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English Spanish 2.6 Dividing Mixed Numbers How can you use division by a mixed number as part of a story? 1 EXAMPLE: Writing a Story 1 Write a story that uses the division problem 6 ÷ 1—. Draw pictures for 2 your story. There are many possible stories. Here is one about a camping trip. Joe goes on a camping trip with his aunt, his uncle, and three cousins. They leave at 5:00 p.m. and drive 2 hours to the campground. Joe helps his uncle put up three tents. His aunt cooks hamburgers on a grill that is over a fire. In the morning, Joe tells his aunt that he is making pancakes for everyone. He decides to triple the recipe so there will be plenty of pancakes for everyone. A single recipe uses 2 cups of water, so he needs a total of 6 cups. Joe’s aunt has a 1-cup measuring cup and a ½-cup measuring cup. The water faucet is about 50 yards from the campsite. Joe tells his cousins that he can get 6 cups of water in only 4 trips. When his cousins ask him how he knows that, he uses a stick to draw a diagram in the dirt. Joe says “This diagram shows that there are four 1½’s in 6.” In other words, 1 2 6 ÷ 1— = 4. Cups 78 Chapter 2 Cups per trip Trips Multiplying and Dividing Fractions English Spanish 2 EXAMPLE: Dividing by a Mixed Number Show how Joe solves the division problem in Example 1. 1 2 6 1 3 2 Rewrite 6 as — and 1 — as —. 6 1 2 3 Multiply by the reciprocal of —, which is —. 6 ÷ 1— = — ÷ — =—×— Multiply fractions. 12 3 Simplify. = — , or 4 1 2 3 2 3 2 6×2 1×3 =— 3 6 1 2 3 ACTIVITY: Writing a Story Work with a partner. Think of a story that uses division by a mixed number. ÷ Whole number, fraction, or mixed number Mixed number a. Write your story. Then draw pictures for your story. b. Solve the division problem and use the answer in your story. Include a diagram of the division problem. 4. IN YOUR OWN WORDS How can you use division by a mixed number as part of a story? In Example 1, the units of the answer are trips. Cups Trips Trips Cups Cups ÷ — = Cups × — Trips Cups = Cups × — = Trips Find the units for the following division problems. Miles Hour Dollars Hour 5. Miles ÷ — 6. Dollars ÷ — 7. Miles ÷ Hour 8. Dollars ÷ Hour Use what you learned about dividing mixed numbers to complete Exercises 5–12 on page 82. Section 2.6 Dividing Mixed Numbers 79 English 2.6 Spanish Lesson Lesson Tutorials Dividing Mixed Numbers Write each mixed number as an improper fraction. Then divide as you would with proper fractions. EXAMPLE 1 Dividing a Mixed Number by a Fraction 1 2 3 8 1 2 Find 4— ÷ —. 1 2 Estimate 5 ÷ — = 10 3 8 9 2 3 8 Write 4— as the improper fraction —. 9 2 8 3 Multiply by the reciprocal of —, which is —. 1 2 4— ÷ — = — ÷ — 3 8 =—×— 3 4 1 1 9×8 =— 2×3 Simplify. Reasonable? 12 ≈ 10 So, the quotient is 12. 2 8 3 Multiply fractions. Divide out common factors. = 12 EXAMPLE 9 2 ✓ Dividing Mixed Numbers 5 6 2 3 Find 3 — ÷ 1—. 5 6 2 3 Estimate 4 ÷ 2 = 2 23 6 5 3 Write each mixed number as an improper fraction. 23 6 3 5 Multiply by the reciprocal of —, which is —. 3— ÷ 1— = — ÷ — 5 3 =—×— 23 × 3 6×5 1 =— 2 3 5 Multiply fractions. Divide out common factors. 23 10 3 10 3 So, the quotient is 2 —. 10 = —, or 2— Simplify. 3 10 ✓ 1 2 4. 6 — ÷ 2 — Reasonable? 2— ≈ 2 Divide. Write the answer in simplest form. Exercises 5 – 20 80 Chapter 2 3 7 2 3 1. 1— ÷ — 1 6 3 4 2. 2 — ÷ — Multiplying and Dividing Fractions 1 4 3. 8 — ÷ 1— 4 5 1 8 English Spanish EXAMPLE Using Order of Operations 3 1 4 1 8 2 3 Evaluate 5 — ÷ 1— − —. Remember 1 4 Be sure to check your answers whenever possible. In Example 3, you can use estimation to check that your answer is reasonable. 1 4 1 8 1 8 2 3 9 8 2 3 Write each mixed number as an improper fraction. 21 4 8 9 2 3 Multiply by the reciprocal of —, which is —. 2 3 Multiply — and —. Divide out common factors. 9 8 =—×—−— 7 2 21 × 8 4×9 21 4 =—−— 2 3 1 5— ÷ 1— − — 3 14 3 2 3 =—−— ≈5÷1−1 =5−1 =4 ✓ EXAMPLE 21 4 5— ÷ 1— − — = — ÷ — − — 8 9 Simplify. 12 3 = —, or 4 4 8 9 Subtract. Real-Life Application 2 3 One serving of tortilla soup is 1— cups. A restaurant cook makes 50 cups of soup. Is there enough to serve 35 people? Explain. 2 3 y 1— to find the number of available servings. Divide 50 by 2 3 50 1 5 3 50 ÷ 1— = — ÷ — Rewrite each number as an improper fraction. ⋅ 35 Multiply by the reciprocal of —, which is —. =— ⋅ ⋅ 1 Multiply fractions. Divide out common factors. = 30 Simplify. 50 1 =— — 10 50 3 1 5 5 3 3 5 No. Because 30 is less than 35, there is not enough soup to serve 35 people. Evaluate the expression. Exercises 29 – 37 1 2 1 6 7 8 1 3 5. 1— ÷ — − — 7. 2 5 4 5 5 6 8 9 6. 3 — ÷ — + — 3 4 — + 2 — ÷ 1— 2 3 4 7 5 7 8. — − 1— ÷ 4 — 9. In Example 4, can 30 cups of tortilla soup serve 15 people? Explain. Section 2.6 Dividing Mixed Numbers 81 English Spanish Exercises 2.6 Help with Homework 1 3 1. VOCABULARY What is the reciprocal of 7 —? 1 4 1 2 1 2 1 4 2. NUMBER SENSE Is 5 — ÷ 3 — the same as 3 — ÷ 5 —? Explain. 3. NUMBER SENSE Is the reciprocal of an improper fraction sometimes, always, or never a proper fraction? Explain. 4. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers. 1 2 1 8 1 2 What is 5 — divided by —? 1 8 Find the quotient of 5— and —. 1 2 1 2 What is 5 — times 8? 1 8 Find the product of 5— and —. 6)=3 9+(- 3)= 3+(- 9)= 4+(- = 1) 9+(- Divide. Write the answer in simplest form. 1 2 1 4 3 4 4 5 5. 2 — ÷ — 1 2 9 10 9. 7 — ÷ 1— 1 3 2 5 1 8 6. 3 — ÷ — 3 4 1 12 10. 3 — ÷ 2 — 5 9 4 7 8. 7 — ÷ — 1 5 12. 8 — ÷ 15 4 7 11. 7 — ÷ 8 13. 8 — ÷ — 2 3 14. 9 — ÷ — 5 6 15. 13 ÷ 10 — 5 6 16. 12 ÷ 5 — 7 8 1 16 18. — ÷ 1— 4 9 7 15 19. 4— ÷ 3 — 5 16 3 8 20. 6 — ÷ 5 — 17. — ÷ 3 — 1 6 5 6 7. 8 — ÷ — 9 11 2 9 5 6 21. ERROR ANALYSIS Describe and correct the error in finding the quotient. ✗ 1 2 2 3 1 2 3 2 7 2 5 2 35 4 3 4 3 — ÷ 1 — = 3 — × 1— = — × — = — = 8 — 1 3 22. DOG FOOD A bag contains 42 cups of dog food. Your dog eats 2 — cups of dog food each day. How many days does the bag of dog food last? 1 4 23. HAMBURGERS How many — -pound hamburgers can be made from 1 2 3 — pounds of ground beef ? 3 5 1 2 24. BOOKS How many 1— -inch thick books can fit on a 14 —-inch long bookshelf? 82 Chapter 2 Multiplying and Dividing Fractions English Spanish 5 8 2 9 ALGEBRA Evaluate the expression when x = 5 — and y = 2 —. 1 6 25. y ÷ 4 — 7 12 26. x ÷ y 27. 4 — ÷ x 28. y ÷ x Evaluate the expression. 5 3 2 3 29. 5 — ÷ 3— − — 6 4 30. 6 — − — ÷ 5 — 1 2 3 5 4 15 33. — × — ÷ 2 — 9 3 5 4 9 32. 3 — + 4 — ÷ — 9 11 7 12 2 3 7 8 11 16 31. 9 — ÷ 5 + 3 — 7 12 7 10 34. 4 — ÷ — × — ( 4 15 35. 1— × 4 — ÷ — 1 6 3 8 3 10 36. 3 — ÷ 8 × 6 — ) 5 14 1 3 3 4 ( 4 7 5 8 3 7 37. 2 — ÷ 2 — × 1— ) 1 2 38. TRAIL MIX You have 12 cups of granola and 8 — cups of peanuts to make trail mix. What is the greatest number of full batches of trail mix you can make? Explain how you found your answer. 1 1 7 ft 8 39. RAMPS You make skateboard ramps by cutting 1 2 pieces from a board that is 12 — feet long. a. Estimate how many ramps you can cut from the board. Is your estimate reasonable? Explain. 7 ft 8 b. How many ramps can you cut from the board? How much wood is left over? 40. At a track meet, the longest shot put throw by a boy is 25 feet and 8 inches. The longest shot put throw by a girl is 19 feet and 3 inches. How many times greater is the longest shot put throw by a boy than by a girl? Write the number as a decimal. SKILLS REVIEW HANDBOOK 41. forty-three hundredths 42. thirteen thousandths 43. three and eight tenths 44. seven and nine thousandths 3 4 SECTION 2.2 45. MULTIPLE CHOICE The winner in a vote for class president received — of the 240 votes. How many votes did the winner receive? A 60 ○ B 150 ○ C 180 ○ Section 2.6 D 320 ○ Dividing Mixed Numbers 83