CHAPTER 3: FRACTIONS CHAPTER 3 CONTENTS

Transcription

1 CHAPTER : FRACTIONS Image from Microsoft Office Clip Art CHAPTER CONTENTS. Introduction to Fractions. Multiplication of Fractions and of Mixed Numbers. Division of Fractions and of Mixed Numbers.4 Addition of Fractions and of Mixed Numbers. Subtraction of Fractions and of Mixed Numbers.6 Order of Operations.7 U.S. Measurement Conversions.8 Application Problems 7

2 CCBC Math 08 Introduction to Fractions Section. pages. Introduction to Fractions Before we begin, we will review some ideas that will be used in our studies of fractions. Prime Number: A prime number is a number greater than that is only divisible by and itself. The number is a prime number; it is only divisible by the number and. The number 4 is not a prime number because it is divisible by, and 4. Prime Factorization: The prime factorization of a number is the product of all of the prime numbers that equals the number. Example : Write the prime factorization of the numbers and 40. One way to determine the prime factorization of a number is to create a factor tree. We will create factor trees for both and 40 below. To create a factor tree, split each number into two factors that multiply to equal the number above. Continue to split the factors into factors until each factor is prime. A prime number is only divisible by and itself. The prime factors in these factor trees have a box around them. The product of all of the prime fractors at the bottom of the factor tree is the prime factorization The prime factorization for is: The prime factorization for 40 is: Practice : Write the prime factorization of the numbers 6 and 4. Answer: Greatest Common Factor: The Greatest Common Factor (GCF) of two numbers is the greatest number that divides both numbers evenly. One way to determine the GCF is to use the prime factorization. The product of all of the prime factors that are common in both numbers will give the GCF. 8

3 CCBC Math 08 Introduction to Fractions Section. pages Example : Determine the Greatest Common Factor of and 40. Denoted as GCF(, 40). The product of the prime factors the two numbers have in common equals the GCF. GCF(,40) = 8 Therefore, 8 is the largest number that divides and 40. Practice : Determine the Greatest Common Factor of 6 and 4. Denoted as GCF(6, 4). Answer: Least Common Multiple: The Least Common Multiple (LCM) of two numbers is the smallest number that both numbers divide evenly. One way to determine the LCM of two numbers is to multiply the two numbers together and divide that product by the GCF. FORMULA FOR DETERMINING THE LEAST COMMON MULTIPLE LCM( a, b) ab GCF( a, b) Example : Determine the Least Common Multiple of the numbers and 40. Denoted LCM(, 40). LCM(, 40) 40 GCF(, 40) Therefore, 60 is the smallest number that both and 40 divide evenly. 9

4 CCBC Math 08 Introduction to Fractions Section. pages Practice : Determine the Least Common Multiple of the numbers 6 and 4. Denoted LCM(6, 4). Answer: 7 We will be using GCF, LCM, and prime factorization as we work with fractions. You may need to refer back to this explanation as you work through this chapter. Fraction: Fractions are used in mathematics for many purposes. Most commonly fractions express the number of equal parts out of a whole (or out of a part), a division problem, a remainder, or a ratio (a comparison of two numbers). PARTS OF A FRACTION Fractions have two numbers that are separated by a fraction bar. Numerator Denominator This number indicates how many equal parts we have. This number indicates how many equal parts make up a whole. If a pizza is cut into eight equal pieces and you take three pieces you have the fraction 8, which is read three eighths or divided by 8 or out of 8. Equivalent Fractions: Changing the numerator and denominator of the fraction without changing its value forms an equivalent fraction. There are two ways to accomplish this: by expanding the fraction or by reducing the fraction. 40

5 CCBC Math 08 Introduction to Fractions Section. pages You can multiply the numerator and denominator by the same number and produce an equivalent fraction, and you can divide the numerator and denominator by the same number and produce an equivalent fraction. EQUIVALENT FRACTIONS a a c and a a c for b, c 0 b b c b b c A fraction whose numerator and denominator are the same number has the value unless the number is 0. For instance, the fraction, 8, means there are 8 pieces that make a whole and we 8 have 8 pieces. Therefore, we have whole. A FRACTION THAT EQUALS a when a is not 0. a Note: The fraction 0 0 is undefined. Reducing Fractions to Lowest Terms In some cases, fractions may be reducible (also called simplifying fractions); however, the numerator and the denominator of the fraction must have a common factor, which will evenly divide both numbers without a remainder. For example, if an egg carton that holds eggs has 4

6 CCBC Math 08 Introduction to Fractions Section. exactly 6 eggs in the container, you might say you have 6 pages of a carton of eggs. However, you might just as easily remark that you have of a carton of eggs. Both observations are correct since 6 is equivalent to. In fact, is 6 reduced to its lowest terms. Let s look at this more closely and determine how to do this mathematically so that we can reduce other fractions that we are not as familiar with. Example 4: Simplify 6 First, determine the prime factorization for the numbers 6 and. We will show how to do this again by factor trees as follows. 6 4 The prime factorization for 6 is: The prime factorization for is: 6 6 Rewrite the fraction using the prime factorization. Divide out a in the numerator and denominator. Note: Divide out a in the numerator and denominator. Note: Note: If all factors in the numerator or denominator are divided out, the number that will remains is a. 4

7 CCBC Math 08 Introduction to Fractions Section. Practice 4: Simplify 8 8. Answer: 7 pages Example : Simplify 6 Another way to simplify fractions is to divide the numerator and the denominator by the Greatest Common Factor (GCF). First determine the GCF for 6 and is 6. To simplify the fraction divide both the numerator and denominator by Since you cannot simplify any further, that is there are no common factors for and other than, the fraction is reduced to its lowest terms. Sometimes at first we do not choose the largest common factor to divide into both the numerator and denominator. In that case, just keep reducing until you cannot reduce any further. This problem could also be simplified in the following way. 6 6 First, divide both the numerator and denominator by. Simplify. 6 Then, divide both the numerator and denominator by. 6 4

8 CCBC Math 08 Introduction to Fractions Section. Practice : Simplify 8 8. Answer: 7 pages Example 6: Simplify 8 4 We will first do this problem by using the prime factorization of the numerator and denominator Determine the prime factorization of the numerator and denominator. Divide out common factors in the numerator and denominator. We can also simplify this fraction by diving the numerator and denominator by common factors = Since both 8 and 4 are even, they can be divided by. without a remainder. So, divide 8 by and 4 by Both 4 and can be divided by 7 without a remainder; so the fraction is not fully simplified. Divide both numbers by 7. The fraction is now fully simplified. Note that you could have gotten the answer in one step by dividing the original numerator and denominator by 4 (the greatest common factor for 8 and 4) Divide the numerator and denominator by the GCF. Unless directions indicate otherwise, you should simplify a fraction to its lowest terms. 44

9 CCBC Math 08 Introduction to Fractions Section. Practice 6: Simplify Answer: pages Expanding Fractions At times, especially when adding or subtracting fractions, we need to rewrite fractions in an equivalent form. To do this we will multiply the numerator and the denominator by the same number (usually a positive integer) EXPANDING FRACTIONS a ac for b, c 0 b b c Example 7: Rewrite by multiplying both the numerator and the denominator by Practice 7: Rewrite 7 by multiplying both the numerator and the denominator by. Answer: 4

10 CCBC Math 08 Introduction to Fractions Section. pages Example 8: Rewrite by multiplying both the numerator and denominator by. 6 0 Practice 8: Rewrite 7 by multiplying both the numerator and denominator by 4. Answer: As long as you multiply both the numerator and the denominator by the same number (except 0), you will obtain a fraction that is equal in value to the original fraction. There are three types of numbers that involve fractions: proper fractions, improper fractions and mixed numbers. Proper fraction: A proper fraction is any fraction whose numerator is smaller than its denominator. Some examples of proper fractions include: 6 4,,, 7 Improper fraction: A fraction whose numerator is equal to or greater than the denominator is an improper fraction. Some examples of improper fractions include: ,,,, Mixed number: When a number consists of both a whole number and a fraction number it is called a mixed number or a mixed fraction. The fraction part of a mixed number is a proper fraction. Some examples of mixed numbers include:,,, PARTS OF A DIVISION STATEMENT 8 4 Dividend Divisor Quotient Dividend Divisor 8 4 Quotient Quotient Divisor Dividend

11 CCBC Math 08 Introduction to Fractions Section. Example 9: Rewrite the division problem as a fraction. is the dividend and is the divisor, so as a fraction we have: pages Practice 9: Rewrite the division problem 6 as a fraction. Answer: 6 Example 0: Rewrite the fraction as you would perform long division. In previous discussions regarding division problems, when there was a remainder you simply wrote the remainder as a whole number; however, we often use fractions to express that remainder. Whole number part Denominator (See, it s as easy as,, ) Numerator What does mean? Picture a whole candy bar that has pieces. You have of these pieces. Practice 0: Rewrite the fraction 6 as you would perform long division. Answer: 6 47

12 CCBC Math 08 Introduction to Fractions Section. pages Example : Rewrite 7 as a mixed number. Divide the denominator into the numerator. Whole number part Denominator 7 Numerator Therefore, 7. Practice : Rewrite 8 as a mixed number. Answer: 8 Example : Rewrite 7 6 as a mixed number. Divide the denominator into the numerator. Whole number part 4 Denominator Numerator This answer is not finished. Look at the fraction on the mixed number. It can be simplified by dividing into the in the numerator and the 6 in the denominator. This gives the final answer: Therefore, Always simplify fractions (when possible) before finishing a problem. 48

13 CCBC Math 08 Introduction to Fractions Section. The previous problem could have been simplified first and then the long division performed as we will show below. pages 7 6 Write prime factorization of numerator and denominator. Divide out a in the numerator and denominator. 9 Simplify, and then complete long division using the simplified fraction. Divide the denominator into the numerator. Whole number part 4 Denominator As we saw before, Numerator Practice : Rewrite 8 as a mixed number. Answer: 6 Integers as Fractions Integers such as 0 or 7 or - or 8 sometimes need to be written as fractions. Since fractions represent division and dividing by does not change the value of the number, whole numbers can be written as a fraction over. That is, divided by. HOW TO WRITE AN INTEGER AS A FRACTION For any integer, a, 49 a a.

14 CCBC Math 08 Introduction to Fractions Section. pages Example : Write 0, 7,, and 8 as fractions. 0 = 0 7 = 7 8 = 8 Practice : Write 0,,, and as fractions. Answer: 0,,, Mixed Numbers When you have a mixed number, you may need to rewrite it as an improper fraction in order to work with it. In this section you will be introduced to an algorithm (procedure) for changing a mixed number to an improper fraction. In section., you will learn why this procedure works. To change a mixed number into an improper fraction, the numerator of the improper fraction will be determined by multiplying the whole number times the denominator of the mixed number and then adding this product to the numerator of the mixed number. The denominator of the improper fraction will be the same as the mixed number. HOW TO CHANGE A MIXED NUMBER TO AN IMPROPER FRACTION b ( Ac) b A c c b b ( Ac) b A A c c c Note: b b A A c c Example 4: Rewrite 8 as an improper fraction. Multiply the denominator with the whole number, and add that product to the numerator. Then place that resulting sum in the numerator, and keep the same denominator. (8)

15 CCBC Math 08 Introduction to Fractions Section. Practice 4: Rewrite 4 as an improper fraction. Answer: 4 pages Example : Rewrite 4 9 as an improper fraction. 4 (9) Practice : Rewrite 7 as an improper fraction. Answer: Example 6: Rewrite as an improper fraction. The number means. We will first determine the improper fraction in the parentheses and then that number will be negative. ( ) 0 Note: Disregard the negative sign until the mixed number has been turned into an improper fraction. Practice 6: Rewrite 4 as an improper fraction. Answer: 4 Mixed numbers are sometimes needed when finding the mean and median value of a set of data. Recall that to compute an arithmetic mean (an average), add all of the values and divide that sum by the number of values. If the answer is not a whole number we can write the result as a mixed number or fraction.

16 CCBC Math 08 Introduction to Fractions Section. Example 7: Find the mean of 7, 4, 80, 94, 6. Add all of the values and divide by the number of values = 9 Practice 7: Determine the mean of 8, 76,, 98. This is the answer as an improper fraction. pages Since 9 does not divide evenly by, we write the mean as a mixed number by performing long division. Answer: 78 Also recall that to find the median of a set of data values, arrange the values in numerical order and locate the middle. If there are an odd number of data values, then the median is the number in the middle of the list. If there is an even number of data values, then there are two data values in the middle of the list: compute the mean (the average) of these two middle values to determine the median. Again, if it is not a whole number we can write the result as a mixed number or a fraction. Example 8: Find the median of 0, 49,, 6, 67, 97. We begin by listing the values in ascending order: 97, 6, 49, 0,, 67. Note that there is an even number (six) of data values. The two middle data values are -49 and 0. We find the mean (average) of these: This is the answer as an improper fraction Since -9 does not divide evenly by, we can write the result as a mixed number by performing long division. A negative divided by a positive is a negative. Therefore our answer is negative. Practice 8: Determine the median of, -9,, -, 6, -6. Answer:

17 CCBC Math 08 Introduction to Fractions Section. Order of Fractions pages When fractions have the same denominator they can be placed in order from least to greatest by comparing their numerators. The following fractions all have the denominator 7. Since they have a common denominator they are arranged from least to greatest according to their numerators. 7, 7, 7, 4 7, 7, 6 7, 7 7 When fractions do not have a common denominator, we will need to first determine the common denominator and then make all of the fractions into equivalent fractions with the common denominator. The Least Common Denominator will be the Least Common Multiple of all the denominators. Example 9: Order the following fractions from least to greatest.,,,, To put fractions in order we must first write all of the fractions with a common denominator. The Least Common Multiple of the denominators and 6 is the smallest number they divide. Multiples of :, 6, 9, Multiples of 6: 6,, 8 The LCM(, 6) = 6 Now, write an equivalent fractions for each fraction with the denominator of ,,,, ,,,, ,,,, Multiply the numerator and denominator by. This fraction already has a denominator of 6. This fraction already has a denominator of 6. Multiply the numerator and denominator by. This fraction already has a denominator of 6. These are the equivalent fraction for,,,, with a common denominator of 6. Put the fractions in order from least to greatest from the smallest numerator to the largest numerator. Then write the fractions in their original form

18 CCBC Math 08 Introduction to Fractions Section. Practice 9: Order the following fractions from least to greatest.,,,, Answer:,,,, Example 0: Determine the median of the following numbers.,,,, pages To determine the median, we must first put the numbers in order. To put fractions in order we must first write all of the fractions with a common denominator. The Least Common Multiple of the denominators, 4 and 8 is the smallest number they all divide. Multiples of :, 4, 6, 8, 0,, 4, 6 Multiples of 4: 4, 8,, 6, 0, 4 Multiples of 8: 8, 6, 4 The LCM(, 4, 8) = 8 Now, write an equivalent fractions for each fraction with the denominator of Multiply the numerator and denominator by 4. 8 This fraction already has a denominator of Multiply the numerator and denominator by Multiply the numerator and denominator by. 8 This fraction already has a denominator of 8. The equivalent fraction for,,,, with a common denominator of 8 are: ,,,, Put the fractions in order from least to greatest from the smallest numerator to the largest numerator ,,,, ,,,, Determine the middle number Therefore, 4 is the median value. 8 4

19 CCBC Math 08 Introduction to Fractions Section. Practice 0: Determine the median of the following numbers. 4,,,, Answer: Example : Determine the mode of the following numbers.,,,, pages To determine the mode we must first look at all of the fractions in order from least to greatest. In Example 0, we have already put this data set in order from least to greatest ,,,, The mode is the data value that is repeated the most often. As you can see the data value 4 8 is repeated the most often. The fraction 4 8 is equal to. Therefore, is the mode. Practice : Determine the mode of the following numbers. 4,,,, Answer: Zero and Fractions Earlier in your studies, you discovered that the number 0 has some special qualities. When you are working with fractions, there are a couple of things that you need to keep in mind. When the numerator is 0 (and the denominator is not 0), the value of that fraction is 0. Thus, 0 = 0. To illustrate why this is true, let s look closely at this equation. 6 Example : Simplify 0 6 We know that this is a division problem of 0 6, which has a quotient of 0. We can express this as 0 6 = 0. On the other hand, if the denominator is 0, you have the case of division by 0. As you learned earlier in this book, you cannot divide by 0: the answer is undefined. Fractions give us a numeric way to demonstrate why you cannot divide by 0.

20 CCBC Math 08 Introduction to Fractions Section. Practice : Simplify 0 Answer: 0 pages Example : Simplify because multiplying the denominator, 0, by the whole number quotient, 7, does not 0 produce, 7: Similarly, 7 0 because multiplying the denominator, 0, by the whole number quotient, 0, does 0 not produce, 7: As you can see, division by 0 is not defined; we say that division by 0 is undefined. Practice : Simplify Answer: undefined RULES FOR FRACTIONS WITH ZERO If 0 is divided by any number (except 0), the answer is n If any number is divided by 0, the answer is undefined. n undefined 0 Watch All: 6

21 CCBC Math 08 Introduction to Fractions Section. pages. Introduction Exercises. The fraction 4 is equivalent to. The fraction 6 is equivalent to. Write 9 as a fraction. 4. Write 98 as a fraction.. Write as a fraction. 6. Write 4 as a fraction. 7. The numerator of 7 0 is and the denominator is. 8. The numerator of 00 is and the denominator is. 9. Simplify 4 0. Simplify 6 4. Simplify 77 by dividing 4 and by 4. by dividing 6 and 4 by 6. by dividing and 77 by.. Simplify Simplify Simplify 6 6. Simplify 60. Simplify Simplify Convert to an improper fraction. 9. Convert 4 7 to an improper fraction. 0. Convert to an improper fraction.. Arrange from smallest to largest:,,,, 4. Arrange from smallest to largest:,,,

22 CCBC Math 08 Introduction to Fractions Section. pages. Write 4 4. Write 78 8 as a mixed number in simplest form. as a mixed number in simplest form.. Find the mean of,,,, 7, Find the mean of, 0, 0, 7,. 7. Find the mean of 8,, 8,. 8. Find the median of,, 4 9. Find the median of 0 4 7,,,, 0. Find the mode of 0 4,,,,,,,

23 CCBC Math 08 Introduction to Fractions Section. pages. Introduction Exercises Answers ,,,, 4 7. numerator: 7, denominator: 0.,,, numerator: 00; denominator: =