Multiplying and Dividing Fractions

1 11 Multiplying and Dividing Fractions2 Overview Fractions and Mixed Numbers Factors and Prime Factorization Simplest Form of a Fraction Multiplying Fractions and Mixed Numbers Dividing Fractions and Mixed Numbers3 Intro to Fractions and Mixed Numbers4 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 75 Example34 The 3 is how many: NumeratorThe 4 is how big how many equal parts: Denominator6 Give the Numerator and Denominator37135111927 Solution3 713511192 Num: 3, Denom: 7 Num: 13, Denom: 5 Num: 11, Denom: 1 Num: 9, Denom: 28 NotationFor typing ease, 3 is written 3/449 Fractions Mean Division3/4 = 3 divided by 49/9 = 1 since 9 x 1 = 911/1 = 11 since 11 x 1 = 1110 What about Zero???0 = 0. There is nothing to count and 0 x 6 = 066 = ??? This makes no sense. How can you have 6 parts of 0something that has no size, no parts?

2 Also, what times 0 = 6????For any number n:0/n = 0; n/0 is undefined it is not a number11 MnemonicIf k and n are numbers: 0/k is OK n/0 is NO; no number12 Write the Fraction Shaded13 Solution3/101/214 Write the Fraction Shaded15 Solution There are 16 circles. 11 are shaded Numerator is 11, denominator is 16 11/16 are shaded16 Improper Fractions and Mixed NumbersIn an improper fraction, the numerator is greater than or equal to the denominator, for example 5/41 1/4 is called a mixed number; it has a whole number and a fraction4/4= 11/4= 5/4 = 1 1/4+17 ExamplesProper Fraction, Improper Fraction or Mixed Number?a. 6/7b. 13/12c. 2/2d. 99/101e. 1 7/8f. 93/7418 SolutionProper Fraction, Improper Fraction or Mixed Number?a. 6/7 Properb. 13/12 Improperc. 2/2 Improperd. 99/101 Propere. 1 7/8 Mixedf. 93/74 Proper19 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 wholeIf we want to write 2 in terms of 1/2, we have two 1/2s in each whole, or 2 x 2 = 4 halves totalIf each small rectangle is 1/2, each large one is 2 halves, and we have 4 halves total = 4/21/21/21/21/220 Rule: Mixed Number to FractionStep 1: Multiply the denominator by the whole numberStep 2: Add the numerator to the result of Step 1 Step 3: Write the result of Step 2 as the numerator of the improper fraction1 2/3 =?

3 Step 1: 3 x 1= 3 Step 2: 2 + 3 = 5 Step 3: 5/3 12/3=5/321 Examplesa. 2 5/7b. 5 1/3c. 9 3/10d. 1 1/522 Solutiona. 2 5/7 = 19/7b. 5 1/3 = 8/3c. 9 3/10 = 93/10d. 1 1/5 = 6/523 Rule: Fraction to Mixed NumberStep 1: Divide the numerator by the denominatorStep 2: The whole number is the quotient, the remainder is the numerator which is written over the denominator5/3 = ?Step 1: 5 3 = 1 remainder 2 Step 2: 5/3 = 1 2/35/3 = 1 2/324 Examplesa. 9/5 =b. 23/9=c. 48/4 =d. 62/13 =e. 51/7 =f. 21/20 =25 Solutiona. 9/5 = 1 4/5b. 23/9= 2 5/9c. 48/4 = 12d. 62/13 = 4 10/13e. 51/7 = 7 2/7f. 21/20 = 1 1/2026 Summary of Terms Fraction Numerator Denominator Proper Fraction Improper Fraction Mixed Number27 Factors and Prime Factorization28 What is a Factor? A factor of a number is a second number that divides the first number evenly:2 is a factor of 49 is a factor of 1810 is a factor of 100100 is not a factor of 10!

4 !! What are the factors of 24?1, 2, 3, 4, 6, 8, 12, 2429 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, What is the next prime number? Numbers that are not prime are composite30 Prime or Composite? 15 23 37 49 5131 Solution 15 = 5 x 3, composite 23 prime 37 prime 49 = 7 x 7, composite 51 prime32 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 333 Factor Trees: Find the prime factorization of 24:24 = 3 x 8; 3 is prime8 = 2 x 4; 2 is prime4 = 2 x 2; 2 is prime24 = 2 x 2 x 2 x 334 Find the Prime Factorization 30 56 72 11735 Solution 30 = 2 x 3 x 5 56 = 2 x 2 x 2 x 7 72 = 2 x 2 x 2 x 3 x 3 117 = 3 x 3 x 1336 Simplest Form of a Fraction37 Equivalent Fractions Equivalent Fractions are Fractions that have the same value Examples:1/2 = 2/4 = 3/6 = 5/104/5 = 8/10 = 20/100 Since n x 1 = n, for any number n,and k/k = 1 for any number kWe can create equivalent Fractions by Multiplying or Dividing by k/k38 Simplifying FractionsTo simplify a fraction, divide out any factors that are in common in the numerator and the denominatorStep 1: Find the prime factorization of the numerator and the denominatorStep 2: Remove common factors from the numerator and denominator.

5 In other words, remove factors of 1 from the fractionExample:3 = 3 = 1 x 3 = 1 x 1 = 16 2 x 3 2 3 2 239 Examples: Write in Simplest Forma. 6/60 =b. 72/26 =c. 30/108 =d. 7/16 =40 Solutionsa. 6/60 = =1/10b. 72/26 = 36/13c. 30/108 = 5/18d. 7/16 = 7/1641 Multiplying Fractions and Mixed Numbers42 Multiplying Fractions 2 x 1/2 = 1/2 x 1/2 = read as 1/2 of 1/2 = 1= = 1/443 Rulea x c = a x cb d b x dExamples:1/3 x 2/5 = 2/152/3 x 5/11 = 10/33 Usually write in simplest form44 Examplesa. 3/8 x 5/7 =b. 1/3 x 1/6 =45 Solutionsa. 3/8 x 5/7 = 15/56b. 1/3 x 1/6 = 1/1846 With Mixed NumbersSame as Fractions , but first change the mixed number to a fractionExamples:2 1/3 x 1/4 = 7/3 x 1/4 = 7/123 1/3 x 7/8 = 10/3 x 7/8 = 70 / 24 (= 35/12 = 2 11/12)47 Examplesa. 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 9/1148 Solutionsa. 2/3 x 18 = 12b.

6 2 1/2 x 8/15 = 4/3c. 3 1/5 x 2 3/4 = 44/5d. 5 x 3 11/15 =56/3e. 0 x 9/11 = 049 Dividing Fractions and Mixed Numbers50 Dividing Fractions 1/2 2 = 1/4 1/2 1/2 = 1 = 1/4 Think: How many 1/2s in 1/2? Just 1 Think: How many 2s in 1/2? Just 1/451 Rule for Dividing FractionsStep 1: Invert the second fractionStep 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 5/1252 Division Examples4/7 3/5=Inverse of 3/5 is 5/3 Multiply: 4/7 x 5/3 = 20/214/15 2/3 =Inverse of 2/3 is 3/2 Multiply: 4/15 x 3/2 = 12/30 = 2/553 Mixed Numbers Just as with multiplication, make the mixed number a fraction Example:3 1/4 1/513/4 1/5= 13/4 x 5 = 65/454 Review To multiply Fractions : Multiply numerators then multiply denominators: 3/4 x 2/3 = (3 x 2) / ( 4 x 3) = 6/12 = To multiply mixed numbers: Change to Fractions then multiply: 1 x 1/3 = 3/2 x 1/3 = 3/6 =1/2 To divide Fractions : Invert the second fraction then multiply.

7 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)55 Some More Examplesa. 4/9 7=b. 8/15 3 4/5=c. 2 2/7 2 3/14=d. 14/17 0=e. 0 2 1/8=56 Solutionsa. 4/9 7= 4/63b. 8/15 3 4/5=8/15 x 6/19 = 8/57c. 2 2/7 2 3/14= 46/31d. 14/17 0= undefinede. 0 2 1/8= 057 Summary Arithmetic with Fractions Multiplying Dividing Mixed Numbers Prime Factoring Simplifying fractions5858 Adding and Subtracting Fractions59 Overview Adding and Subtracting Fractions Greatest Common Multiple Unlike Fractions Mixed numbers Order, exponents, order of operations60 The Key to Adding and Subtracting Fractions You have to add and subtract equal objects:1/4s and 1/4s, 1/6s and 1/6sIt s like adding cats and dogs and monkeys you can t do it unless you make them all animals, all mammals, etc.

8 In other words, they have to all be the same Once you have equal objects, just add the numerators and simplify the fraction1/4 + 3/4 = 4/4 = 13/5 + 1/5 = 4/561 RuleStep 1: Find a common denominatorSept 2: Use equivalent Fractions to make the Fractions have the same denominatorStep 3: Add or subtract the numeratorsStep 4: Simplify1/4 + 1/2 = Step 1: denominator = 4 Step 2: 1/4 + 2/4 Step 3: = 3/42/5 - 1/10 = 4/10 1/10 = 3/1062 Finding the Common DenominatorNeed to find a number that is a multiple of both denominators1/5 and 1/6? Need 1/303/4 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 workIdeally, you want to work with the smallest number possible2. Find the least common multiple through prime factoring63 Finding the Least Common Multiple (LCM)Say have denominators of 12 and 1812 = 2 x 2 x 318 = 2 x 3 x 3To be a multiple of both denominators, need 2 x 2 x 3 x 3 = 3612 x 3 = 3618 x 2 = 36 Note: Sometimes you can figure it out in your head feel free64 Examples: Find the LCM 8 and 10 8 and 16 25 and 30 20, 24 and 4565 Solutions 8 and 10; 40 8 and 16; 16 25 and 30; 150 20, 24 and 45; 36066 Add the Fractions 1/6 + 5/18 = 5/6 + 2/9 = 2/5 + 4/9 7/12 5/24 9/10 3/767 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 = 33/7068 Mixed NumberStep 1: Add or subtract the fractionsStep 2: Add or subtract the whole numbersStep 3: Combine and simplify4 2/5 + 5 1/6 = 4 + 5 = 92/5 + 1/6 = 12/30 + 5/30 = 17/30= 9 17/3069 Subtraction Follow same steps, but3 1/4 1 1/2 = ?

9 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: 2 1 + 4/4 1/43 1/4 1 1/2 = 1 3/470 Examples:a. 4 2/5 + 5 1/6 =b. 2 5/14 = 5 6/7c. 10 + 2 6/7 + 3 1/5 =d. 29 7/9 13 5/18e. 9 7/15 5 3/571 Solutionsa. 4 2/5 + 5 1/6 = 9 17/30b. 2 5/14 = 5 6/7= 8 3/14c. 10 + 2 6/7 + 3 1/5 = 16 2/35d. 29 7/9 13 5/18=16 1/2e. 9 7/15 5 3/5 = 3 13/1572 Only One Whole Number or Fraction?73 Comparing Fractions Which is larger 1/4 or 1/3? 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)74 Examples: Which is Largera. 1/7 or 1/9b. 8/9 or 10/11c. 3/5 or 2/9 75 Solution: Which is Largera. 1/7 or 1/9: 1/7b. 8/9 or 10/11: 10/11c. 3/5 or 2/9 : 3/576 Raising Fractions to Powers a2= a x a b4= b x b x b x bWe can always count the number of terms (a/b) 2= (a/b) x (a/b) = (axa) /(bxb) = a2/b277 Examplesa.

10 (1/5) 2=b. (2/3) 3=c. (1/4) 2(2/3) 3=78 Solutionsa. (1/5) 2= 1/25b. (2/3) 3=8/27c. (1/4) 2(2/3) 3=1/5479 Order of OperationsThink of this as mathematical grammar1. Perform everything inside parentheses2. Evaluate exponents3. Multiply or divide, left to right4. Add or subtract, left to rightPEMDAS: Please Excuse My Dear Aunt SallyP parentheses, E exponents, M &D multiply and divideA & S add and subtract80 Examplesa. 2/9 4/7 x 3/10 =b. (2/5)2 (3/5 11/25) =81 Solutiona. 2/9 4/7 x 3/10 =7/60b. (2/5)2 (3/5 11/25) = 182 Summary Arithmetic with Fractions Multiplying Dividing Mixed Numbers Prime Factoring Simplifying fractions83 Fraction Review1) 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 1/4 + 3 1/5 =8) 3 1/3 2 4/5 =9) 1 1/2 2 1/3 =10) Simplify 65/52