Multiplying and Dividing Fractions
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1 Multiplying and Dividing Fractions 1
2 Overview Fractions and Mixed Numbers Factors and Prime Factorization Simplest Form of a Fraction Multiplying Fractions and Mixed Numbers Dividing Fractions and Mixed Numbers 2
3 Intro to Fractions and Mixed Numbers 3
4 What is a Fraction? Has two parts: Numerator: How many Denominator: What: think of as how big 4 of 7 parts are shaded Numerator is 4 Denominator is 7 4
5 Example The 3 is how many: Numerator The 4 is how big how many equal parts: Denominator 3 4 5
6 Give the Numerator and Denominator
7 Solution 3 7 Num: 3, Denom: Num: 13, Denom: Num: 11, Denom: Num: 9, Denom: 2 7
8 Notation For typing ease, 3 is written 3/4 4 8
9 3/4 = 3 divided by 4 Fractions Mean Division 9/9 = 1 since 9 x 1 = 9 11/1 = 11 since 11 x 1 = 11 9
10 What about Zero??? 0 = 0. There is nothing to count and 0 x 6 = =??? This makes no sense. How can you have 6 parts of 0 something that has no size, no parts? Also, what times 0 = 6???? For any number n: 0/n = 0; n/0 is undefined it is not a number 10
11 Mnemonic If k and n are numbers: 0/k is OK n/0 is NO; no number 11
12 Write the Fraction Shaded 12
13 Solution 3/10 1/2 13
14 Write the Fraction Shaded 14
15 Solution There are 16 circles. 11 are shaded Numerator is 11, denominator is 16 11/16 are shaded 15
16 Improper Fractions and Mixed Numbers In an improper fraction, the numerator is greater than or equal to the denominator, for example 5/4 4/4= 1 + 1/4 = 5/4 = 1 1/4 1 1/4 is called a mixed number; it has a whole number and a fraction 16
17 Examples Proper Fraction, Improper Fraction or Mixed Number? a. 6/7 b. 13/12 c. 2/2 d. 99/101 e. 1 7/8 f. 93/74 17
18 Solution Proper Fraction, Improper Fraction or Mixed Number? a. 6/7 Proper b. 13/12 Improper c. 2/2 Improper d. 99/101 Proper e. 1 7/8 Mixed f. 93/74 Proper 18
19 Mixed Numbers as Fractions For any mixed number, we have a whole number and a fraction A whole number can be written as the number of parts in the whole If we want to write 2 in terms of 1/2, we have two 1/2s in each whole, or 2 x 2 = 4 halves total 1/2 1/2 1/2 1/2 If each small rectangle is 1/2, each large one is 2 halves, and we have 4 halves total = 4/2 19
20 Rule: Mixed Number to Fraction Step 1: Multiply the denominator by the whole number Step 2: Add the numerator to the result of Step 1 Step 3: Write the result of Step 2 as the numerator of the improper fraction 1 2/3 =? Step 1: 3 x 1= 3 Step 2: = 5 Step 3: 5/3 1 2/3 =5/3 20
21 Examples a. 2 5/7 b. 5 1/3 c. 9 3/10 d. 1 1/5 21
22 Solution a. 2 5/7 = 19/7 b. 5 1/3 = 8/3 c. 9 3/10 = 93/10 d. 1 1/5 = 6/5 22
23 Rule: Fraction to Mixed Number Step 1: Divide the numerator by the denominator Step 2: The whole number is the quotient, the remainder is the numerator which is written over the denominator 5/3 =? Step 1: 5 3 = 1 remainder 2 Step 2: 5/3 = 1 2/3 5/3 = 1 2/3 23
24 Examples a. 9/5 = b. 23/9= c. 48/4 = d. 62/13 = e. 51/7 = f. 21/20 = 24
25 Solution a. 9/5 = 1 4/5 b. 23/9= 2 5/9 c. 48/4 = 12 d. 62/13 = 4 10/13 e. 51/7 = 7 2/7 f. 21/20 = 1 1/20 25
26 Summary of Terms Fraction Numerator Denominator Proper Fraction Improper Fraction Mixed Number 26
27 Factors and Prime Factorization 27
28 What is a Factor? A factor of a number is a second number that divides the first number evenly: 2 is a factor of 4 9 is a factor of is a factor of is not a factor of 10!!! What are the factors of 24? 1, 2, 3, 4, 6, 8, 12, 24 28
29 Prime Numbers A prime number is a number that has only two unique factors (1 and itself, but note that the definition excludes 1 as prime!) Prime Numbers: 2, 3, 5, 7, 11, 13, 17 What is the next prime number? Numbers that are not prime are composite 29
30 Prime or Composite?
31 Solution 15 = 5 x 3, composite 23 prime 37 prime 49 = 7 x 7, composite 51 prime 31
32 Prime Factorization Prime factorization means writing a number in terms of its factors, where each factor is prime 15 = 3 x 5 21 = 3 x 7 25 = 5 x 5 24 = 2 x 2 x 2 x 3 32
33 Factor Trees: Find the prime factorization of 24: 24 = 3 x 8; 3 is prime 8 = 2 x 4; 2 is prime 4 = 2 x 2; 2 is prime 24 = 2 x 2 x 2 x 3 33
34 Find the Prime Factorization
35 Solution 30 = 2 x 3 x 5 56 = 2 x 2 x 2 x 7 72 = 2 x 2 x 2 x 3 x = 3 x 3 x 13 35
36 Simplest Form of a Fraction 36
37 Equivalent Fractions Equivalent fractions are fractions that have the same value Examples: 1/2 = 2/4 = 3/6 = 5/10 4/5 = 8/10 = 20/100 Since n x 1 = n, for any number n, and k/k = 1 for any number k We can create equivalent fractions by multiplying or dividing by k/k 37
38 Simplifying Fractions To simplify a fraction, divide out any factors that are in common in the numerator and the denominator Step 1: Find the prime factorization of the numerator and the denominator Step 2: Remove common factors from the numerator and denominator. In other words, remove factors of 1 from the fraction Example: 3 = 3 = 1 x 3 = 1 x 1 = x
39 Examples: Write in Simplest Form a. 6/60 = b. 72/26 = c. 30/108 = d. 7/16 = 39
40 Solutions a. 6/60 = =1/10 b. 72/26 = 36/13 c. 30/108 = 5/18 d. 7/16 = 7/16 40
41 Multiplying Fractions and Mixed Numbers 41
42 Multiplying Fractions 2 x 1/2 = = = 1 1/2 x 1/2 = read as 1/2 of 1/2 = 1/4 42
43 a x c = a x c b d b x d Rule Examples: 1/3 x 2/5 = 2/15 2/3 x 5/11 = 10/33 Usually write in simplest form 43
44 Examples a. 3/8 x 5/7 = b. 1/3 x 1/6 = 44
45 Solutions a. 3/8 x 5/7 = 15/56 b. 1/3 x 1/6 = 1/18 45
46 With Mixed Numbers Same as fractions, but first change the mixed number to a fraction Examples: 2 1/3 x 1/4 = 7/3 x 1/4 = 7/12 3 1/3 x 7/8 = 10/3 x 7/8 = 70 / 24 (= 35/12 = 2 11/12) 46
47 Examples a. 2/3 x 18 = b. 2 1/2 x 8/15 = c. 3 1/5 x 2 3/4 = d. 5 x 3 11/15 = e. 0 x 9/11 47
48 Solutions a. 2/3 x 18 = 12 b. 2 1/2 x 8/15 = 4/3 c. 3 1/5 x 2 3/4 = 44/5 d. 5 x 3 11/15 =56/3 e. 0 x 9/11 = 0 48
49 Dividing Fractions and Mixed Numbers 49
50 Dividing Fractions 1/2 2 = 1/4 = 1/4 Think: How many 2s in 1/2? Just 1/4 1/2 1/2 = 1 Think: How many 1/2s in 1/2? Just 1 50
51 Rule for Dividing Fractions Step 1: Invert the second fraction Step 2: Multiply the first fraction by the inverse (Invert and Multiply) Inverses: Inverse of 5/4 is 4/5 Inverse of 9/8 is 8/9 Inverse of 12/5 is 5/12 51
52 Division Examples 4/7 3/5= Inverse of 3/5 is 5/3 Multiply: 4/7 x 5/3 = 20/21 4/15 2/3 = Inverse of 2/3 is 3/2 Multiply: 4/15 x 3/2 = 12/30 = 2/5 52
53 Mixed Numbers Just as with multiplication, make the mixed number a fraction Example: 3 1/4 1/5 13/4 1/5 = 13/4 x 5 = 65/4 53
54 Review To multiply fractions: Multiply numerators then multiply denominators: 3/4 x 2/3 = (3 x 2) / ( 4 x 3) = 6/12 = ½ To multiply mixed numbers: Change to fractions then multiply: 1 ½ x 1/3 = 3/2 x 1/3 = 3/6 =1/2 To divide fractions: Invert the second fraction then multiply: 1/2 3/5 = 1/2 x 5/3 = 5/6 To divide mixed numbers, change to a fraction first A prime number has only two unique factors, 1 and itself Every composite number can be factored by primes Primes are useful for simplifying fractions (and other things) 54
55 Some More Examples a. 4/9 7= b. 8/15 3 4/5= c. 2 2/7 2 3/14= d. 14/17 0= e /8= 55
56 Solutions a. 4/9 7= 4/63 b. 8/15 3 4/5=8/15 x 6/19 = 8/57 c. 2 2/7 2 3/14= 46/31 d. 14/17 0= undefined e /8= 0 56
57 Arithmetic with fractions Multiplying Dividing Mixed Numbers Prime Factoring Simplifying fractions Summary 57
58 Adding and Subtracting Fractions 58
59 Overview Adding and Subtracting Fractions Greatest Common Multiple Unlike Fractions Mixed numbers Order, exponents, order of operations 59
60 The Key to Adding and Subtracting Fractions You have to add and subtract equal objects: 1/4s and 1/4s, 1/6s and 1/6s It s like adding cats and dogs and monkeys you can t do it unless you make them all animals, all mammals, etc. In other words, they have to all be the same Once you have equal objects, just add the numerators and simplify the fraction 1/4 + 3/4 = 4/4 = 1 3/5 + 1/5 = 4/5 60
61 Rule Step 1: Find a common denominator Sept 2: Use equivalent fractions to make the fractions have the same denominator Step 3: Add or subtract the numerators Step 4: Simplify 1/4 + 1/2 = Step 1: denominator = 4 Step 2: 1/4 + 2/4 Step 3: = 3/4 2/5-1/10 = 4/10 1/10 = 3/10 61
62 Finding the Common Denominator Need to find a number that is a multiple of both denominators 1/5 and 1/6? Need 1/30 3/4 and 1/2? Need 1/4 (or 1/8) 2/9 and 3/2? Need 1/18 How to find? 1. Can always multiply the denominators, but that can be a big number and create extra work Ideally, you want to work with the smallest number possible 2. Find the least common multiple through prime factoring 62
63 Finding the Least Common Multiple (LCM) Say have denominators of 12 and = 2 x 2 x 3 18 = 2 x 3 x 3 To be a multiple of both denominators, need 2 x 2 x 3 x 3 = x 3 = x 2 = 36 Note: Sometimes you can figure it out in your head feel free 63
64 Examples: Find the LCM 8 and 10 8 and and 30 20, 24 and 45 64
65 Solutions 8 and 10; 40 8 and 16; and 30; , 24 and 45;
66 Add the Fractions 1/6 + 5/18 = 5/6 + 2/9 = 2/5 + 4/9 7/12 5/24 9/10 3/7 66
67 Solution 1/6 + 5/18 = 4/9 5/6 + 2/9 = 19/18 2/5 + 4/9 = 38/45 7/12 5/24 = 3/8 9/10 3/7 = 33/70 67
68 Mixed Number Step 1: Add or subtract the fractions Step 2: Add or subtract the whole numbers Step 3: Combine and simplify 4 2/ /6 = = 9 2/5 + 1/6 = 12/30 + 5/30 = 17/30 = 9 17/30 68
69 Follow same steps, but 3 1/4 1 1/2 =? Subtraction Fractions: 1/4 1/2 = 1/4-2/4 = -1/4 Whole Numbers: 3 1 = 2 Combining: How do we combine 2 and 1 1/4? Make the 2 = 1 + 4/4 then combine: /4 1/4 3 1/4 1 1/2 = 1 3/4 69
70 Examples: a. 4 2/ /6 = b. 2 5/14 = 5 6/7 c / /5 = d. 29 7/9 13 5/18 e. 9 7/15 5 3/5 70
71 Solutions a. 4 2/ /6 = 9 17/30 b. 2 5/14 = 5 6/7= 8 3/14 c / /5 = 16 2/35 d. 29 7/9 13 5/18=16 1/2 e. 9 7/15 5 3/5 = 3 13/15 71
72 Only One Whole Number or Fraction? 72
73 Which is larger 1/4 or 1/3? Comparing Fractions The larger the number of pieces, the smaller each piece is Can always make sure by getting a common denominator The Hamburger Fiasco (why you need to study math) 73
74 Examples: Which is Larger a. 1/7 or 1/9 b. 8/9 or 10/11 c. 3/5 or 2/9 74
75 a. 1/7 or 1/9: 1/7 Solution: Which is Larger b. 8/9 or 10/11: 10/11 c. 3/5 or 2/9 : 3/5 75
76 Raising Fractions to Powers a 2 = a x a b 4 = b x b x b x b We can always count the number of terms (a/b) 2 = (a/b) x (a/b) = (axa) /(bxb) = a 2 /b 2 76
77 Examples a. (1/5) 2 = b. (2/3) 3 = c. (1/4) 2 (2/3) 3 = 77
78 Solutions a. (1/5) 2 = 1/25 b. (2/3) 3 =8/27 c. (1/4) 2 (2/3) 3 =1/54 78
79 Order of Operations Think of this as mathematical grammar 1. Perform everything inside parentheses 2. Evaluate exponents 3. Multiply or divide, left to right 4. Add or subtract, left to right PEMDAS: Please Excuse My Dear Aunt Sally P parentheses, E exponents, M &D multiply and divide A & S add and subtract 79
80 Examples a. 2/9 4/7 x 3/10 = b. (2/5) 2 (3/5 11/25) = 80
81 Solution a. 2/9 4/7 x 3/10 =7/60 b. (2/5) 2 (3/5 11/25) = 1 81
82 Arithmetic with fractions Multiplying Dividing Mixed Numbers Prime Factoring Simplifying fractions Summary 82
83 Fraction Review 1) 4/5 x 3/4 = 2) 2/3 5/2 = 3) 3 1/4 x 2 2/3 = 4) 5 1/6 3 1/3 = 5) 3/4 + 2/9 = 6) 5/7 2/3 = 7) 2 1/ /5 = 8) 3 1/3 2 4/5 = 9) 1 1/2 2 1/3 = 10) Simplify 65/52 83
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