English 2.2 Spanish Multiplying Fractions and Whole Numbers What does it mean when a whole number is multiplied by a fraction? Will the product be greater than or less than the whole number? 1 EXAMPLE: Multiplying a Fraction and a Whole Number 3 You have 3 gallons of paint. You use — of the paint. How many gallons 4 did you use? THINK ABOUT THE QUESTION: One way to think about this question is to rewrite the question. Words: 3 4 What is — of 3? 3 4 —×3=? Numbers: Here is one way to get the answer. ● Draw a length of 3. 3 3 Because you want to find 4 of the length, divide it into 4 equal sections. + ? + ? + ? ? = 3 Now, you need to think of a way to divide 3 into 4 equal parts. ● Rewrite the number 3 as a fraction whose numerator is divisible by 4. Because the length is divided into 4 equal sections, multiply the numerator and denominator by 4. 3 3 × 4 12 = = 1 1×4 4 0 12 4 3 4 In this form, you see that — can be divided into four equal parts of —. ● 3 4 Each part is — gallon and you used three of them. Written as multiplication, you have 3 4 9 4 — × 3 = —. 9 4 So, you used — gallons of paint. 50 Chapter 2 Multiplying and Dividing Fractions 0 3 4 + 3 4 6 4 + 3 4 9 4 + 3 4 = 12 4 9 4 English Spanish 2 EXAMPLE: Multiplying a Whole Number and a Fraction 2 5 Find 4 × —. 2 Add 5 four times. + 2 5 + 2 5 + 2 5 + 2 5 8 End at 5 . Start at 0. 1 5 0 2 5 8 5 2 5 3 5 4 5 1 6 5 7 5 8 5 9 5 2 3 5 So, 4 × — = —, or 1 —. Inductive Reasoning Work with a partner. Complete the table using a number line. Exercise Repeated Addition 3 4 3 4 1 3. — × 3 2 4. 4 × — 3 4 3 4 9 4 —+—+—=— 2 5 2 5 2 5 2 5 2 5 8 5 —+—+—+—=— 7 6 5. — × 5 9 5 6. 3 × — 1 3 7. — × 12 2 3 8. a. Write a real-life problem that is related to the product — × 5. 2 3 b. Write a different real-life problem that is related to the product 5 × —. c. Are the two products equal? How is your answer related to the Commutative Property of Multiplication? 9. IN YOUR OWN WORDS What does it mean when a whole number is multiplied by a fraction? Will the product be greater than or less than the whole number? 10. Write a general rule for multiplying fractions and whole numbers. Use what you learned about multiplying fractions and whole numbers to complete Exercises 4 –11 on page 54. Section 2.2 Multiplying Fractions and Whole Numbers 51 English 2.2 Spanish Lesson Lesson Tutorials Multiplying Fractions and Whole Numbers Multiply the numerator of the fraction by the whole number. Then write the product over the denominator. Words 1 8 9 ⋅ ⋅ bc a b c a — = —, where c ≠ 0 Algebra EXAMPLE 2×4 9 4 9 2× —= —= — Numbers Multiplying a Whole Number and a Fraction 7 8 Find 3 × —. Estimate 3 × 1 = 3 Multiply the numerator and whole number. 7 3×7 3×—=— 8 8 Write the product over the denominator. 21 8 5 8 = —, or 2 — Simplify. 5 8 So, the product is 2 —. EXAMPLE 2 ✓ Multiplying a Fraction and a Whole Number 11 12 Find — × 6. Remember A common factor is a factor that is shared by two or more whole numbers. For example, 3 and 9 share a common factor of 3. 5 8 Reasonable? 2 — ≈ 3 Estimate 1 × 6 = 6 Multiply the numerator and whole number. 11 × 6 12 11 12 —×6=— Write the product over the denominator. 1 11 × 6 12 =— Divide out the common factor, 6, from 6 and 12. 2 11 2 1 2 = —, or 5 — Simplify. 1 2 So, the product is 5 —. 1 2 Reasonable? 5 — ≈ 6 ✓ Multiply. Write the answer in simplest form. Exercises 4 –19 52 Chapter 2 1 5 1. 4 × — 3 8 2. — × 7 Multiplying and Dividing Fractions 2 3 3. 9 × — 7 10 4. — × 5 English Spanish EXAMPLE 3 Standardized Test Practice ⋅ 1 3 2 9 What is the value of — + x y when x = 3 and y = — ? A ○ 2 3 — B ○ 1 3 —+x Remember Be sure to use the order of operations when evaluating numerical expressions. 20 27 — ⋅ y = —13 + 3 ⋅ —29 1 3 2 9 Substitute 3 for x and — for y. ⋅ Multiply 3 and —. ⋅ Divide out the common factor 3. 3 2 9 2 9 =—+— 1 2 3 D 1— ○ C 1 ○ 1 3 2 =—+— 3 9 3 1 3 2 3 =—+— Simplify. =1 Add. The correct answer is ○ C . EXAMPLE 4 Real-Life Application 9 10 About — of the weight of a watermelon is water. How many pounds of water are in the watermelon? 9 10 Estimate 1 × 20 = 20 To find — of 20, multiply. 9 10 9 × 20 10 — × 20 = — 9 × 20 10 2 =— Divide out the common factor 10. 1 = 18 So, 18 pounds of water are in the watermelon. Reasonable? 18 ≈ 20 Exercises 32–39 ✓ ⋅ 1 2 5 8 5. Find the value of a b − — when a = — and b = 4. 6. WHAT IF? In Example 4, a watermelon weighs 15 pounds. How many pounds of water are in the watermelon? Section 2.2 Multiplying Fractions and Whole Numbers 53 English Spanish Exercises 2.2 Help with Homework 1. WRITING Describe how to multiply a fraction by a whole number. 7 8 2. NUMBER SENSE Use repeated addition to find the product 6 × —. 1 3 3. NUMBER SENSE Without multiplying, which is greater, — × 24 1 4 or — × 24? Explain. 6)=3 9+(- 3)= 3+(- 9)= 4+(- = 1) 9+(- Multiply. Write the answer in simplest form. 1 2 4. 3 × — 1 7 5. — × 5 6. — × 7 3 4 7. 4 × — 8. — × 8 6 5 9. 9 × — 5 3 10. — × 8 5 6 11. 25 × — 2 3 13. 7 × — 3 10 14. 15 × — 3 20 17. — × 24 3 8 18. 9 × — 12. — × 5 16. — × 5 1 8 4 9 7 10 13 14 2 3 15. — × 28 13 24 3 4 19. 18 × — ERROR ANALYSIS Describe and correct the error in finding the product. ✗ 20. 21. 2 2+8 —×8=— 5 5 ✗ 3 7 3 63 9×—=— 10 5 1 21 =—=2 =— 3 22. CDS Your friend has 12 CDs and — of them are pop music. How many CDs 4 are pop music? 2 3 23. OATMEAL MUFFINS A batch of oatmeal muffins calls for — cup of oats. How many cups of oats do you need to make four batches of muffins? 2 24. RAIN There are 365 days in a year and rain falls on — of the days. How many 5 days does it rain during the year? Use compatible numbers to estimate the product. 1 3 25. — × 17 54 Chapter 2 1 8 26. 35 × — Multiplying and Dividing Fractions 5 6 27. 22 × — 4 5 28. — × 7 English Spanish Tell how the Commutative and Associative Properties of Multiplication can help find the product mentally. Then find the product. 2 5 29. 25 × 6 × — 30. ( ) 3 7 5 9 8 × — × 18 31. — × 13 × 14 3 16 ALGEBRA Evaluate the expression when x = 6, y = —, and z = 20. 3 4 ⋅ 3 32. — x 2 5 36. —xz ⋅ 7 10 ⋅ 33. 8 y 34. — z 37. xyz 38. — + xy ⋅ 49 35. x — 3 8 5 12 39. — + yz 40 40. AREA A rectangular picnic shelter is 75 feet long by 1 15 1 15 60 feet wide. A model of the shelter is — as wide and — as long as the actual shelter. What is the area, in square feet, of the model of the shelter? 1 3 1 2 41. REASONING You spend — of your money and your friend spends — of her money shopping. Is it possible that you spend more money than your friend? Explain your reasoning. 42. NECKLACES You make bead necklaces using the beads shown in the table. Each necklace has a clasp that is Bead Length — centimeter long. 9 10 Bugle — cm a. How long is a necklace with 48 bugle beads and 24 tube beads? Crow — cm b. How long is a necklace with 18 bugle beads, 18 crow beads, and 18 tube beads? Tube — cm c. 9 20 9 10 2 5 You want to make a necklace 38 centimeters or longer. You have 26 bugle beads, 18 crow beads, and 16 tube beads. Do you have enough beads? If not, what is the smallest number of bugle beads you need to add to make the necklace? Explain how you found your answer. Evaluate the expression. SKILLS REVIEW HANDBOOK 3×5 2×4 43. — 9×2 1×5 44. — 4×8 7 × 25 45. — 10 × 13 3 × 11 46. — 47. MULTIPLE CHOICE What is the area of a parallelogram with a base of 12 inches and a height of 4 inches? SECTION 1.5 A 24 in. ○ B 48 in. ○ Section 2.2 C 24 in.2 ○ D 48 in.2 ○ Multiplying Fractions and Whole Numbers 55