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Transcript
4.1 – Fractions and Mixed Numbers Fractions a b Defn: Numbers that show the number of parts existing compared to the number of parts in a whole. Numerator (a): the top number of a fraction that describes the number of parts existing. Denominator (b): the bottom number of the fraction that describes the number of parts that make a whole. 4.1 – Fractions and Mixed Numbers Write a fraction to represent the shaded portion of each figure. 2 5 5 8 7 12 4.1 – Fractions and Mixed Numbers Write a fraction to represent the shaded portion of each figure. 7 10 3 5 5 6 4.1 – Fractions and Mixed Numbers Draw and shade each fraction. 3 7 4.1 – Fractions and Mixed Numbers Proper Fractions Defn: A fraction whose numerator is smaller than its denominator. Improper Fractions Defn: A fraction whose numerator is larger than its denominator. Mixed Numbers Defn: A number which is made up of an integer and a fraction. 4.1 – Fractions and Mixed Numbers Classify each of the following fractions: 15 7 23 29 proper 47 33 improper 5 4 9 proper 27 7 mixed number 61 62 85 improper mixed number 4.1 – Fractions and Mixed Numbers Converting Mixed Numbers to Improper Fractions 1. Multiply the denominator by the integer. 2. Add the numerator to the product of the denominator and the integer. 3. Write the sum as the numerator over the original denominator. 2 7 5 2 35 2 37 5 7 7 7 7 2 3 6 2 18 2 6 3 3 3 20 3 4.1 – Fractions and Mixed Numbers Converting Mixed Numbers to Improper Fractions 7 10 10 7 100 7 107 10 10 10 10 10 11 12 8 11 96 11 107 8 12 12 12 12 4.1 – Fractions and Mixed Numbers Converting Improper Fractions to Mixed Numbers 1. Divide the numerator by the denominator. 2. The quotient is the integer of the mixed number. 3. The remainder is the numerator over the original denominator. 9 5 1 5 9 5 4 4 1 5 4.1 – Fractions and Mixed Numbers Converting Improper Fractions to Mixed Numbers 2 23 9 23 9 18 5 4 62 13 62 13 52 10 5 2 9 10 4 13 4.2 – Factors and Simplest Form Divisibility Tests 1. A whole number is divisible by 2 if the number is even. 354 968 140 2. A whole number is divisible by 3 if the sum of the digits is divisible by 3. 126 1 2 6 9 (is divisible by 3) 24831 2 4 8 3 1 18 (is divisible by 3) 3. A whole number is divisible by 4 if the last 2 digits are divisible by 4. 236 (36 is divisible by 4) 10,528 (28 is divisible by 4) 4.2 – Factors and Simplest Form Divisibility Tests 4. A whole number is divisible by 5 if the number ends in a 0 or a 5. 140 1265 345 5. A whole number is divisible by 6 if it is divisible by both 2 and 3. 126 1 2 6 9 (is divisible by 2 and 3) 24834 2 4 8 3 4 21 (is divisible by 2 and 3) 6. A whole number is divisible by 9 if the sum of the digits is divisible by 9. 936 9 3 6 18 (is divisible by 9) 4.2 – Factors and Simplest Form A Number as a Product of Prime Numbers Factor Trees 24 24 2 3 12 2 2 6 2 8 3 3 2 3 2 2 23 4 2 2 3 2 2 23 2 3 4.2 – Factors and Simplest Form A Number as a Product of Prime Numbers Factor Trees 72 2 210 36 2 105 2 18 2 9 3 3 2 2 2 3 3 2 3 32 5 21 3 7 2 35 7 4.2 – Factors and Simplest Form Simplest Form Defn: A fraction is in simplest form when the numerator and denominator have no other common factors other than 1. 30 45 15 2 3 2 2 35 3 335 45 30 5 5 9 3 3 4.2 – Factors and Simplest Form Simplest Form Write in Simplest Form. 49 112 7 77 22227 112 49 7 2 56 2 28 14 2 2 7 7 16 4.2 – Factors and Simplest Form Simplest Form Write in Simplest Form. 64 common factor is 4 20 16 16 64 5 20 5 4.2 – Factors and Simplest Form Simplest Form Write in Simplest Form. 7a 3 2 common factor is 7a 56a 2 a 7a 3 56a 2 8 a 8 4.2 – Fractions and Simplest Form Equivalent Fractions – Two Methods 7 21 Are and equivalent ? 9 27 1. Simplify each fraction. 7 7 21 9 27 9 7 7 9 9 2. Cross Multiply. 7 21 9 21 7 27 9 27 189 189 Fractions are equivalent. 4.2 – Fractions and Simplest Form Equivalent Fractions – Two Methods 6 34 Are and equivalent ? 15 85 1. Simplify each fraction. 2. Cross Multiply. 6 34 2 3 2 17 15 85 3 5 5 17 6 34 15 34 6 85 15 85 2 2 5 5 510 510 Fractions are equivalent. 4.2 – Fractions and Simplest Form Equivalent Fractions – Two Methods 12 10 Are and equivalent ? 39 36 1. Simplify each fraction. 12 10 39 36 12 10 39 36 2 23 25 3 13 2 2 33 5 4 13 18 2. Cross Multiply. 12 36 39 10 432 390 Fractions are not equivalent. 4.3 – Multiplying and Dividing Fractions Multiplying Fractions 1. Multiply the numerators. 2. Multiply the denominators. 3. The product of the numerators remains as the numerator as the product of the denominators remains as the denominator. 3 5 15 3 5 7 11 77 7 11 1 1 3 9 1 1 39 1 27 4.3 – Multiplying and Dividing Fractions Multiplying Fractions 3 1 6 7 6 7 3 1 11 4 77 8 77 8 11 4 1 1 4 3 43 27 8 27 8 9 2 1 18 1 1 92 3 44 4.3 – Multiplying and Dividing Fractions Multiplying Fractions 4 33 11 16 3 1 1 3 4 33 1 4 11 16 1 4 1 1 2 3 y 2 3 y 1 1 y 3 2 1 1 3 2 1 1 3 4 y 4.3 – Multiplying and Dividing Fractions Multiplying Fractions a3 b 2 2 b a 3 4 3 a 3 1 a b 2 2 b a b 1 a 1 b 1 3 3 3 4 4 4 33 43 a b 27 64 4.3 – Multiplying and Dividing Fractions Multiplying Fractions 1 5 1 3 25 1 3 25 1 1 5 6 10 16 2 2 16 6 10 16 2 2 5 64 4.3 – Multiplying and Dividing Fractions Dividing Fractions 1. Write the reciprocal of the second fraction (the divisor). 2. Change the division operator to multiplication. 3. Work the problem as a multiplication problem. 4 2 8 9 1 9 4 8 5 45 8 2 7 1 7 2 7 5 1 4 1 9 2 20 7 4.3 – Multiplying and Dividing Fractions Dividing Fractions 5 10 9 45 10 2 59 4 4 2 4 9 4 1 1 1 3 1 3 1 1 3y 3 3y 5y 3 2 2 20 y 4 5y2 45 y 4 y 4.3 – Multiplying and Dividing Fractions Dividing Fractions 1 3 2 9 7 2 9 15 1 3 15 45 3 14 7 1 7 7 3 14 15 49 1 7 3 9 Evaluate the expression x y, if x and y . 4 2 1 1 1 1 1 3 2 3 9 6 23 4 9 4 2 2 3 4.3 – Multiplying and Dividing Fractions Dividing Fractions 9 9 Is a solution of the equation 2 x ? YES 8 4 12 9 1 9 9 9 9 9 2 1 4 4 1 8 4 4 8 4 9 9 4 4 1 3 2 9 7 2 9 15 1 3 15 45 3 14 7 1 7 7 3 14 15 49 1 7 4.3 – Multiplying and Dividing Fractions Dividing Fractions Hershey Park is located in Pennsylvania. Of its sixty rides, one-sixth of them are roller coasters. How many roller coasters are in Hershey Park? 1 of 60 6 10 1 60 1 10 1 10 roller coasters 60 6 1 1 1 6 1