Basic Math Review. Chapter. Objectives. Key Terms

Transcription

1 Chapter 3 Basic Math Review Objectives After reading this chapter, you will be able to: Identify whole numbers. Perform basic calculations using addition and subtraction. Perform basic calculations using multiplication and division. Set up and use fractions. Compare fractions, express them as decimals, and find common denominators. Perform basic mathematical calculations using fractions and decimals. Understand and use percentages. Perform basic mathematical calculations using percentages. Key Terms Addition Common denominator Cross-multiplication Decimal number Denominator Difference Division Factor Fractions Improper fraction Mixed number Multiplication Number Numerator Percentages Product Proper fraction Quotient Subtraction Sum Whole numbers

2 60 UNIT One Foundation 77Chapter Overview This chapter will review basic mathematical skills that are necessary to perform more complex calculations. Calculations in the pharmacy setting require a basic understanding of addition, subtraction, division, multiplication, fractions, percentages, and decimals. 77 Numbers A number is a quantity or amount that is made up of one or more numerals. There are several kinds of numbers: whole numbers, fractions, and decimal numbers. Examples of whole numbers include 2, 10, 100, 2,000, and 5,000,000. Examples of fractions include 1 4, 3 /10, and 7 /8. Examples of decimals include 0.2, 0.75, and 2.6. A number is represented by one or more numerals. 77Whole Numbers Our number system is a base-10 number system. That is, it is based on the number 10. We use 10 digits, also called whole numbers (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). Whole numbers are counting numbers. We use whole numbers to count. 77Addition and Subtraction Many dosage calculations involve addition and subtraction. With addition, you add something to a given number Figure 3.1. The sum is the amount obtained when adding numbers (see Table 3.1 ). With subtraction, you take something away from a given number Figure 3.2. The difference is the amount obtained when subtracting numbers (see Table 3.2 ). There is an inverse relationship between addition and subtraction. Addition means that you are adding or increasing the quantity by a given number. Adding a zero to a number does not change the value of the original number. The value of zero is nothing. Examples: = 5; = 10; = 210. Subtraction means that you are taking away or reducing the quantity by a given number. Subtracting a zero from a number does not change the value of the original number. The value of zero is nothing. Examples: 7 0 = 7; = 128; 21 0 = Figure 3.1 Addition example using a number line.

3 CHAPTER 3 Basic Math Review 61 Table 3.1 Addition Strategy Fact Strategy Example Answer (Sum) Plus Plus Plus Plus Plus Plus Plus Plus Plus Plus Plus Figure 3.2 Subtraction example using a number line. Table 3.2 Subtraction Strategy Fact Strategy Example Answer (Difference) Minus Minus Minus Minus Minus Minus Minus Minus Minus Minus Minus Multiplication and Division Multiplication and division are used constantly in performing pharmacy calculations. Sometimes, you must enlarge or reduce the quantity or determine a part or portion of a quantity needed. Multiplication and division are the mathematical processes used to do this. The product is the amount obtained by multiplying numbers. The quotient is the amount obtained by dividing one number by another number. Table 3.3 shows a multiplication chart.

4 62 UNIT One Foundation Table 3.3 Multiplication Chart Division is the inverse relationship of multiplication. Example: 9 5 = = = 9 Example: 7 6 = = = 7 When multiplying numbers, any number multiplied by zero always equals zero. Example: = 0 Example: = 0 Example: 4 0 = 0 When dividing numbers, any number divided by zero always equals zero. Example: = 0 Example: 69 0 = 0 Example: 78 0 = 0 77 Fractions Many dosage calculations use amounts other than whole numbers for example, fractions. Pharmacy technicians work with these amounts to fill orders and complete pharmaceutical calculations. Pharmacy technicians must be able to recognize fractions and understand what amounts they represent. Fractions express a quantity or portion that is a part of a whole number Figure 3.3. It is a ratio of a part to the whole. The number above the fraction line is called the numerator. The number below the fraction line is called the denominator. The denominator tells how many equal parts are in the whole or set. The numerator tells you the number of those parts you are expressing.

5 CHAPTER 3 Basic Math Review 63 Fractions express a quantity that is a part of a whole number. The numerator is the number above the fraction line. The denominator is the number below the fraction line. Another way to think about fractions is by dividing a pie into parts. For example, imagine a pie that has been divided into eight pieces or parts. Each piece is a fraction, or 1 8 of the whole pie Figure 3.4. A number line can be a useful tool when thinking about fractions Figure 3.5. A proper fraction has a numerator that is smaller than the denominator. The value of this type of fraction is always less than 1. Examples of a proper fraction include 1 5, 2 3, and Numerator Denominator Figure 3.3 Defining a fraction Figure 3.5 Fraction example using a number line. The value of a proper fraction is always less than 1. An improper fraction has a numerator that is equal to or larger than the denominator. The value of this type of fraction is always equal to or greater than 1. When the numerator and denominator are the same, the value of the fraction is always 1, because any number divided by itself always equals 1. Examples of an improper fraction include 2 2, 4 /3, and 9 /7. The value of an improper fraction is always equal to or greater than 1. A mixed number contains both whole numbers and fractions. Examples of mixed numbers include and 2 4 /7. A mixed number has whole numbers and fractions. You can convert a mixed number to an improper fraction by multiplying the denominator by the whole number and adding the numerator Figure 3.6. You can rewrite a fraction greater than 1 as a mixed number or as a whole number Figure 3.7. For example, to write 7 /4 as a mixed number, follow these steps: 1. Divide the numerator by the denominator. 2. Use the remainder to write the fraction part of the quotient. A fraction is in its simplest form when its numerator and denominator have no common factor other than 1 Figure 3.8. A factor is a number that you multiply by another number, which is also a factor, to make yet another number. For example, the simplest form of 6 /18 is 1 3. Figure 3.4 Fraction example using a pie = (( 3 1) + 2) = Figure 3.6 Converting a mixed fraction to an improper fraction. 7 4 = Figure 3.7 Division example using fractions. 6 3 = Figure 3.8 Simplifying a fraction. When a fraction is in its simplest form, the numerator and denominator have no common factor other than 1. You can divide the numerator and denominator by common factors until the only common factor is 1. To convert a mixed number to an improper fraction, multiply the denominator by the whole number. Then add the numerator to this calculated amount and place it over the same denominato

6 64 UNIT One Foundation Reduce every mixed number and improper fraction. To simplify a fraction, divide the numerator and denominator by a common factor until the only common factor is 1. Comparing Fractions Comparing fractions helps to ensure that correct calculations are used in the practice of pharmacy and in the normal work tasks for pharmacy technicians. Fractions can be compared using denominators that are the same or different, as well as numerators that are the same or different. Using fractions in pharmacy work is a simple and helpful mathematical concept to be mastered by the pharmacy technician. Fractions with Like Denominators Comparing fractions with the same denominators is similar to comparing whole numbers Figure 3.9. When comparing fractions that have the same denominator, the fraction with the larger numerator will have the larger value. For example, 3 5 is greater than 2 5. When two fractions have the same denominator, the fraction with the larger numerator has a larger value. Fractions with Like Numerators When two fractions have the same numerator, the fraction with the smaller denominator has the larger value Figure When two fractions have the same numerator, the fraction with the smaller denominator has a larger the value. If the numerator is the same, the smaller the denominator, the larger the value Figure If the denominator is the same, the smaller the numerator, the smaller the value Figure Figure 3.9 Same denominator fraction graph.

7 CHAPTER 3 Basic Math Review 65 Figure 3.10 Different denominator fraction pie graph. Figure 3.11 Which tablet is bigger? Figure 3.12 Which tablet is smaller? Adding and Subtracting Fractions When you add and subtract fractions, it often helps to find the lowest common denominator. A common If both fractions have the same denominators, add the numerators and keep denominator is a number into which both denominators can divide evenly. the denominator the same. When adding fractions that have the same denominators, add the numerators and keep the denominator the same. Example: = 3 7 Rules for Adding Fractions When adding fractions, first convert the fractions to the lowest common denominator (if necessary). Then, add the numerators, placing the sum over denominator. Finally, reduce the resulting fraction.

8 66 UNIT One Foundation In order to add fractions with different denominators, you must first find the lowest common denominator. For example, suppose you want to add The lowest number that can be divided evenly by both 3 and 4 is 12. Therefore, the common denominator is 12. Example: /4 = 4 / = 6 /12 Answer: 4 / /12 = 10 /12 When adding fractions with different denominators, first find the lowest common denominator. If both fractions have the same denominator, subtract the numerators and keep the denominator the same. You can further simplify this to 5 /6 by dividing both the numerator and the denominator by their common factor of 2: 10 / = 5 /6 When subtracting fractions that have the same denominators, subtract the numerators and keep the denominator the same. Example: 5 /7 2 7 = 3 7 In order to subtract fractions with different denominators, you must first find the lowest common denominator. For example, suppose you want to subtract 8 / The lowest number that can be divided evenly by both 10 and 5 is 10. Therefore, the common denominator is 10. Example: 8 / / = 8 / = 4 /10 When subtracting fractions with different denominators, first find the lowest common denominator. Answer: 8 /10 4 /10 = 4 /10 You can further simplify this answer to 2 5 by dividing both the numerator and the denominator by the least common factor, which is 2: 4 / = 2 5 Rules for Subtracting Fractions When subtracting fractions, first convert the fractions to the lowest common denominator (if necessary). Next, subtract the numerators, placing the difference over the denominator. Finally, reduce the resulting fraction. Multiplying and Dividing Fractions It is important to understand the mathematical concepts used in multiplying and dividing fractions. Many dosage calculations will require the pharmacy technician to calculate the amount of drug required using multiplication and division with fractions prior to a final check by the pharmacist. Multiplying Fractions To multiply fractions, multiply the numerators of each fraction and the denominators of each fraction.

9 CHAPTER 3 Basic Math Review 67 Example: /5 2 4 = = 15 Answer: 8 /15 Rules for Multiplying Fractions When multiplying fractions, first convert any mixed numbers into improper fractions (if necessary). Next, multiply the numerators to obtain a new numerator. Then multiply the denominators to obtain a new denominator. Finally, reduce the resulting fraction. Dividing Fractions To divide fractions, invert or reverse the numbers of the second fraction in the equation. Then multiply the numerators of each fraction and the denominators of each fraction, as you do when multiplying fractions. This is known as invert and multiply. Example: / Answer: 8 /3, which simplifies to Rules for Dividing Fractions When dividing fractions, first convert any mixed numbers into improper fractions (if necessary). Next, find the reciprocal of the divisor (the second fraction). Then multiply the first number by the reciprocal. Finally, reduce the resulting fraction. 77 Decimals Not every number is a whole number. Decimal numbers are numbers that are written using place value. The decimal system includes 10 digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). Using these digits, you can express numbers of all sizes using a decimal point. The decimal point, which looks like a period, is the central character in the decimal system. Any value can be expressed using a combination of these numerals. Digits to the left of the decimal point indicate whole numbers, and digits to the right indicate fractions of a whole. The value of a digit increases by a multiple of 10 each time it moves one space to the left. Equally, the value of a digit decreases by a multiple of 10 each time it moves one space to the right. The scheme of the decimal system is depicted in Table 3.4. In this case, the number 63, is used to illustrate decimal-system notation. The total value of a number expressed by the decimal system is the sum of all its digits according to their place to the right or left of the decimal point. Therefore, in the sample number, 63, is equal to the following: 77 60, (6 ten-thousands) (3 thousands) (2 hundreds) (0 tens) (7 ones)

10 68 UNIT One Foundation Table 3.4 Decimal System Notation Value Ten-thousands Thousands Hundreds Tens Ones Decimal point Tenths Hundredths Thousandths Ten-thousandths Sample Number (5 tenths) (1 hundredth) (8 thousandths) (4 ten-thousandths) Any fraction can be written in decimal form. The position of a number in relation to the decimal point indicates the value of that number. Digits to the left of the decimal point indicate whole numbers, and digits to the right of the decimal point indicate fractions of a whole. Each place value is multiplied by 10 as you move left from the decimal point. Each place value is divided by 10 as you move right from the decimal point. In pharmacy practice, when using the decimal system to express a number less than 1, a leading zero (0) is placed to the left of the decimal point. Trailing zeros, or zeros that are placed to the right of the final digit, are not used in pharmacy. These techniques help eliminate errors when reading decimals. If a leading zero is not included or a trailing zero is included, and the decimal point may not be read. That means a dose may be misinterpreted by a factor of 10, 100, or even 1,000. For example, consider the following: 2 5 = 2 5 = 0.4. This would not be written as.4,.40, or In this case, if 0.4 represented a dose, and a leading zero were not included, the value could be misinterpreted as 4 or 40, resulting in respective overdoses of 10 or 100 times the intended dose. Or, if a Think of money to help understand decimals and their place values. One dollar can be thought of as having 100 parts, with each part being equal to one penny. A dollar can also be thought of as having 10 parts, with each part then being equal to one dime. Always include a leading zero when expressing doses that are less than 1. dose of 10mg was written as 10.0, the value could easily be interpreted as 100mg. A leading zero is placed to the left of the decimal point when expressing a number less than 1. Do not include a trailing zero in pharmacy calculations, except when it is critical to the accuracy of the measurement. Never use a trailing zero when a dose is expressed as a whole number. The exception to the rule of no trailing zeros is when the zero is considered significant to the dose or calculation, or when rounding results in a trailing zero for example, rounded to the nearest hundredth is 0.80 (written with a trailing zero). Another example is the use of a trailing zero in calculations involving currency Figure For

11 CHAPTER 3 Basic Math Review 69 THIS IS NOT LEGAL TENDER AND SHALL NOT BE USED FOR SUCH 6 F B 6 6 F B = W ASH INGTON 1 dollar dollar dollar = 1.11 dollars Figure 3.13 Practical decimal demonstration. Table 3.5 Different Ways to Express a Decimal Form Example Standard Word forty-five and thirty-eight hundredths Expanded (4 10) + (5 1) + (3 0.1) + (8 0.01) example, 10 dollars and 70 cents is written mathematically as $10.70, not as $10.7, without the trailing zero. There are many ways to write the same decimal, as shown in Table 3.5. Adding and Subtracting Decimals When adding and subtracting decimals, the numbers should be placed in columns so that the decimal points are all aligned. For example: Align the decimal points before adding and subtracting decimal numbers Multiplying Decimals When multiplying decimals, you multiply them as whole numbers, and then move the decimal the total number of places that were in the two numbers being multiplied Figure There is an implied decimal point for all whole numbers, which is placed at the end of the smallest or last digit in the number. For Multiply the decimals as whole numbers first and then move the decimal over the total number of places. example, the whole number 76 is also the same as 76.0, with the decimal point and zero being the implied decimal. Thus, when multiplying decimals, the starting point for moving the decimal begins with the implied decimal point at the end of the whole number. For example, to multiply , you first multiply = 1,068,844. The implied decimal point is 1,068,844.0 and the movement of the decimal places begins after the number four in the ones digit place. Next, count from right to left three decimal places (because there are three total decimals in 23.6 and 45.29) and place the decimal point there: 1, Figure 3.14 Moving decimals.

12 70 UNIT One Foundation Change each decimal to a whole number by multiplying each number by the same factor of 10 before dividing decimals. Dividing Decimals When dividing decimals, first change each decimal to a whole number by multiplying each number by the same factor of 10. Then proceed with the division operation. Example: = = 210 Answer: = Example: = = 125 Answer: = 2 77 Percentages Pharmacy calculations often require the use of percentages. A percentage is represented by the % symbol and is used to express the number of parts of one hundred. It represents the same number as a fraction whose denominator is 100. Examples include the following: 7 0.9% = 77 2% = 2 / /100 = 5% / Tech Math Practice Question: What is the sum of ? Answer: 354 Question: What is the sum of ? Answer: 62 Question: What is the difference of ? Answer: 91 Question: What is the difference of ? Answer: 357 Question: What is 8 6? Answer: 48

13 CHAPTER 3 Basic Math Review 71 Question: What is 12 11? Answer: 132 Question: What is 99 9? Answer: 11 Question: What is 108 9? Answer: 12 Question: What type of fraction is 4 /3? Answer: Improper fraction Question: What type of fraction is 2 7? Answer: Proper fraction Question: What type of fraction is 1 7 8? Answer: Mixed number Question: What is the simplest form of the fraction 20 /40? Answer: 20 / = 10 / = 1 2 Question: What is the simplest form of the fraction 8 /24? Answer: 8 /24 4 /4 = = 1 3 Question: What is the simplest form of the fraction 12 /16? Answer: 12 /16 4 /4 = 3 4 Question: Which of the following doses of drug is the smallest dose? 3 10 mg tablet, 7 10 mg tablet, or 2 10 mg tablet? Answer: The 2 10 mg tablet is the smallest dose because when fractions have the same denominator, the fraction with the smaller numerator has the smallest value. Question: Which of the following doses of drug is the largest dose? 1 /15 mg tablet, 3 /15 mg tablet, or 7 /15 mg tablet? Answer: The 7 /15 mg tablet is the largest dose because when fractions have the same denominator, the fraction with the larger numerator has the largest value.

14 72 UNIT One Foundation Question: Which of the following tablets is the smallest dose? 3 /10 mg tablet, 3 /12 mg tablet, or 3 /20 mg tablet? Answer: The 3 /20 mg tablet is the smallest dose because when fractions have the same numerator, the fraction with the larger denominator has the smallest value. Question: Which of the following tablets is the largest dose? 4 /7 mg tablet, 4 /12 mg tablet, or 4 /15 mg tablet? Answer: The 4 /7 mg tablet is the largest dose because when fractions have the same numerator, the fraction with the smallest denominator has the largest value. Question: What is the sum of /6? Answer: 5 /6 Question: What is the sum of ? Answer: 5 /8 Question: What is the sum of /10? Answer: To obtain the sum, first determine the lowest common denominator: = 2 /10. Then complete the addition operation: 2 / /10 = 4 /10, which simplifies to 2 5. Question: What is the sum of ? Answer: To obtain the sum, first determine the lowest common denominator: /7 = 14 /21 and = 9 /21. Then complete the addition operation: 14 / /21 = 23 /21. Note that 23 /21 is an improper fraction, and can be simplified to 1 12 /21. Question: What is the difference of 8 /9 3 9? Answer: 5 /9 Question: What is the difference of ? Answer: /9 3 /9 = 3 /9 = 1 /3 Question: What is the sum of /12? Answer: To obtain the sum, first determine the lowest common denominator: = 9 /24 and 3 / = 6 /24. Then complete the addition operation: 9 / /24 = 15 /24. Question: What is the difference of 4 /12 2 6? Answer: To obtain the difference, first determine the lowest common denominator: 4 / = 4 /12 and = 4 /12. Then complete the subtraction operation: 4 /12 4 /12 = 0.

15 CHAPTER 3 Basic Math Review 73 Question: What is the product of ? Answer: 6 /20, which simplifies to 3 /10. Question: What is the product of /7? Answer: 12 /35 Question: What is the product of 4 /12 3 8? Answer: 12 /96, which simplifies to 1 8 Question: What is the quotient of 4 /7 3 7? Answer: 28 /21, which simplifies to Question: What is the quotient of 2 /11 1 2? Answer: 4 /11 Question: What is the quotient of 4 /9 5 /8? Answer: 32 /45 Question: What is the value of the digit 4 in ? Answer: The digit 4 is in the hundredths place. It has a value of 0.04, or 4 hundredths. Question: What is the value of the digit 2 in ? Answer: The digit 2 is in the ten-thousandths place. It has a value of , or 2 tenthousandths. Question: What is the sum of ? Answer: Question: What is the sum of ? Answer:

16 74 UNIT One Foundation Question: What is the difference of ? Answer: Question: What is the quotient of ? Answer: To determine the quotient, first multiply the dividend (the first number) by 10: = 66. Next, multiply the divisor (the second number) by 10: = 22. Finally, solve the problem: = 3. Question: What is the quotient of ? Answer: To determine the quotient, first multiply the dividend by 10: = 48. Next, multiply the divisor by 10: = 12. Finally, solve the problem: = 4. Question: What is the quotient of ? Answer: To determine the quotient, first multiply the dividend by 10: = 49. Next, multiply the divisor by 10: = 7. Finally, solve the problem: 49 7 = 7. Question: How do you write ¼ as a percent? Answer: To write ¼ as a percent, first find the equivalent fraction with a denominator of 100: /25 = 25 / /100 = 25%. Another method is to divide the top of the fraction (the numerator) by the bottom of the fraction (the denominator) and multiply the result by 100: 1 4 = = 25%. Question: How do you write 2 /20 as a percent? Answer: To write 2 /20 as a percent, first find the equivalent fraction with a denominator of 100: 2 /20 5 /5 = 10 / /100 = 10%.

17 CHAPTER 3 Basic Math Review 75 WRAP UP 77Chapter Summary 77 Numbers can be represented as whole numbers, fractions, and decimals. 77 Each type of number can be used in addition, subtraction, multiplication, and division. 77 Addition and subtraction involve adding numbers to each other or taking numbers away from each other, respectively. 77 Multiplication involves enlarging the quantity of number by multiplying a number by another number. Division involves reducing the quantity of a number to determine a part or portion of a quantity needed. Cross-multiplication is a mathematical concept that is very useful to the pharmacy technician when performing calculations. Fractions are used to express a quantity or portion that is a part of a whole number. It is a ratio of a part to the whole. The numerator is the number above the fraction line, while the denominator is the number below the fraction line. The denominator tells you how many equal parts are in the whole or set. The numerator tells you the number of those parts you are expressing. Decimals are numbers that are used to repre- sent a place value. Any fraction can be represented using a decimal point. Percentages are used to express the number of parts of 100. A percentage represents the same number as a fraction whose denominator is 100, and is denoted by the % symbol. Fractions, decimals, and percentages can be converted to each other and used in addition, subtraction, multiplication, and division equations. When adding fractions, first convert the frac- tions to the lowest common denominator (if necessary). Then, add the numerators, placing the sum over denominator. Finally, reduce the resulting fraction. When subtracting fractions, first convert the fractions to the lowest common denominator (if necessary). Next, subtract the numerators, placing the difference over the denominator. Finally, reduce the resulting fraction. 77 When multiplying fractions, first convert any mixed numbers into improper fractions (if necessary). Next, multiply the numerators to obtain a new numerator. Then multiply the denominators to obtain a new denominator. Finally, reduce the resulting fraction. 77 When dividing fractions, first convert any mixed numbers into improper fractions (if necessary). Next, find the reciprocal of the divisor (the second fraction). Then multiply the first number by the reciprocal. Finally, reduce the resulting fraction. 77 Pharmacy technicians need to be comfortable performing basic mathematical equations with all the concepts in this chapter. 77Learning Assessment Questions 1. Which of the following fractions has the highest value? A B. 2 3 C. 4 /9 D. 5 /9 2. What is a fraction with a value of less than 1 called? A. Mixed number B. Proper fraction C. Improper fraction D. Whole number 3. What is a fraction with a value greater than 1 called? A. Mixed number B. Proper fraction C. Improper fraction D. Whole number 4. What is 12 /16, reduced to its simplest form? A. 12 /16 B. 6 /8 C. 3 4 D. 4 /7

18 76 UNIT One Foundation 5. Which dose is larger? A. 1 4 of a 300mg tablet B. 1 2 of a 300mg tablet C. 1 3 of a 300mg tablet D. 1 /10 of a 300mg tablet 6. What is the value of the digit 3 in 2, ? A. 0.3 B C D A percentage represents the same number as a fraction whose denominator is? A. 50 B. 25 C. 100 D What is the product of 9 7? A. 54 B. 63 C. 72 D What is the sum of ? A B C D What is the value of the digit 7 in 3, ? A. 0.7 B C D What is the value of the digit 4 in the number ? A. 4 tenths B. 4 hundredths C. 4 thousandths D What is the product of ? A B. 50 C. 0.5 D /20 is the same as what percent? A. 50% B. 60% C. 30% D. 20% is the same as what percent? A. 25% B. 55% C. 75% D. 33% 15. Convert 0.40 to a fraction. A. 8 /10 B. 1 5 C. 4 /10 D. 40 / Convert 4 /8 to a decimal. A. 0.5 B. 0.2 C. 0.4 D. 0.8

19 CHAPTER 3 Basic Math Review 77 WRAP UP 17. What is the difference in ? A. 9.4 B. 9.3 C. 6.6 D What is the product of ? A B C D What is the sum of ? A. 0 B. 978 C. 980 D What is the quotient of 978,657 0? A. 978,657 B. 987,756 C. 0 D. 978,665

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