Fraction Operations Review for Mastery: Least Common Multiple The smallest number that is a multiple of two or more numbers is called the least common multiple (LCM) of those numbers. To find the least common multiple of 3, 6, and 8, list the multiples for each number and put a circle around the LCM in the three lists. Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24 Multiples of 6: 6, 12, 18, 24, 30, 36, 42 Multiples of 8: 8, 16, 24, 32, 40, 48, 56 So 24 is the LCM of 3, 6, and 8. List the multiples of each number to help you find the least common multiple of each group. 1. 3 and 4 2. 5 and 7 3. 8 and 12 Multiples of 3: ________ Multiples of 5: ________ Multiples of 8: ________ Multiples of 4: ________ Multiples of 7: ________ Multiples of 12: _______ LCM: _______________ LCM: _______________ LCM: ______________ 4. 2 and 9 5. 4 and 6 6. 4 and 10 Multiples of 2: ________ Multiples of 4: ________ Multiples of 4: ________ Multiples of 9: ________ Multiples of 6: ________ Multiples of 10: _______ LCM: _______________ LCM: _______________ LCM: ______________ 7. 2, 5, and 6 8. 3, 4, and 9 9. 8, 10, and 12 Multiples of 2: ________ Multiples of 3: ________ Multiples of 8: ________ Multiples of 5: ________ Multiples of 4: ________ Multiples of 10: _______ Multiples of 6: ________ Multiples of 9: ________ Multiples of 12: _______ LCM: _______________ LCM: _______________ LCM: ______________ Holt McDougal Mathematics Name _______________________________________ Date __________________ Class __________________ Fraction Operations Review for Mastery: Adding and Subtracting with Unlike Denominators Unlike fractions have different denominators. To add and subtract fractions, you must have a common denominator. The least common denominator (LCD) is the least common multiple of the denominators. To add or subtract unlike fractions, first find the LCD of the fractions. 2 + 1 3 4 Multiples of 4: 4, 8, 12, … Multiples of 3: 3, 6, 9, 12, … The LCD is 12. Next, use fraction strips to find equivalent fractions. Then use fraction strips to find the sum or difference. 8 + 3 = 11 12 12 12 So, 2 + 1 = 11 . 3 4 12 Use fraction strips to find each sum or difference. Write your answer in simplest form. 1. 1 + 1 2. 5 − 2 3. 3 − 1 4. 3 + 3 4 8 6 3 4 3 5 10 _______________ _______________ _______________ ________________ 5. 3 + 1 6. 1 + 3 7. 2 − 1 8. 1 − 1 4 6 2 8 3 6 3 4 _______________ _______________ _______________ ________________ Holt McDougal Mathematics Name _______________________________________ Date __________________ Class __________________ Fraction Operations Review for Mastery: Regrouping to Subtract Mixed Numbers You can use fraction strips to regroup and subtract mixed numbers. To find 31 − 1 3 , first model the first mixed number in the expression. 4 4 There are not enough 1 pieces to subtract, so you have to regroup. 4 Trade one one-whole strip for four 1 pieces, because 4 = 1. 4 4 Now there are enough 1 pieces to subtract. Take away 13 . 4 4 The remaining pieces represent the difference. Write the difference in simplest form. 3 1 − 13 = 12 = 11 4 4 4 2 Use fraction strips to find each difference. Write your answer in simplest form. 1. 3 1 − 2 3 2. 3 1 − 15 3. 4 3 − 17 4. 3 1 − 2 2 4 4 6 6 8 8 3 3 _______________ _______________ _______________ ________________ 5. 5 5 − 2 7 6. 3 3 − 1 9 7. 5 1 − 15 8. 4 − 11 12 12 10 10 8 8 3 _______________ _______________ _______________ ________________ 9. 3 1 − 13 10. 2 1 − 17 11. 3 − 11 12. 6 3 − 2 5 8 8 8 8 4 8 8 _______________ _______________ _______________ ________________ Holt McDougal Mathematics Name _______________________________________ Date __________________ Class __________________ Fraction Operations Review for Mastery: Solving Fraction Equations: Addition and Subtraction You can write related facts using addition and subtraction. 3 + 4 = 7 7 − 4 = 3 You can use related facts to solve equations. A. x + 2 1 = 4 2 Think: 4 − 2 1 = x 2 x = 4 − 2 1 2 x = 3 2 − 2 1 Regroup 4 as 3 2 . 2 2 2 x = 11 2 B. x − 4 1 = 3 1 3 2 Think: 3 1 + 4 1 = x 2 3 x = 3 1 + 4 1 2 3 x = 7 + 13 Write the mixed numbers as improper fractions. 2 3 x = 21 + 26 Write the fractions using a common denominator. 6 6 x = 47 = 7 5 Write the sum as a mixed number. 6 6 Use related facts to solve each equation. 1. x + 3 1 = 7 2. x − 2 1 = 4 1 3. x + 3 = 5 1 4. x − 5 = 2 1 3 4 2 8 4 12 2 x = 7 − 3 1 x = 4 1 + 2 1 x = 5 1 − 3 x = 2 1 + 5 3 2 4 4 8 2 12 x = 6 3 − 3 1 x = 9 + 9 x = 21 − 3 x = 5 + 5 3 3 2 4 4 8 2 12 x = 21 − 10 x = 18 + 9 x = 42 − 3 x = 30 + 5 3 3 4 4 8 8 12 12 x = _______ x = _______ x = _______ x = _______ 5. x − 13 = 7 1 6. x − 3 2 = 11 7. x + 3 1 = 6 1 8. x − 2 2 = 1 3 4 2 3 3 2 4 5 10 _______________ _______________ _______________ ________________ Holt McDougal Mathematics Name _______________________________________ Date __________________ Class __________________ Fraction Operations Review for Mastery: Multiplying Mixed Numbers To find 1 of 2 1 , first change 2 1 to an improper fraction. 3 2 2 2 1 = 5 2 2 Then multiply as you would with two proper fractions. Check to see if you can divide by the GCF to make the problem simpler. Then multiply the numerators and multiply the denominators. The problem is now 1 • 5 . 3 2 1• 5 = 5 3• 2 6 So, 1 • 2 1 is 5 . 3 2 6 Rewrite each mixed number as an improper fraction. Is it possible to simplify before you multiply? If so, what is the GCF? 1. 1 • 11 2. 1 • 2 1 3. 1 • 11 4. 1 • 12 4 3 6 2 8 2 3 5 = 1 • ______ = 1 • ______ = 1 • ______ = 1 • ______ 4 6 8 3 _______________ _______________ _______________ ________________ 5. 11 • 12 6. 11 • 11 7. 13 • 2 1 8. 11 • 2 2 3 3 2 3 4 2 6 3 • • • • 3 3 2 3 4 2 6 3 _______________ _______________ _______________ ________________ 9. 3 1 • 2 10. 2 1 • 1 11. 13 • 2 1 12. 3 1 • 11 3 5 2 5 4 2 3 5 _______________ _______________ _______________ ________________ Holt McDougal Mathematics Name _______________________________________ Date __________________ Class __________________ Fraction Operations Review for Mastery: Dividing Fractions and Mixed Numbers Two numbers are reciprocals if their product is 1. 2 and 3 are 3 2 reciprocals because 2 • 3 = 6 = 1. 3 2 6 Dividing by a number is the same as multiplying by its reciprocal. 1 ÷ 2 = 1 1 • 1 = 1 4 8 4 2 8 So, you can use reciprocals to divide by fractions. To find 2 ÷ 4, first rewrite the expression as a multiplication 3 expression using the reciprocal of the divisor, 4. 2 • 1 3 4 Then use canceling to find the product in simplest form. 2 ÷ 4 = 2 • 1 = 1 • 1 = 1 3 3 4 3 2 6 To find 3 1 ÷ 11 , first rewrite the expression using improper fractions. 4 2 13 ÷ 3 4 2 Next, write the expression as a multiplication expression. 13 • 2 4 3 3 1 ÷ 11 = 13 ÷ 3 = 13 • 2 = 13 • 1 = 13 = 2 1 4 2 4 2 4 3 2 3 6 6 Divide. Write each answer in simplest form. 1. 1 ÷ 3 2. 11 ÷ 11 3. 3 ÷ 2 4. 2 1 ÷ 13 4 2 4 8 3 4 1 ÷ 3 ÷ 3 ÷ ÷ 4 1 2 4 8 1 3 4 _______ • _______ _______ • _______ _______ • _______ _______ • _______ ________________ ________________ ________________ ________________ 5. 1 ÷ 2 6. 11 ÷ 2 2 7. 1 ÷ 4 8. 3 1 ÷ 1 5 6 3 8 8 2 _______________ _______________ _______________ ________________ Holt McDougal Mathematics Name _______________________________________ Date __________________ Class __________________ Fraction Operations Review for Mastery: Solving Fraction Equations: Multiplication and Division You can write related facts using multiplication and division. 3 • 4 = 12 4 = 12 ÷ 3 You can use related facts to solve equations. A. 2 • x = 12 3 Think: 12 ÷ 2 = x Check: 2 • x = 12 3 3 x = 12 • 3 2 • 18 =? 12 Substitute 2 3 x = 12 • 3 2 • 18 =? 12 1 2 3 1 x = 36 36 =? 12 2 3 x = 18 12 = 12 V B. 2x = 3 5 2 • x = 3 5 Think: 3 ÷ 2 = x Check: 2x = 3 5 5 x = 3 • 5 2 • x =? 3 2 5 x = 3 • 5 2 • 15 =? 3 Substitute 1 2 5 2 x = 15 30 =? 3 2 10 x = 7 1 3 = 3 V 2 Use related facts to solve each equation. Then check each answer. 1. 1 • x = 3 2. 3x = 2 3. 3 • x = 2 4 4 5 3 _______________________ ________________________ ________________________ 4. 1 • x = 6 5. 2x = 1 6. 1 • x = 3 3 5 8 _______________________ ________________________ ________________________ Holt McDougal Mathematics Holt McDougal Mathematics .