; and the 14 and 15. Because there are 6 spaces, we know that 15 9 = 6. To work out the problem 15 9, you simply borrow a pack of 10 ones from the tens column and add it to the 5. Now, you are subtracting 9 from 15. Let s look at another example: 21 2 The way to work through this is by borrowing a pack of ones from the tens column. This reduces the 2 in 21 to a 1 and makes the 1 actually worth 11. We know that 11 2 is 9. So, we have 9 ones in our answer. The 1 in the tens column stands alone. That means you simply drop it down so there is 1 in the tens column and 9 in the ones column for our answer (19) Here s a similar problem: 72 5 You borrow a pack of ten ones from the tens column so you decrease the 7 to a 6 and make the 2 a 12. That enables you to subtract 5 from 12 to get 7 in the ones column. Then you drop the 6 down in the tens column and come up with your answer Page 10

16 Let s look at one final example: To solve this equation, you must borrow a pack of ones from the tens column. This decreases the 8 to a 7 and increases the 5 to a 15. We ve already seen the answer to 15 9 = 6, so we can write in the 6 in the ones column. The top 7 must now be figured in, however. You subtract the bottom 7 from the top 7, and that gives you zero. So, in this problem, there is no number in the tens column, and the solved equation looks like this: The philosophy of borrowing is the same if you are borrowing from the tens column or the hundreds, or thousands, or further over in the tenths or ones column. Each time you borrow, it decreases the number on the left by 1 and increases the number on the right by 10. Borrowing from the tens column increases the ones by 10 and decreases the tens by 1. If you borrow from the hundreds column, your tens increase by 10 and your hundreds decrease by 1. Look at the chart below. Borrowed From Result Tenths Tenths decrease by 1, hundredths increase by 10 Ones Ones decrease by 1, tenths increase by 10 Tens Tens decrease by 1, ones increase by 10 Hundreds Hundreds decrease by 1, tens increase by 10 Thousands Thousands decrease by 1, hundreds increase by 10 Page 11

17 Complete this practice review: Circle the letter representing the best answer for each question. 1. On a number line, 4 is to the left of 6. Because of this, we can say. a. 4 is greater than 6 b. the number line is backwards c. 4 is less than 6 d. none of the above 2. In the number 45,987, the underlined 9 has what place value? a. tens b. thousands c. hundredths d. hundreds 3. Solve this equation: ,046 =. a. 7,732 b. 8,734 c. 14,907 d. 6, Solve this equation: a. 1, b c. 1, d. 1, Page 12

18 5. Solve this equation: a b c d For questions 6-10, choose the correct term that will complete the equation or sentence. 6. is when you take one number away from another. a. addition b. number line c. place value d. subtraction 7. Insert the term that solves this equation: = 155 a. addition b. minus c. plus d. less than 8. is the opposite of subtraction. a. Addition b. Less than c. Greater than d. Division 9. If a number (x) is located to the right of number (y) on the number line, then number (x) number (y). a. less than b. equal to c. greater than d. not enough information to solve Page 13

19 10. In the number: 19,875.46, the 4 is in the. a. hundreds place b. middle c. tenths place d. tens place Now, check your answers. 1. c 2. d 3. b 4. d 5. b 6. d 7. c 8. a 9. c 10. c Next, you will be learning about multiplication, division, and the order of operations. This section covers the basics of multiplication and division. It is designed to give you a working knowledge of these functions. You will probably use a calculator for any calculations you perform. As you move through these sections, remember the methods and the reasoning behind multiplication and division. You need to know when to use each function and how to use it. On an adding machine, multiplication is represented by the x symbol and division is the symbol. If you are using a computer, there are two different symbols: multiplication is the asterisk (*) and division is a slash (/). Whatever method you use, become familiar with it and the underlying principles involved. Page 14

20 Multiplication Multiplication refers to the operation of taking multiples of a number. A multiple is the number itself added to your answer. For example, saying three times four is the same as saying three multiples of four. In either case, the answer is twelve. Look at the following examples: a. 3 5 = 15 b. 7 2 = 14 c = 42 These can also be written like this: a b c Example (a) and (b) both show simple multiplication using only the ones column. These can be interpreted as three multiples of five and seven multiples of two, respectively. Example (c) shows a more common occurrence of multiplication. A double-digit number (14) is multiplied by a single digit number (3). Try these examples: = = Page 15

21 Check your answers: ,470 Notice that in examples 2 and 3, you needed to CARRY numbers from one column to the next. Carrying is a very important concept to remember when you multiply two numbers together when one of the numbers has more than one digit. Look at this example: 5, a. 26,170 b. 314,040 c. 523,400 d. 863,610 The answer to this example is 863,610. How did we find that? There are some very simple rules to multiplying multi-digit numbers. Most of these rules have to do with place values. Look at the example we just worked through. The bottom number in the problem, 165, can be broken down into its place values: 1 hundred + 6 tens + 5 ones. When you multiply 5,234 by 1 hundred + 6 tens + 5 ones, you can break down the problem into three steps. The first step, represented by the a above, shows the result of multiplying 5,234 by 5 ones. That answer is 26,170. Now, step b above shows the result of multiplying 5,234 by 6 tens (which can also be written as 60). That result is 314,040. Why isn t it just 31, 404? You must add the extra zero to move into the correct place value. If you didn t, you would be multiplying by 6 instead of 60. Finally, in step c you see the result of multiplying 5,234 by 1 hundred (written also as 100). That is 523,400. You add the two zeros at the right to have the correct place value. Now you have the problem in an expanded answer. You have the ones result: 26,170; the tens result: 314,040; and the hundreds result: 523,400. To find the correct answer d, simply add the a, b, and c together. Page 16

22 A simple way to remember how many zeroes to add to each answer is to count up. The ones column gets no zero. Tens get one zero. Hundreds get two zeroes. Thousands get three, and so on. In many ways, multiplication is like addition. And just as addition has an opposite subtraction multiplication has an opposite division. Division Division is the function that splits a number into parts. To divide a number, you simply determine how many parts you want. If you have 4 apples and you need to divide them into 2 parts, you know you should have 2 apples in each part. That can be written in equation form like this: 4 2 = 2. The same method applies to many business problems. If Bill s business has 14 customers in 7 days, he might want to know how many customers he has each day on average. To find the average, Bill will divide the total number of customers by the number of days in question. In this case, 14 7 = 2. So, Bill s business averaged 2 customers per day for that 7-day period. Let s hope those customers were good ones! What do you do if the numbers do not come out evenly? For example, if Bill had 15 customers, he would average 2 customers per day, but there would be 1 left over. That leftover number is called the remainder. You use the remainder in a fraction that has the remainder as the top number and the number you were dividing by as the bottom number. In this second example, Bill would average 2 1/7 customers each day. Now, you try these examples: 1. Joy owns a candy factory. If the candy factory puts out a total of 208 pounds of chocolate in 4 days, how many pounds per day did the factory put out? 2. Divide 146 by Harry writes 24 checks in 3 days. How many checks does he write per day? Page 17

23 Now check your answers: pounds of chocolate / checks per day In order to perform these operations, you might have to use long division. Long division is simply a way of setting up and figuring out a problem. To set up a long division problem like 155 5, write down the number you are dividing (155). Then put a horizontal line above it connected to a vertical line to the left of it. It should resemble an L lying on its side. Next, put the number you are dividing by (the 5) to the left of the vertical line. Your problem should look like this: Look at the problem on the previous page as you read through the next paragraph. Carefully follow the steps as described below. To divide 155 by 5, work from left to right. Find the first portion of the 155 that the 5 will go into. This, from left to right, is the first two numbers the 15 portion of 155. You know that 5 goes into 15 three times. Write the 3 directly above the 5 in 15 (the number that is farthest right). Next, multiply the 3 by the number you are dividing by the 5. That answer is 15 and needs to be written directly below the number you divided, the first 2 numbers in 155 (15). Finally, subtract the 15 from the 15 in 155 and write that answer below the 2 numbers. In this case, the answer is 0. The next step is to move to the right and bring down the next numeral in the number you are dividing. Put that number next to the 0. Now you have a 05 at the bottom of your problem. Take that number and divide it by the original 5. That answer, 1, goes next to the 3 on the top of the problem. Finally, multiply the 5 by the 1 in the answer. Subtract that number from the 5 at the bottom of the problem. The result of that subtraction is your remainder (in this case, 0). Page 18

24 Remember, in business, you will most often be using calculators to perform your basic math operations. We are reviewing these basic operations so that you will have insight into what the calculator is doing, so you will better understand how each function key works. Order of Operation As you look at your four basic math functions, you need to remember the Order of Operations. This refers to which function you perform first. If the equation looks something like this, for example: = you need to know where to start. The easiest way to remember is My Dear Aunt Sue. This is a memory aid for the order of operations: multiplication first, then division, then addition, and finally, subtraction. So, our example equation would be figured like this: 1. Multiply the 4 2 (the answer is 8). Put this answer in place of the 4 2 in the equation. 2. Add the 5 and 8 (the answer is 13). Insert this into the equation. 3. Finally, perform the remaining operation subtraction = 9 The correct answer is 9. This Order of Operations principle is important. Look at the same problem and figure it just by working from left to right = 14 Your answer in this method (14) is NOT correct. By remembering your order of operations, you will get the correct answer. Page 19

25 Try these samples: =? =? x 2 =? Check your answers: Complete these review exercises: 1. The four basic math functions are. a. less than, greater than, equal to, percent b. addition, subtraction, multiplication, division c. decimals, fractions, whole numbers, number line d. none of the above 2. If a number X is to the left of number Y on the number line, then we can say that number X is number Y. a. less than b. greater than c. equal to d. less than or equal to For questions 3-9 solve the equation and choose the correct answer from the choices a. 2,053 b. 1,943 c. 2,153 d. 1,755 Page 20

26 a. 1, b. 1, c. 1, d. 1, , a. 3, b. 3, c. 3, d. 3, =. a. 418 b. 455 c. 318 d. 1, =. a. 6,219 b. 603 c. 6,199 d. 519 Page 21

27 =. a. 1,676 b. 2,508 c. 1,698 d. 2, =. a. 255 b. 790 c. 164 d. 244 Now check your answers: 1. b 2. a 3. a 4. c 5. d 6. a 7. b 8. a 9. c Part I of your Business Math Tutor has reviewed the four basic math functions: addition, subtraction, multiplication, and division. In the remaining two parts of your Business Math Tutor, you will be learning how to use a calculator to help you perform these and other functions quickly and accurately. Also, you will be applying what you have learned to real-life situations from the business world. Page 22