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Name Reteaching 9-1 Divide a fraction by a whole number. Find _18 4 4. Use a model to show _18 . Divide each eighth into 4 equal parts. 1 1. 5 __ Each section shows _____ 32 (8 3 4) 1 4 4 5 __ 1. _ 8 32 1 1 1 1 1 1 1 1 8 8 8 8 8 8 8 8 1 32 Divide a fraction by a fraction. Find _34 4 _14. Use a number line. Count the number of _14s in _34. 0 1 4 2 4 3 4 Use repeated subtraction. Subtract _14 from _34 until the difference is 0. 3 2_ 1 5_ 2 _ 4 4 4 2 2_ 1 5_ 1 _ 1 4 4 4 1 2_ 1 50 _ 4 4 Count the number of times you subtracted to find the quotient. 3 4 _ 1 53 _ 4 4 There are three _14s. 3 4_ 1 53 _ 4 4 Solve each division sentence. Use a model if you wish. 1. 3 4 _13 5 9 2. _15 4 4 5 __ 1 20 Find each quotient. Simplify if possible. 3. 3 4 _12 5 3 4 __ 1 5 6. __ 16 16 9. 8 4 _14 5 6 3 32 9 5 4 _13 5 15 7 4 __ 1 5 __ 7 12 12 9 4 __ 1 5 4. __ 10 10 5. _15 4 3 5 7. 8. _12 4 6 5 10. 11. _67 4 _17 5 __ 1 15 __ 1 12 6 12. Draw a Picture The square dancing club meets for 3 hours. Every _34 hour, the dancers change partners. How many different partners will each dancer have in one meeting? Draw a picture to show your solution. 3 4 _34 5 4; 4 partners. Check students’ drawings. 13. Writing to Explain Explain why the quotient of two fractions less than 1 is always greater than either fraction. The quotient is always greater than the fractions because you are dividing by a number less than one. 26 Topic 9 © Pearson Education, Inc. 6 Reteaching 9-1 Understanding Division of Fractions Name Practice 9-1 Understanding Division of Fractions Solve each division sentence using the models provided. 9 1. 3 ⫼ _13 = 1 2 1 4 3 4. 8 ⫼ _14 ⫽ 7. _78 ⫼ _18 ⫽ 10. 6 ⫼ _23 ⫽ 1 6 0 Find each quotient. Simplify if possible. 32 7 __ 1 5. _17 ⫼ 4 ⫽ 11 ⫼ __ 1 ⫽ 8. __ 12 12 9 11. 7 ⫼ _13 ⫽ 5 3. _56 ⫼ _16 ⫽ 24 2 6 3 6 4 6 28 6. 5 ⫼ _12 ⫽ 10 11 1 ⫼3⫽ 9. __ 12 36 21 15 ⫼ __ 1 ⫽ 12. __ 16 16 Practice 9-1 0 __ 1 2. _14 ⫼ 6 ⫽ 5 6 __ 1 15 13. Draw a Picture Olivia has a piece of ribbon _12 yard long. If she cuts it into 6 equal pieces, what will be the length of each piece, in yards? 1 yard Check students pictures: __ 12 14. Geometry A regular polygon has a perimeter of 12 units. If each side measures _34 unit, how many sides does the polygon have? 16 15. Which division expression is shown by this model? 1 2 3 4 5 6 7 8 9 10 0 9 ⫼ __ 1 A __ 10 10 9 1 B 1⫼ __ 10 9 D 10 ⫼ __ 10 9 ⫼1 C __ 10 © Pearson Education, Inc. 6 16. Writing to Explain When you divide a whole number by a fraction less than 1, will the quotient be greater than or less than the whole number? Explain, and give an example. The quotient will be greater because you are dividing by a number less than 1. 4 ÷ _13 = 12: There are more fraction parts than wholes. 1 2 3 4 5 6 1 7 2 1 Each section 3 8 9 10 11 12 3 Topic 9 27 Name Reteaching 9-2 Dividing a Whole Number by a Fraction Number Reciprocal Product 3 1 _ 1 1 _ 1 1 8 2 _ 3 3 8 _ 1 3 _ 2 Find 14 _47. Step 1 Step 2 Step 3 Rewrite the division as multiplication using the reciprocal of the divisor. Divide out common factors if possible. Then multiply. If your answer is an improper fraction, change it to a mixed number. The reciprocal of _47 is _74 . 14 __ 7 1 _74 49 __ 2 49 24_ 1 __ 2 2 2 14 ÷ _47 = 14 × _74 Find the reciprocal of each fraction or whole number. _7 5 1. _57 2. 11 __ 1 11 3. _92 _2 9 Find each quotient. Simplify if possible. 4. 12 _45 7. 24 _34 10. 36 _89 15 32 40_12 5. 2 ÷ _14 8. 18 _89 11. 42 _78 8 20_14 48 8 6. 16 ÷ __ 10 10 9. 25 ÷ __ 11 12. 40 ÷ _45 20 27_12 50 13. Karolyn makes rolls for a friend’s dinner party. She uses 3 lb of butter. Each stick of butter weighs _14 lb. How many sticks of butter does Karolyn need to make her rolls? 12 sticks 32 Topic 9 © Pearson Education, Inc. 6 Reteaching 9-2 To divide a whole number by a fraction, you can multiply the whole number by the reciprocal of the fraction. The reciprocal of a number has the numerator and the denominator reversed. The product of a number and its reciprocal is 1. Name Practice 9-2 Dividing a Whole Number by a Fraction Find the reciprocal of each fraction or whole number. 1. _59 _9 5 2. 8 _1 3. _73 8 Find each quotient. Simplify if possible. 24 42 4 _79 5 54 _ 22 4 _56 5 262 5 15 4 _13 5 45 48 10 4 _56 5 12 _ 7 4 _23 5 101 2 6 4 _15 5 30 4. 8 4 _25 5 5. 4 4 _16 5 6. 18 4 _38 5 7. 8. 9. 10. 13. 11. 14. 12. 15. Practice 9-2 20 12 4 _12 5 24 20 4 _34 5 26_23 94 _18 5 72 _37 16. Writing to Explain Will the quotient of 5 4 _78 be greater than or less than 5? Explain. Thequotientwillbegreaterthan5becauseyouare dividingbyanumberthatislessthan1. 17. Reasoning How many times will you need to fill a _12 cup measuring cup to measure 4 cups of flour? 8 18. Geometry The distance around a circular flower bed is 36 feet. Jasper wants to put stakes every 8 inches ( _23 of a foot) around the bed. How many stakes does he need? 54stakes 19. Algebra Which expression is equal to 9 3 _32 ? A 2 4 _39 B 3 4 _92 C 9 4 _23 © Pearson Education, Inc. 6 D 9 4 _32 Topic 9 33 Name Reteaching 9-3 Dividing Fractions To divide by a fraction, you can multiply by its reciprocal. The reciprocal of a number has the numerator and the denominator reversed. Step 1 Step 2 Step 3 Rewrite the division as multiplication using the reciprocal of the divisor. Divide out common factors if possible. Then multiply. If your answer is an improper fraction, change it to a mixed number. 3 is __ 10 . The reciprocal of __ 10 3 4 _ 2 5 1 3 _ 10 4 __ 4 × __ _ 5 5 10 3 10 ___ 3 3 Find each quotient. Simplify if possible. 4 1. _12 _14 _12 2 8 = 2. _47 __ 21 Reciprocal of _14 3. _13 _12 6. _37 3 9. _59 _12 9 _ 4 12. __ 5 10 _2 3 4. _25 ÷ _23 7 7. _13 _89 _1 1_19 1_18 10. _35 _34 13. _13 _38 8 2_ 2 _ 3 3 8 _ _3 _4 7 __ 21 8 1_12 8 Reciprocal of __ 21 5 7 5. _58 ÷ __ 10 8 _4 5 _8 9 8. _56 ÷ _18 _3 11. _34 ÷ _56 14. _47 ÷ _34 __ 25 28 6_23 __ 9 10 __ 16 21 3 -gallon servings 15. Aaron has _78 gallon of bottled water. How many __ 16 can he pour? 4_23 servings 16. Draw a Picture Show how Rebecca can divide _34 of a cake into 9 pieces. What fraction of the whole cake will each piece be? 1 Sample answer: Each piece is __ 12 of the cake. © Pearson Education, Inc. 6 Reteaching 9-3 3. Find _45 ÷ __ 10 38 Topic 9 Name Practice 9-3 Dividing Fractions Find each quotient. Simplify if possible. 1. _13 _56 4. _59 5 3 7. _12 __ 10 10. _13 _34 _2 5 _1 9 2. _38 _12 5. _67 _34 1_23 5 _ 2 8. __ 12 3 _4 9 11. _38 4 _3 4 1_17 _5 8 __ 3 32 7 3. _78 __ 12 1_12 6. _23 _34 9 14 _ 2 9. __ 5 15 9 _ 3 12. __ 5 10 _8 2_13 1_12 13. Writing to Explain Serena said that by looking for common factors 5 1__ 9 . Do you 4 _ and simplifying the expression, she found that __ 10 8 16 agree with Serena? Why or why not? Practice 9-3 5 10 __ 4 5 _ 8 4 25 __ 16 9 1__ 16 No, Serena needs to use the reciprocal of the divisor, not the dividend, when she rewrites the expression as 5 ⴝ __ 8 , which 4 ⴜ_ 4 ⴛ_ a multiplication expression: __ 5 1016 8 10 can be simplified as _45 ⴛ _45 ⴝ __ . 25 1 yard wide. 14. A _56-yard piece of fencing is made of boards that are __ 12 How many boards make up the fence? 10 boards 3 -lb servings does he 15. Nathan has _78 lb of hummus. How many __ 10 have? 11 servings 2__ 12 16. Algebra Which equation can you use to find the number of _14 −inch pieces that can be cut from a piece of metal _58 of an inch long? A _58 _14 n © Pearson Education, Inc. 6 B _14 _58 n C _58 _14 n D _14 _85 n Topic 9 39 Name Reteaching 9-4 Estimating Quotients When you are working with fractions and mixed numbers, you can estimate using rounding and compatible numbers. Estimate 23_56 8_37 . 23_56 8_37 Round each mixed number to the nearest whole number. 24 8 3 Divide. 23_56 8_37 ⬇ 3 4. 35_18 5_45 7. 86_34 10_56 9 2__ 3 10. 7__ 10 10 30 5 6 31_16 4_58 艐 6 Estimate each quotient. 1. 11_12 6_14 31_16 4_58 2 6 8 4 Change 31_16 and 4_58 to the nearest compatible whole numbers. Think: 31_16 and 4_58 are close to 30 and 5. Sample answers are shown. 2. 19_13 3_23 5 3. 41_79 7_15 5. 61_38 8_59 7 6. 72_29 7_78 9 2_ 5 9 8. 26__ 9. 11_27 3_35 10 8 6 7__ 5 5_ 1 7 4 11. 47__ 12. 60__ 10 12 12 9 6 9 3 10 13. Critical Thinking Which of these two estimates is closer to the actual quotient? How do you know? Lisa’s estimate: 55_12 6_34 艐 54 6 9 Hayden’s estimate: 55_12 6_34 艐 56 7 8 Sample answer: Hayden’s estimate is closer because he used numbers closer to the original numbers. 14. Patrick uses wire to make wreaths. He has 31_12 feet of wire left on a spool. Estimate how many 3_34 pieces can he cut from the longer piece of wire. Sample answer: About 8 pieces © Pearson Education, Inc. 6 Reteaching 9-4 Estimate 31_16 4_58 . 44 Topic 9 Name Practice 9-4 Estimating Quotients Estimate each product. 1. 37_13 ⫼ 5_78 ⫽ 4. 12_34 ⫼ 5_59 ⫽ 7. 67_15 ⫼ 7_27 ⫽ 10. 71_45 ⫼ 7_89 ⫽ 13. 26_14 ⫼ 2_38 ⫽ 6 2 10 9 13 Sample answers are shown. 2. 25_12 ⫼ 6_14 ⫽ 4 3. 49_45 ⫼ 6_12 ⫽ 7 9 ⫽ 5. 43_23 ⫼ 5_25 ⫽ 9 6. 8_13 ⫼ 2__ 3 10 8. 55_59 ⫼ 7_16 ⫽ 8 9. 19_67 ⫼ 4_18 ⫽ 5 7 ⫼ 3_ 4 ⫽ 11. 15__ 12. 79_47 ⫼ 8_58 ⫽ 10 4 10 9 14. 40_89 ⫼ 7_35 ⫽ 6 15. 58_13 ⫼ 19_56 ⫽ 3 16. Number Sense Tran wants to cut strips of paper that are 2_14 in. wide. His sheet of paper is 11_12 in. wide. He estimates that 11_12 ⫼ 2_14 ⫽ 6, so he can cut 6 strips from each sheet of paper. Is his estimate an overestimate or an underestimate? Explain. Practice 9-4 Sample answer: 6 is an overestimate because Tran rounded the width of the paper up and the width of the strips down. 17. Writing to Explain Eliza uses 2_78 feet of yarn in each gift basket she makes. Explain how to estimate how many baskets Eliza can make if she has 22 feet of yarn. Sample answer: Divide 22 by 2_78 . Use compatible numbers to estimate: 21 ⴜ 3 ⴝ 7, so she has enough yarn to make at least 7 gift baskets. 18. Geometry The area of this rectangle is 257_14 sq in. What is the best estimate of side length w? A 66,000 in. B 50 in. C 25 in. 257 14 sq in. 1 10 2 in. w D 5 in. © Pearson Education, Inc. 6 19. Critical Thinking What estimation method did you use to find the length of side w in Problem 18? Sample answer: I used compatible numbers: 250 ⴜ 10 ⴝ 25. Topic 9 45 Name Reteaching 9-5 Dividing Mixed Numbers You can follow these steps to find 5_13 4 1_13 and 21 4 2_13. Step 1 Step 2 Step 3 First estimate. Then write each number as an improper fraction. Find the reciprocal of the divisor. Rewrite as a multiplication problem. Look for common factors. Simplify, then multiply. 1 1 1 . 3 3 Estimate 5 1 5. 16 4 3 3 16 3 3 4 16 3 3 4 Find 5 5 1 1 1 3 3 Find 21 2 1 . 3 1 Estimate 21 2 10 . 2 1 1 1 3 7 21 3 1 21 3 1 7 3 21 7 1 3 9 21 3 9 1 1 7 1 1 9 is close to 10 , 2 so the answer is reasonable. 21 7 1 3 Find each quotient. Simplify if possible. 1. 2_23 4 3_14 5 3. 2_15 4 2_13 5 5. 10 4 3_14 5 __ 32 39 __ 33 30422_T10_10_6RT-1 35 1 3__ 13 2. 1_34 4 4_18 5 4. 5_14 4 3 5 6. 7_14 4 2_18 5 __ 14 33 1_34 7 3__ 17 7. Writing to Explain Paper needs to be cut for voting ballots. Each piece of paper is 10_12 in. long. Each ballot should be 1_34 in. long. How many ballots can be cut from one piece of paper? Six ballots can be cut. © Pearson Education, Inc. 6 Reteaching 9-5 1 4 is close to 5, so the answer is reasonable. 16 4 3 3 21 2 4 16 3 4 4 3 4 1 50 Topic 9 Name Practice 9-5 Dividing Mixed Numbers Find each quotient. Simplify if possible. 1. 1_12 ⫼ 2_13 ⫽ 4. 3_12 ⫼ 2_14 ⫽ 7. 8 ⫼ 2_34 ⫽ __ 9 14 1_59 10 2__ 11 2. 4_14 ⫼ 3_18 ⫽ 5. 3_34 ⫼ 2 ⫽ 8. 2_12 ⫼ 1_38 ⫽ 9 1__ 25 _ 178 9 1__ 11 3. 2_14 ⫼ 5_12 ⫽ 6. 1_12 ⫼ 2_14 ⫽ 9. 4_23 ⫼ 1_34 ⫽ __ 9 22 _ 2 3 2_23 10. Reasoning Is it possible to divide 15 by a mixed number and get a quotient that is greater than 15? Explain. No, because you are dividing by a number that is greater than 1. Room Gallons of Paint 2_12 Bedroom 3_34 Living room 4_13 Practice 9-5 Kitchen Max is painting the inside of an apartment complex. The table shows how many gallons of paint are needed to paint each type of room. 11. How many kitchens can Max paint with 20 gal? 12. How many living rooms can Max paint with 26 gal? 13. How many bedrooms can Max paint with 60 gal? 8 kitchens 6 living rooms 16 bedrooms 14. Find 4_12 ⫼ 2_14 . A 1 B 2 C 3 D 4 Sample answer: Write each mixed number as an improper fraction. Find the reciprocal of the second term and multiply the two fractions. Simplify the answer. 1_45 © Pearson Education, Inc. 6 15. Writing to Explain Explain how you would find 4_15 ⫼ 2_13. Topic 9 51 Name Reteaching 9-6 Solving Equations Here is how to solve addition, subtraction, multiplication, and division equations with fractions. Addition Subtraction 4PMWFO O 9 3 O 5 O 8 0XOWLSOLFDWLRQ 6ROYHy y y 'LYLVLRQ Solve a a a a 3 . 3 3 ( ) ( ) y Reteaching 9-6 Solve Y 2 6 . Y 2 6 Y22 x6 2 x8 a Z 2 Solve each equation and check your answer. 1. z ⫹ 2_13 ⫽ 3_16 3. x ⫺ 1 ⫽ 4_23 5. _38 ⫹ n ⫽ 10 _ 5 6 5_23 9_58 2. 6n ⫽ _34 _ 1 8 4. y ⫼ _12 ⫽ 2_18 1 1__ 16 6. 2_29 ⫼ 5 ⫽ x 9 _ 4 9 in. Yesterday it 7. Algebra The rainfall total for this month is 4__ 10 9 1 1 in. Use the equation n ⫹ 2__ ⫽ 4__ to calculate how rained 2__ 10 10 10 much rainfall was received before yesterday. © Pearson Education, Inc. 6 2_45 in. 56 Topic 9 Name Practice 9-6 Solving Equations Solve each equation and check your answer. 1. y 1 1_14 5 2_38 2. w 2 2 5 3_12 3. z 4 _34 5 4_14 4. _13 5 _78q 5. 6_12 5 _56b 6. 2_14 5 p 2 _38 7. 2_14 5 x 4 _12 y 5 _98 or 1_18 11 or 5_ 1 w 5 __ 2 2 __ 3 z 5 3 16 8 q 5 __ 21 b 5 7_45 p 5 2_58 x 5 _98 or 1_18 8. Number Sense Is the solution of m 4 _23 5 9 greater than or less than the solution of m 4 _14 5 9? Explain. m 4 _23 5 9 is greater because you are dividing by a larger number. 10. Alex had a ball of string. He cut the string into 26 equal pieces. Each piece measured 3_14 in. Use the equation m 4 26 5 3_14 to find the length of the ball of string. Practice 9-6 9. The bakery used 42_13 c of flour. There were 10_13 c left in the flour bin. Use the equation x 2 42_13 5 10_13 to find out how many cups of flour the bakery had to start with. Sample answer: 52_23 c 84_12 in. 11. Solve 12y 5 2_14 . A 1_12 B 1_18 7 C __ 36 3 D __ 16 © Pearson Education, Inc. 6 12. Writing to Explain Write the steps you would use to solve the equation z 1 3_15 5 6_35 . Solve. Change the mixed numbers to improper 16 5 __ 33 . Then subtract fractions and write z 1 __ 5 5 __ 16 from both sides of the equation. So, 5 17 , or 3_ 2. z 5 __ 5 5 Topic 9 57 Name Reteaching 9-7 Problem Solving: Look for a Pattern Sometimes you can solve a problem by identifying a pattern. Here are two types of patterns. Ask yourself: Are the numbers increasing? Are they decreasing? Do they change by the same amount each time? Do you add, subtract, multiply, or divide to find the next number? Patterns in sets of numbers 15 , __ 13 , __ 9, _ 5, _ 3 11 , _ 7, _ __ 4 4 4 4 4 4 4 Ask yourself: How is the first figure modified to make the second figure? How is the second figure modified to make the third? Patterns in groups of figures 30422_T10_10-8RT-1 Remember: Once you have identified a possible number pattern, check at least two other consecutive numbers to make sure that the pattern is true for all of the numbers. Find the missing numbers. Describe the pattern. 1_34 , 2 , 2_14 , 2_12 , 2_34 89, 78, 67, 56 , 45 , 34 , 23 , 12, 1 Add _41 Subtract 11 Add _35 2. 10 , 3. _15, _45, _75, __ 5 __ 13 5 , __ 16 5 , __ 19 5 , __ 22 5 25 , __ 5 4. Draw the next figure in the pattern below. 5. Number Sense The table below shows the number of cells in a 30422_T10_10-8RT-2 culture. How many cells will there be at 4:30? Time Number of Cells 128 cells 62 Topic 9 1:00 1:30 2:00 2:30 3:00 1 2 4 8 16 © Pearson Education, Inc. 6 Reteaching 9-7 1. _34, 1, 1_14, 1_12, Name Practice 9-7 Problem Solving: Look for a Pattern Find the missing numbers. Describe the pattern. __ 21 16 , 11 , __ 1. _18, _68, __ 8 8 8 2. _14, _12, 1, 2 , __ 26 , 8 , , 8 4 __ 31 8 , , 16 __ 36 8 41 , __ 8 , 32, 64 3. 1.1, 1.1, 2.2, 6.6, 26.4 , 132 , 792 , 5,544 4. 14_12, 12_34 , 11, 5. 27, 9, 3, 1, _13, 6. 3, 5, 9, 15, 9_14 , 9 , _ 1 23 , 7_12 , 27 , __ 1 33 , 5_34 , 81 , __ 1 , 2_14 , _12 4 ___ 1 243 45 , 59 , 75 7. Number Sense In the figure, the sum of each row forms a pattern. What is the sum of the seventh row? 49 Add _58 Multiply by 2 Multiply by 1, then by 2, then by 3, etc. Subtract 1_34 Multiply by _13 or divide by 3 Add 2, add 4, add 6, etc. 1 2 2 3 3 3 4 4 4 4 5 5 5 5 5 8. Which figure completes this pattern? A Practice 9-7 B C D © Pearson Education, Inc. 6 9. Writing to Explain How can you find the answer to 7 without finding the sum of the numbers in a row? Sample answer: You can square the number of the row: row 2 ⴝ 22 or 4, row 3 ⴝ 32 or 9, row 7 ⴝ 72 or 49, and so on. Topic 9 63