my homework lesson 5 multiply by 9

Common Core Grade 5 Math (Worksheets, Homework, Lesson Plans)

Looking for video lessons that will help you in your Common Core Grade 5 Math classwork or homework? Looking for Common Core Math Worksheets and Lesson Plans that will help you prepare lessons for Grade 5 students?

The following lesson plans and worksheets are from the New York State Education Department Common Core-aligned educational resources. The Lesson Plans and Worksheets are divided into six modules.

Related Pages Common Core Math Resources, Lesson Plans And Worksheets Common Core Math Video Lessons, Math Worksheets and Games for Grade 5 Common Core Math Video Lessons, Math Worksheets and Games for all grades

Grade 5 Homework, Lesson Plans And Worksheets

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McGraw-Hill My Math Grade 4 Volume 1

Textbook: mcgraw-hill my math grade 4 volume 1 isbn: 9780021150236.

Use the table below to find videos, mobile apps, worksheets and lessons that supplement McGraw-Hill My Math Grade 4 Volume 1 book.

Chapter 1: Place Value

Lesson 1: place value, lesson 2: read and write multi-digit numbers, lesson 3: compare numbers, lesson 4: order numbers, lesson 5: use place value to round, chapter 2: add and subtract whole numbers, lesson 1: addition properties and subtraction rules, lesson 2: addition and subtraction patterns, lesson 3: add and subtract mentally, lesson 4: estimate sums and difference, lesson 5: add whole numbers, lesson 6: subtract whole numbers, lesson 7: subtract across zeros, lesson 8: solve multi-step word problems, chapter 3: understand multiplication and division, lesson 1: relate multiplication and division, lesson 2: relate division and subtraction, lesson 3: multiplication as comparison, lesson 4: compare to solve problems, lesson 5: multiplication properties and division rules, lesson 6: the associative property of multiplication, lesson 7: factors and multiples, chapter 4: multiply with one-digit numbers, lesson 1: multiples of 10, 100, and 1,000, lesson 2: round to estimate products, lesson 3: hands on: use place value to multiply, lesson 4: hands on: use models to multiply, lesson 5: multiply by a two-digit number, lesson 6: hands on: model regrouping, lesson 7: the distributive property, lesson 8: multiply with regrouping, lesson 9: multiply by a multi-digit number, lesson 10: multiply across zeros, chapter 5: multiply with two-digit numbers, lesson 1: multiply by tens, lesson 2: estimate products, lesson 3: hands on: use the distributive property to multiply, lesson 4: multiply by a two-digit number, lesson 5: solve multi-step word problems, chapter 6: divide by a one-digit number, lesson 1: divide multiples of 10, 100, and 1,000, lesson 2: estimate quotients, lesson 3: hands on: use place value to divide, lesson 4: divide with remainders, lesson 5: interpret remainders, lesson 6: place the first digit, lesson 7: hands on: distributive property and partial quotients, lesson 8: divide greater numbers, lesson 9: quotients with zeros, lesson 10: solve multi-step word problems, chapter 7: patterns and sequences, lesson 1: nonnumeric patterns, lesson 2: numeric patterns, lesson 3: sequences, lesson 4: addition and subtraction rules, lesson 5: multiplication and division rules, lesson 6: order of operations, lesson 7: hands on: equations with two operations, lesson 8: equations with multiple operations.

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Curriculum  /  Math  /  4th Grade  /  Unit 2: Multi-Digit Multiplication  /  Lesson 4

Multi-Digit Multiplication

Lesson 4 of 18

Criteria for Success

Tips for teachers, anchor tasks.

Problem Set

Target Task

Additional practice.

Multiply 10, 100, and 1,000 by one- and two-digit numbers.

Common Core Standards

Core standards.

The core standards covered in this lesson

Number and Operations in Base Ten

4.NBT.B.5 — Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Foundational Standards

The foundational standards covered in this lesson

3.NBT.A.3 — Multiply one-digit whole numbers by multiples of 10 in the range 10—90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations.

4.NBT.A.1 — Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.

Operations and Algebraic Thinking

3.OA.C.7 — Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

The essential concepts students need to demonstrate or understand to achieve the lesson objective

• Multiply 10, 100, and 1,000 by one-digit numbers.
• Multiply 10, 100, and 1,000 by two-digit numbers.
• Identify patterns in multiplication of 10, 100, and 1,000 by one- and two-digit numbers (MP.8).

Suggestions for teachers to help them teach this lesson

• Throughout this topic and then next one, students rely on many math practices in their work with multiplication. For example, “when students decompose numbers into sums of multiples of base-ten units to multiply them, they are seeing and making use of structure (MP.7). As they illustrate and explain the calculation by using physical or drawn models, they are using appropriate drawn tools strategically (MP.5) and attending to precision (MP.6) as they use base-ten units in the appropriate places” ( PARCC Model Content Frameworks, Mathematics, Grades 3–11 ). Further, “by reasoning repeatedly (MP.8) about the connection between math drawings and written numerical work, students can come to see multiplication and division algorithms as abbreviations or summaries of their reasoning about quantities” (NBT Progression, p. 14).
• It is important to preserve place value when talking about products of multi-digit numbers. For example, when discussing the computation 1,768 × 4, the partial product of the 6 digit and the 4 digit should be referenced as “six tens times four” or “sixty times four” rather than “six times four.” 

Lesson Materials

• Optional : Base ten blocks (3 ones, 3 tens, 3 hundreds, and 3 thousands per student or small group) — Students might not need these depending on their reliance on concrete materials.

Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress.

Tasks designed to teach criteria for success of the lesson, and guidance to help draw out student understanding

a.   Solve. 

• $$3\times1=$$  ________
• $$3\times10=$$  ________
• $$3\times100=$$  ________
• $$3\times1,000=$$  _______

b.   What do you notice about Part (a)? What do you wonder? 

Guiding Questions

Grade 4 Mathematics > Module 3 > Topic B > Lesson 4 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds . © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US  license. Accessed Dec. 2, 2016, 5:15 p.m..

a.   4 × 1,000 = _____

b.   _____ = 8 × 100

Grade 3 Mathematics > Module 3 > Topic B > Lesson 4 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds . © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US  license. Accessed Dec. 2, 2016, 5:15 p.m..

a.   15 × 10 = _______

b.   ______ = 28 × 100

c.   1,000 × 30 = ______

Unlock the answer keys for this lesson's problem set and extra practice problems to save time and support student learning.

Discussion of Problem Set

• What is the difference between saying “10 more” and “10 times as many”?
• What is another expression that has the same value as 10 x 800 and 1,000 x 8?

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Solve. 

a.   7 × 10 = _______

b.   100 × 7 = _______ 

c.   _______ = 7 × 1,000

d.   10 × 3 = _______

e.   1,000 × 30 = _______

f.   _______ = 10 × 30

g.   100 × 14 = _______

h.   _______ = 10 × 25

i.   54 × 1,000 = _______

Student Response

The Extra Practice Problems can be used as additional practice for homework, during an intervention block, etc. Daily Word Problems and Fluency Activities are aligned to the content of the unit but not necessarily to the lesson objective, therefore feel free to use them anytime during your school day.

Extra Practice Problems

Answer keys for Problem Sets and Extra Practice Problems are available with a Fishtank Plus subscription.

Word Problems and Fluency Activities

Help students strengthen their application and fluency skills with daily word problem practice and content-aligned fluency activities.

Topic A: Multiplicative Comparison

Solve multiplicative comparison problems with a larger unknown. Distinguish multiplicative comparison from additive comparison.

4.OA.A.1 4.OA.A.2

Solve multiplicative comparison problems with a smaller unknown.

Solve multiplicative comparison problems with an unknown multiplier. Interpret a multiplication equation as a comparison. 

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Topic B: Multiplication of up to Four-Digit Whole Numbers by One-Digit Whole Numbers

Multiply multiples of 10, 100, and 1,000 by one-digit numbers. Estimate multi-digit products by rounding numbers to their largest place value.

Multiply two-, three-, and four-digit numbers by one-digit numbers using a variety of mental strategies.

Multiply two-digit numbers by one-digit numbers.

Multiply three-digit numbers by one-digit numbers.

Multiply four-digit numbers by one-digit numbers.

Multiply two-, three-, and four-digit numbers by one-digit numbers and assess the reasonableness of the product. 

Topic C: Multiplication of Two-Digit Whole Numbers by Two-Digit Whole Numbers

Multiply two-digit multiples of 10 by two-digit multiples of 10. Estimate multi-digit products by rounding numbers to their largest place value.

Multiply two-digit multiples of 10 by two-digit numbers.

Multiply two-digit numbers by two-digit numbers using a variety of mental strategies.

Multiply two-digit by two-digit numbers using four partial products.

Multiply two-digit by two-digit numbers using two partial products and assess the reasonableness of the product.

Topic D: Multi-Step Word Problems

Abstract the formulas for the area and perimeter of a rectangle and apply those formulas in real-world and mathematical problems involving multiplication, addition, and subtraction.

4.MD.A.3 4.OA.A.3

Solve two-step word problems involving multiplication, addition, and subtraction.

4.OA.A.2 4.OA.A.3

Solve multi-step word problems involving multiplication, addition, and subtraction.

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My Math - 4th Grade - Chapter 9 - Operations with Fractions Worksheets

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• 4th Grade Math
• Title: My Math
• Author: McGraw Hill
• Edition: Volume 1

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Big Ideas Math Answers Grade 5 Chapter 9 Multiply Fractions

Are you in the search of Big Ideas Math Answers Grade 5 Chapter 9 Multiply Fractions ? If yes, you are on the correct page. Check out the detailed explanation for all the problems available on Big Ideas Grade 5 Chapter 9 Multiply Fractions Math Answers. You can easily become a math expert by referring to the BIM Grade 5 Chapter 9 Math Answers. Download Big Ideas Math Book 5th Grade Answer Key Chapter 9 Multiply Fractions now and start your preparation for the exam. You must prepare all the concepts and verify answers to test your preparation level.

Big Ideas 5th Grade Chapter 9 Multiply Fractions Math Book Answer Key

Every student must prepare each and every topic of ch 9 to score good marks in the exam. Also, take the math assignments and homework to get the perfect preparation. Prepare well by referring to our Big Ideas Math Book 5th Grade Answer Key Chapter 9 Multiply Fractions. Download Big Ideas Math Answers Grade 5 Chapter 9 Multiply Fractions for free . The below links have every topic problem along with explanations. So, check out every link and practice more where you lag.

Lesson: 1 Multiply Whole Numbers by Fractions

Lesson 9.1 Multiply Whole Numbers by Fractions

Multiply whole numbers by fractions homework & practice 9.1.

Lesson: 2 Use Models to Multiply Fractions by Whole Numbers

Lesson 9.2 Use Models to Multiply Fractions by Whole Numbers

Use models to multiply fractions by whole numbers homework & practice 9.2.

Lesson: 3 Multiply Fractions and Whole Numbers

Lesson 9.3 Multiply Fractions and Whole Numbers

Multiply fractions and whole numbers homework & practice 9.3.

Lesson: 4 Use Models to Multiply Fractions

Lesson 9.4 Use Models to Multiply Fractions

Use models to multiply fractions homework & practice 9.4.

Lesson: 5 Multiply Fractions

Lesson 9.5 Multiply Fractions

Multiply fractions homework & practice 9.5.

Lesson: 6 Find Areas of Rectangles

Lesson 9.6 Find Areas of Rectangles

Find areas of rectangles homework & practice 9.6.

Lesson: 7 Multiply Mixed Numbers

Lesson 9.7 Multiply Mixed Numbers

Multiply mixed numbers homework & practice 9.7.

Lesson: 8 Compare Factors and Products

Lesson 9.8 Compare Factors and Products

Compare factors and products homework & practice 9.8.

Chapter: 9 – Multiply Fractions

Multiply Fractions Performance Task

Multiply fractions activity, multiply fractions chapter practice.

Explore and Grow

Write any proper fraction that is not a unit fraction. Draw a model to represent your fraction. Draw a model to find a multiple of your fraction? Answer: The proper fraction that is not a unit fraction is:  \(\frac{5}{8}\) Let the proper fraction be multiplied by 5 So, We have to find the value of 5 × \(\frac{5}{8}\) Now, We know that, a = \(\frac{a}{1}\) a × \(\frac{a}{b}\) = \(\frac{a × a}{b × 1}\) So, 5 × \(\frac{5}{8}\) = \(\frac{5}{1}\) × \(\frac{5}{8}\) = \(\frac{5 × 5}{1 × 8}\) = \(\frac{25}{8}\) Hence, from the above, We can conclude that the multiple of your proper fraction is: \(\frac{25}{8}\)

Reasoning How can you use a model to multiply a whole number by a fraction? Explain. Answer: We can multiply a whole number by a fraction using the properties of multiplication. They are: A) a = \(\frac{a}{1}\) B) a × \(\frac{a}{b}\) = \(\frac{a × a}{b × 1}\) Hence, by using the above properties, we can multiply the whole number by a fraction.

Think and Grow: Multiply Whole Numbers by Fractions

Show and Grow

Multiply. Question 1. 2 × \(\frac{3}{4}\) = ______ Answer: 2 × \(\frac{3}{4}\) = \(\frac{6}{4}\)

Explanation: The given numbers are: 2 and \(\frac{3}{4}\) We know that, a = \(\frac{a}{1}\) a × \(\frac{a}{b}\) = \(\frac{a × a}{b × 1}\) So, 2 × \(\frac{3}{4}\) = \(\frac{2}{1}\) × \(\frac{3}{4}\) = \(\frac{2 × 3}{1 × 4}\) = \(\frac{6}{4}\) Hence, 2 × \(\frac{3}{4}\) = \(\frac{6}{4}\)

Question 2. 4 × \(\frac{5}{8}\) = ____ Answer: 4 × \(\frac{5}{8}\) = \(\frac{20}{8}\)

Explanation: The given numbers are: 4 and \(\frac{5}{8}\) We know that, a = \(\frac{a}{1}\) a × \(\frac{a}{b}\) = \(\frac{a × a}{b × 1}\) So, 4 × \(\frac{5}{8}\) = \(\frac{4}{1}\) × \(\frac{5}{8}\) = \(\frac{4 × 5}{1 × 8}\) = \(\frac{20}{8}\) Hence, 4 × \(\frac{5}{8}\) = \(\frac{20}{8}\)

Apply and Grow: Practice

Multiply. Question 3. 5 × \(\frac{7}{10}\) = ______ Answer: 5 × \(\frac{7}{10}\) = \(\frac{35}{10}\)

Explanation: The given numbers are: 5 and \(\frac{7}{10}\) We know that, a = \(\frac{a}{1}\) a × \(\frac{a}{b}\) = \(\frac{a × a}{b × 1}\) So, 5 × \(\frac{7}{10}\) = \(\frac{5}{1}\) × \(\frac{7}{10}\) = \(\frac{5 × 7}{1 × 10}\) = \(\frac{35}{10}\) Hence, 5 × \(\frac{7}{10}\) = \(\frac{35}{10}\)

Question 4. 8 × \(\frac{2}{3}\) = ______ Answer: 8 × \(\frac{2}{3}\) = \(\frac{16}{3}\)

Explanation: The given numbers are: 8 and \(\frac{2}{3}\) We know that, a = \(\frac{a}{1}\) a × \(\frac{a}{b}\) = \(\frac{a × a}{b × 1}\) So, 8 × \(\frac{2}{3}\) = \(\frac{8}{1}\) × \(\frac{2}{3}\) = \(\frac{8 × 2}{1 × 3}\) = \(\frac{16}{3}\) Hence, 8 × \(\frac{2}{3}\) = \(\frac{16}{3}\)

Question 5. 7 × \(\frac{5}{6}\) = ______ Answer: 7 × \(\frac{5}{6}\) = \(\frac{35}{6}\)

Explanation: The given numbers are: 7 and \(\frac{5}{6}\) We know that, a = \(\frac{a}{1}\) a × \(\frac{a}{b}\) = \(\frac{a × a}{b × 1}\) So, 7 × \(\frac{5}{6}\) = \(\frac{7}{1}\) × \(\frac{5}{6}\) = \(\frac{7 × 5}{1 × 6}\) = \(\frac{35}{6}\) Hence, 7 × \(\frac{5}{6}\) = \(\frac{35}{6}\)

Question 6. 9 × \(\frac{1}{2}\) = ______ Answer: 9 × \(\frac{1}{2}\) = \(\frac{9}{2}\)

Explanation: The given numbers are: 9 and \(\frac{1}{2}\) We know that, a = \(\frac{a}{1}\) a × \(\frac{a}{b}\) = \(\frac{a × a}{b × 1}\) So, 9 × \(\frac{1}{2}\) = \(\frac{9}{1}\) × \(\frac{1}{2}\) = \(\frac{9 × 1}{1 × 2}\) = \(\frac{9}{2}\) Hence, 9 × \(\frac{1}{2}\) = \(\frac{9}{2}\)

Question 7. 6 × \(\frac{3}{100}\) = ______ Answer: 6 × \(\frac{3}{100}\) = \(\frac{18}{100}\)

Explanation: The given numbers are: 6 and \(\frac{3}{100}\) We know that, a = \(\frac{a}{1}\) a × \(\frac{a}{b}\) = \(\frac{a × a}{b × 1}\) So, 6 × \(\frac{3}{100}\) = \(\frac{6}{1}\) × \(\frac{3}{100}\) = \(\frac{6 × 3}{1 × 100}\) = \(\frac{18}{100}\) Hence, 6 × \(\frac{3}{100}\) = \(\frac{18}{100}\)

Question 8. 15 × \(\frac{4}{7}\) = ______ Answer: 15 × \(\frac{4}{7}\) = \(\frac{60}{7}\)

Explanation: The given numbers are: 15 and \(\frac{4}{7}\) We know that, a = \(\frac{a}{1}\) a × \(\frac{a}{b}\) = \(\frac{a × a}{b × 1}\) So, 15 × \(\frac{4}{7}\) = \(\frac{15}{1}\) × \(\frac{4}{7}\) = \(\frac{15 × 4}{1 × 7}\) = \(\frac{60}{7}\) Hence, 15 × \(\frac{4}{7}\) = \(\frac{60}{7}\)

Question 9. 10 × \(\frac{5}{3}\) = ______ Answer: 10 × \(\frac{5}{3}\) = \(\frac{50}{3}\)

Explanation: The given numbers are: 10 and \(\frac{5}{3}\) We know that, a = \(\frac{a}{1}\) a × \(\frac{a}{b}\) = \(\frac{a × a}{b × 1}\) So, 10 × \(\frac{5}{3}\) = \(\frac{10}{1}\) × \(\frac{5}{3}\) = \(\frac{10 × 5}{1 × 3}\) = \(\frac{50}{3}\) Hence, 10 × \(\frac{5}{3}\) = \(\frac{50}{3}\)

Question 10. 4 × \(\frac{5}{2}\) = ______ Answer: 4 × \(\frac{5}{2}\) = 10

Explanation: The given numbers are: 4 and \(\frac{5}{2}\) We know that, a = \(\frac{a}{1}\) a × \(\frac{a}{b}\) = \(\frac{a × a}{b × 1}\) So, 4 × \(\frac{5}{2}\) = \(\frac{4}{1}\) × \(\frac{5}{2}\) = \(\frac{4 × 5}{1 × 2}\) = \(\frac{20}{2}\) = 10 Hence, 4 × \(\frac{5}{2}\) = 10

Question 11. 3 × \(\frac{11}{8}\) = ______ Answer: 3 × \(\frac{11}{8}\) = \(\frac{33}{8}\)

Explanation: The given numbers are: 3 and \(\frac{11}{8}\) We know that, a = \(\frac{a}{1}\) a × \(\frac{a}{b}\) = \(\frac{a × a}{b × 1}\) So, 3 × \(\frac{11}{8}\) = \(\frac{3}{1}\) × \(\frac{11}{8}\) = \(\frac{11 × 3}{1 × 8}\) = \(\frac{33}{8}\) Hence, 3 × \(\frac{11}{8}\) = \(\frac{33}{8}\)

Explanation: The given fractions are: \(\frac{3}{20}\) and \(\frac{9}{20}\) Let the missing number be X So, X × \(\frac{3}{20}\) = \(\frac{9}{20}\) When multiplication goes from left to right or from right to left, it will become the division When division goes from left to right or from right to left, it will become multiplication. So, X = \(\frac{9}{20}\) ÷ \(\frac{3}{20}\) We know that, \(\frac{a}{b}\) ÷ \(\frac{x}{y}\) = \(\frac{a}{b}\) × \(\frac{y}{x}\) So, \(\frac{9}{20}\) ÷ \(\frac{3}{20}\) = \(\frac{9}{20}\) × \(\frac{20}{3}\) = \(\frac{9 × 20}{20 × 3}\) = 3 Hence, The missing number is: 3

Explanation: The given fractions are: \(\frac{4}{9}\) and \(\frac{24}{9}\) Let the missing number be X So, X × \(\frac{4}{9}\) = \(\frac{24}{9}\) When multiplication goes from left to right or from right to left, it will become the division When division goes from left to right or from right to left, it will become multiplication. So, X = \(\frac{24}{9}\) ÷ \(\frac{4}{9}\) We know that, \(\frac{a}{b}\) ÷ \(\frac{x}{y}\) = \(\frac{a}{b}\) × \(\frac{y}{x}\) So, \(\frac{24}{9}\) ÷ \(\frac{4}{9}\) = \(\frac{24}{9}\) × \(\frac{9}{4}\) = \(\frac{9 × 24}{9 × 4}\) = 6 Hence, The missing number is: 6

Explanation: The given fractions are: \(\frac{5}{12}\) and \(\frac{50}{12}\) Let the missing number be X So, X × \(\frac{5}{12}\) = \(\frac{50}{12}\) When multiplication goes from left to right or from right to left, it will become the division When division goes from left to right or from right to left, it will become multiplication. So, X = \(\frac{50}{12}\) ÷ \(\frac{5}{12}\) We know that, \(\frac{a}{b}\) ÷ \(\frac{x}{y}\) = \(\frac{a}{b}\) × \(\frac{y}{x}\) So, \(\frac{50}{12}\) ÷ \(\frac{5}{12}\) = \(\frac{50}{12}\) × \(\frac{12}{5}\) = \(\frac{12 × 50}{12 × 5}\) = 10 Hence, The missing number is: 10

Explanation: It is given that a recipe calls for \(\frac{3}{4}\) cup of dried rice noodles. It is also given that you make 44 batches of the recipe So, The number of cups of dried rice noodles you used = ( The number of batches of the recipe ) × ( The number of dried rice noodles for each batch ) = 44 × \(\frac{3}{4}\) Now, We know that, a = \(\frac{a}{1}\) a × \(\frac{a}{b}\) = \(\frac{a × a}{b × 1}\) So, 44 × \(\frac{3}{4}\) = \(\frac{44}{1}\) × \(\frac{3}{4}\) = \(\frac{44 × 3}{1 × 4}\) = \(\frac{33}{1}\) = 33 Hence, from the above, We can conclude that the number of cups of dried noodles you used is: 33 cups

Question 16. YOU BE THE TEACHER Your 5 × \(\frac{3}{5}\) friend says that 5 is equal to \(\frac{3}{5}\) + \(\frac{3}{5}\) + \(\frac{3}{5}\) + \(\frac{3}{5}\) + \(\frac{3}{5}\). Is your friend correct? Explain. Answer: Yes, your friend is correct.

Explanation: It is given that your 5 × \(\frac{3}{5}\) friend says that 5 is equal to \(\frac{3}{5}\) + \(\frac{3}{5}\) + \(\frac{3}{5}\) + \(\frac{3}{5}\) + \(\frac{3}{5}\). Now, We know that, n × a = a + a + a + a + …………….. + n times Where, ‘n’ is the multiple of a Here, ‘a’ will be the whole number or the fraction So, According to the above property, 5 × \(\frac{3}{5}\) = \(\frac{3}{5}\) + \(\frac{3}{5}\) + \(\frac{3}{5}\) +\(\frac{3}{5}\) + \(\frac{3}{5}\) Hence, from the above, We can conclude that your friend is correct.

Think and Grow: Modeling Real Life

Question 18. An excavator is moving 4 piles of dirt that are the same size. Each pile requires \(\frac{3}{4}\) hour to move. Can the excavator move all of the piles in 2 hours? Answer: No, the excavator can’t move all of the piles in 2 hours

Explanation: It is given that an excavator is moving 4 piles of dirt that are of the same size and each pile requires \(\frac{3}{4}\) hour to move. So, We know that, 1 hour = 60 minutes So, \(\frac{3}{4}\) of 1 hour = \(\frac{3}{4}\) × 60 = \(\frac{3}{4}\) × \(\frac{60}{1}\) = \(\frac{3 × 60}{4 × 1}\) = 45 minutes So, The time is taken to move 4 piles of dirt = ( The time is taken to move each pile )  × ( The total number of piles ) = 45 × 4 =180 minutes It is given that you have to move the piles of dirt in 2 hours but the time taken is 4 hours. Hence, from the above, We can conclude that the excavator can’t move all the piles in 2 hours.

Explanation: It is given that you walk dogs \(\frac{5}{4}\) miles two times each day So, The total number of miles you walk dogs in 1 day = 2 × \(\frac{5}{4}\) = \(\frac{5}{4}\) × \(\frac{2}{1}\) = \(\frac{5 × 2}{4 × 1}\) = \(\frac{5}{2}\) miles We know that, 1 week = 7 days So, The total number of miles you walk dogs in 1 week = ( The number of miles you walk dogs in 1 day ) × 7 = \(\frac{5}{2}\) × 7 = \(\frac{5}{2}\) × \(\frac{7}{1}\) = \(\frac{5 × 7}{2 × 1}\) = \(\frac{35}{2}\) Hence, from the above, We can conclude that the number of miles you walk dogs in 1 week is: \(\frac{35}{2}\) miles

Question 20. DIG DEEPER You have10 feet of string. You need \(\frac{5}{3}\) feet of string to make 1 necklace. You make 5 necklaces. Do you have enough string to make another necklace? Explain. Answer: Yes, we have enough string to make another necklace

Explanation: It is given that you have 10 feet of string and you need \(\frac{5}{3}\) feet of string to make 1 necklace. and you make 5 necklaces. So, The length of the string required for 5 necklaces = ( The length of string required for 1 necklace ) × ( The total number of necklaces ) = \(\frac{5}{3}\) × 5 = \(\frac{5}{3}\) × \(\frac{5}{1}\) = \(\frac{5 × 5}{3 × 1}\) = \(\frac{25}{3}\) feet We can write 10 feet as \(\frac{30}{3}\) feet So, When we compare the total length of the string and the length of the string to make 5 necklaces, We can observe that the length of the string to make 5 necklaces is less than the total length of the string. Hence, from the above, We can conclude that we have enough string to make another necklace.

Multiply Question 1. 5 × \(\frac{2}{3}\) = ______ Answer: 5 × \(\frac{2}{3}\) = \(\frac{10}{3}\)

Explanation: The given numbers are: 5 and \(\frac{2}{3}\) We know that, a = \(\frac{a}{1}\) a × \(\frac{a}{b}\) = \(\frac{a × a}{b × 1}\) So, 5 × \(\frac{2}{3}\) = \(\frac{5}{1}\) × \(\frac{2}{3}\) = \(\frac{2 × 5}{1 × 3}\) = \(\frac{10}{3}\) Hence, 5 × \(\frac{2}{3}\) = \(\frac{10}{3}\)

Question 2. 9 × \(\frac{7}{8}\) = ______ Answer: 9 × \(\frac{7}{8}\) = \(\frac{63}{8}\)

Explanation: The given numbers are: 9 and \(\frac{7}{8}\) We know that, a = \(\frac{a}{1}\) a × \(\frac{a}{b}\) = \(\frac{a × a}{b × 1}\) So, 9 × \(\frac{7}{8}\) = \(\frac{9}{1}\) × \(\frac{7}{8}\) = \(\frac{9 × 7}{1 × 8}\) = \(\frac{63}{8}\) Hence, 9 × \(\frac{7}{8}\) = \(\frac{63}{8}\)

Question 3. 4 × \(\frac{11}{12}\) = ______ Answer: 4 × \(\frac{11}{12}\) = \(\frac{44}{12}\)

Explanation: The given numbers are: 4 and \(\frac{11}{12}\) We know that, a = \(\frac{a}{1}\) a × \(\frac{a}{b}\) = \(\frac{a × a}{b × 1}\) So, 4 × \(\frac{11}{12}\) = \(\frac{4}{1}\) × \(\frac{11}{12}\) = \(\frac{4 × 11}{1 × 12}\) = \(\frac{44}{12}\) Hence, 4 × \(\frac{11}{12}\) = \(\frac{44}{12}\)

Question 4. 8 × \(\frac{35}{100}\) = ______ Answer: 8 × \(\frac{35}{100}\) = \(\frac{280}{100}\)

Explanation: The given numbers are: 8 and \(\frac{35}{100}\) We know that, a = \(\frac{a}{1}\) a × \(\frac{a}{b}\) = \(\frac{a × a}{b × 1}\) So, 8 × \(\frac{35}{100}\) = \(\frac{8}{1}\) × \(\frac{35}{100}\) = \(\frac{8 × 35}{1 × 100}\) = \(\frac{280}{100}\) Hence, 8 × \(\frac{35}{100}\) = \(\frac{280}{100}\)

Question 5. 3 × \(\frac{1}{2}\) = ______ Answer: 3 × \(\frac{1}{2}\) = \(\frac{3}{2}\)

Explanation: The given numbers are: 3 and \(\frac{1}{2}\) We know that, a = \(\frac{a}{1}\) a × \(\frac{a}{b}\) = \(\frac{a × a}{b × 1}\) So, 3 × \(\frac{1}{2}\) = \(\frac{3}{1}\) × \(\frac{1}{2}\) = \(\frac{1 × 3}{1 × 2}\) = \(\frac{3}{2}\) Hence, 3 × \(\frac{1}{2}\) = \(\frac{3}{2}\)

Question 6. 7 × \(\frac{2}{5}\) = ______ Answer: 7 × \(\frac{2}{5}\) = \(\frac{14}{5}\)

Explanation: The given numbers are: 7 and \(\frac{2}{5}\) We know that, a = \(\frac{a}{1}\) a × \(\frac{a}{b}\) = \(\frac{a × a}{b × 1}\) So, 7 × \(\frac{2}{5}\) = \(\frac{7}{1}\) × \(\frac{2}{5}\) = \(\frac{2 × 7}{1 × 5}\) = \(\frac{14}{5}\) Hence, 7 × \(\frac{2}{5}\) = \(\frac{14}{5}\)

Question 7. 6 × \(\frac{7}{4}\) = ______ Answer: 6 × \(\frac{7}{4}\) = \(\frac{42}{4}\)

Explanation: The given numbers are: 6 and \(\frac{7}{4}\) We know that, a = \(\frac{a}{1}\) a × \(\frac{a}{b}\) = \(\frac{a × a}{b × 1}\) So, 6 × \(\frac{7}{4}\) = \(\frac{6}{1}\) × \(\frac{7}{4}\) = \(\frac{6 × 7}{1 × 4}\) = \(\frac{42}{4}\) Hence, 6 × \(\frac{7}{4}\) = \(\frac{42}{4}\)

Question 8. 12 × \(\frac{8}{7}\) = ______ Answer: 12 × \(\frac{8}{7}\) = \(\frac{96}{7}\)

Explanation: The given numbers are: 12 and \(\frac{8}{7}\) We know that, a = \(\frac{a}{1}\) a × \(\frac{a}{b}\) = \(\frac{a × a}{b × 1}\) So, 12 × \(\frac{8}{7}\) = \(\frac{12}{1}\) × \(\frac{8}{7}\) = \(\frac{12 × 8}{1 × 7}\) = \(\frac{96}{7}\) Hence, 12 × \(\frac{8}{7}\) = \(\frac{96}{7}\)

Question 9. 25 × \(\frac{10}{9}\) = ______ Answer: 25 × \(\frac{10}{9}\) = \(\frac{250}{9}\)

Explanation: The given numbers are: 25 and \(\frac{10}{9}\) We know that, a = \(\frac{a}{1}\) a × \(\frac{a}{b}\) = \(\frac{a × a}{b × 1}\) So, 25 × \(\frac{10}{9}\) = \(\frac{25}{1}\) × \(\frac{10}{9}\) = \(\frac{25 × 10}{1 × 9}\) = \(\frac{250}{9}\) Hence, 25 × \(\frac{10}{9}\) = \(\frac{250}{9}\)

Explanation: The given fractions are: \(\frac{5}{7}\) and \(\frac{25}{7}\) Let the missing number be X So, X × \(\frac{5}{7}\) = \(\frac{25}{7}\) When multiplication goes from left to right or from right to left, it will become the division When division goes from left to right or from right to left, it will become multiplication. So, X = \(\frac{25}{7}\) ÷ \(\frac{5}{7}\) We know that, \(\frac{a}{b}\) ÷ \(\frac{x}{y}\) = \(\frac{a}{b}\) × \(\frac{y}{x}\) So, \(\frac{25}{7}\) ÷ \(\frac{5}{7}\) = \(\frac{25}{7}\) × \(\frac{7}{5}\) = \(\frac{7 × 25}{7 × 5}\) = 5 Hence, The missing number is: 5

Explanation: The given fractions are: \(\frac{9}{10}\) and \(\frac{63}{10}\) Let the missing number be X So, X × \(\frac{9}{10}\) = \(\frac{63}{10}\) When multiplication goes from left to right or from right to left, it will become the division When division goes from left to right or from right to left, it will become multiplication. So, X = \(\frac{63}{10}\) ÷ \(\frac{9}{10}\) We know that, \(\frac{a}{b}\) ÷ \(\frac{x}{y}\) = \(\frac{a}{b}\) × \(\frac{y}{x}\) So, \(\frac{63}{10}\) ÷ \(\frac{9}{10}\) = \(\frac{63}{10}\) × \(\frac{10}{9}\) = \(\frac{63 × 10}{10 × 9}\) = 7 Hence, The missing number is: 7

Explanation: The given fractions are: \(\frac{3}{5}\) and \(\frac{27}{5}\) Let the missing number be X So, X × \(\frac{3}{5}\) = \(\frac{27}{5}\) When multiplication goes from left to right or from right to left, it will become the division When division goes from left to right or from right to left, it will become multiplication. So, X = \(\frac{27}{5}\) ÷ \(\frac{3}{5}\) We know that, \(\frac{a}{b}\) ÷ \(\frac{x}{y}\) = \(\frac{a}{b}\) × \(\frac{y}{x}\) So, \(\frac{27}{5}\) ÷ \(\frac{3}{5}\) = \(\frac{27}{5}\) × \(\frac{5}{3}\) = \(\frac{5 × 27}{5 × 3}\) = 9 Hence, The missing number is: 9

Question 13. You make 5 servings of pancakes. You top each serving with \(\frac{1}{4}\) cup of strawberries. How many cups of strawberries do you use? Answer: The number of cups of strawberries you used is: 20 cups

Explanation: It is given that you make 5 servings of pancakes and you top each serving with \(\frac{1}{4}\) cup of strawberries. So, The number of cups of strawberries = \(\frac{The number of servings of pancakes}{Each serving in pancake}\) = 5 ÷ \(\frac{1}{4}\) = \(\frac{5}{1}\) ÷ \(\frac{1}{4}\) = \(\frac{5}{1}\) × \(\frac{4}{1}\) = \(\frac{5 × 4}{1 × 1}\) = 20 cups hence, from the above, We can conclude that the number of cups of strawberries is: 20 cups

Question 14. Which One Doesn’tBelong? Which one does not belong with the other three? 4 × \(\frac{3}{8}\) 3 × \(\frac{1}{8}\) \(\frac{3}{8}\) + \(\frac{3}{8}\) + \(\frac{3}{8}\) + \(\frac{3}{8}\) 1 + \(\frac{1}{2}\) Answer: Let the expresseons be named A., B., C., D. So, A) 4 × \(\frac{3}{8}\) B) 3 × \(\frac{1}{8}\) C) \(\frac{3}{8}\) + \(\frac{3}{8}\) + \(\frac{3}{8}\) + \(\frac{3}{8}\) D) 1 + \(\frac{1}{2}\) So, from the above four expressions, Expressions A), B), C) are in the normal form and C) is the expanded form of expression A) Hence, from the above, We can conclude that expression D) does not belong to the other three.

Explanation: It is given that you complete \(\frac{5}{2}\) inches of weaving each day for 5 days. So, The total number of inches you completed in 5 days = ( The number of inches you weaved each day ) × 5 = \(\frac{5}{2}\) × 5 = \(\frac{5}{2}\) × \(\frac{5}{1}\) = \(\frac{5 × 5}{2 × 1}\) = \(\frac{25}{2}\) It is also given that the weaving needs to be at least 11 inches long. So, 11 inches can also be written as \(\frac{22}{2}\) So, When we compare the total number of inches needed for weaving and the total number of inches you weaved in 5 days, we can observe that You weaved more than the amount of the weaving. Hence, from the above, We can conclude that your weaving is completed.

Explanation: It is given that you spend \(\frac{7}{2}\) hours each day playing drums for 2 days So, The time you played drums for 2 days = The time you played drums for each day × 2 = \(\frac{7}{2}\) × 2 = \(\frac{7}{2}\) × \(\frac{2}{1}\) = \(\frac{7 × 2}{2 × 1}\) = \(\frac{14}{2}\) = \(\frac{28}{4}\) = 7 hours It is also given that Your friend spends \(\frac{5}{4}\) hours playing drums each day for 6 days. So, The time your friend played drums for 6 days = The time your friend played drums for each day × 6 = \(\frac{5}{4}\) × 6 = \(\frac{5}{4}\) × \(\frac{6}{1}\) = \(\frac{5 × 6}{4 × 1}\) = \(\frac{30}{4}\) hours Now, When we compare the time played drums by you and your friends, your friend played more time than you So, The amount of time more your friend played than you = \(\frac{30}{4}\) – \(\frac{28}{4}\) = \(\frac{30 – 28}{4}\) = \(\frac{2}{4}\) = \(\frac{1}{2}\) hour Hence, from the above, We can conclude that Your friend spends more time playing drums than you The amount you played drums more than your friend is: \(\frac{1}{2}\) hours

Review & Refresh

Explanation: The given decimal numbers are: 0.6 and 0.4 the representation of the decimal numbers in the fraction form is: \(\frac{6}{10}\) and \(\frac{4}{10}\) Now, \(\frac{6}{10}\) × \(\frac{4}{10}\) = \(\frac{6 × 4}{10 × 10}\) = \(\frac{24}{100}\) So, The representation of \(\frac{24}{100}\) in the decimal form is: 0.24 Hence, 0.6 × 0.4 = 0.24

Explanation: The given decimal numbers are: 2.37 and 1.9 the representation of the decimal numbers in the fraction form is: \(\frac{237}{100}\) and \(\frac{19}{10}\) Now, \(\frac{237}{100}\) × \(\frac{19}{10}\) = \(\frac{237 × 19}{100 × 10}\) = \(\frac{4,503}{1000}\) So, The representation of \(\frac{4503}{1000}\) in the decimal form is: 4.503 Hence, 2.37 × 1.9 = 4.503

Explanation: The given decimal numbers are: 52.8 and 0.75 the representation of the decimal numbers in the fraction form is: \(\frac{6}{10}\) and \(\frac{4}{10}\) Now, \(\frac{528}{10}\) × \(\frac{75}{100}\) = \(\frac{528 × 75}{10 × 100}\) = \(\frac{3960}{1000}\) So, The representation of \(\frac{3960}{1000}\) in the decimal form is: 39.6 Hence, 52.8 × 0.75 = 39.6

You need to give water to \(\frac{2}{3}\) of the dogs at a shelter. There are 12 dogs at the shelter. How many dogs need water? Draw a model to support your answer. Answer: The number of dogs that need water is: 8 dogs

Explanation: It is given that you need to give water to \(\frac{2}{3}\) of the dogs at a shelter and there are 12 dogs at the shelter So, The number of dogs that need water = ( The fraction of dogs that need water ) × ( The total number of dogs ) = \(\frac{2}{3}\) × 12 We know that, \(\frac{a}{b}\) × x = \(\frac{a}{b}\) × \(\frac{x}{1}\) = \(\frac{a × x}{b × 1}\) So, \(\frac{2}{3}\) × 12 = \(\frac{2}{3}\) × \(\frac{12}{1}\) = \(\frac{2 × 12}{3 × 1}\) = \(\frac{8}{1}\) = 8 Hence, from the above, We can conclude that there are 8 dogs that need water

Reasoning How can you use a model to multiply a fraction by a whole number? Explain. Answer: We can multiply a fraction by a whole number by using the following multiplication properties. They are: A) \(\frac{a}{b}\) × x = \(\frac{a}{b}\) × \(\frac{x}{1}\) B) x = \(\frac{x}{1}\)

Think and Grow: Multiply Fractions by Whole Numbers

Question 1. Find \(\frac{2}{5}\) of 10. Answer: \(\frac{2}{5}\) × 10 = 4

Explanation: The given numbers are: \(\frac{2}{5}\) and 10 We know that, \(\frac{a}{b}\) × x = \(\frac{a}{b}\) × \(\frac{x}{1}\) = \(\frac{a × x}{b × 1}\) So, \(\frac{2}{5}\) × 10 = \(\frac{2}{5}\) × \(\frac{10}{1}\) = \(\frac{2 × 10}{5 × 1}\) = \(\frac{4}{1}\) = 4 Hence, \(\frac{2}{5}\) × 10 = 4

Question 2. Find \(\frac{7}{12}\) × 6. Answer: \(\frac{7}{12}\) × 6 = \(\frac{7}{2}\)

Explanation: The given numbers are: \(\frac{7}{12}\) and 6 We know that, \(\frac{a}{b}\) × x = \(\frac{a}{b}\) × \(\frac{x}{1}\) = \(\frac{a × x}{b × 1}\) So, \(\frac{7}{12}\) × 6 = \(\frac{7}{12}\) × \(\frac{6}{1}\) = \(\frac{7 × 6}{12 × 1}\) = \(\frac{7}{2}\) Hence, \(\frac{7}{12}\) × 6 = \(\frac{7}{2}\)

Multiply. Use a model to help. Question 3. \(\frac{5}{6}\) of 12 Answer: \(\frac{5}{6}\) × 12 = 10

Explanation: The given numbers are: \(\frac{5}{6}\) and 12 We know that, \(\frac{a}{b}\) × x = \(\frac{a}{b}\) × \(\frac{x}{1}\) = \(\frac{a × x}{b × 1}\) So, \(\frac{5}{6}\) × 12 = \(\frac{5}{6}\) × \(\frac{12}{1}\) = \(\frac{5 × 12}{6 × 1}\) = \(\frac{10}{1}\) = 10 Hence, \(\frac{5}{6}\) × 12 = 10

Question 4. \(\frac{2}{3}\) × 9 Answer: \(\frac{2}{3}\) × 9 = 6

Explanation: The given numbers are: \(\frac{2}{3}\) and 9 We know that, \(\frac{a}{b}\) × x = \(\frac{a}{b}\) × \(\frac{x}{1}\) = \(\frac{a × x}{b × 1}\) So, \(\frac{2}{3}\) × 9 = \(\frac{2}{3}\) × \(\frac{9}{1}\) = \(\frac{2 × 9}{3 × 1}\) = \(\frac{6}{1}\) = 6 Hence, \(\frac{2}{3}\) × 9 = 6

Question 5. \(\frac{1}{5}\) × 10 Answer: \(\frac{1}{5}\) × 10 = 2

Explanation: The given numbers are: \(\frac{1}{5}\) and 10 We know that, \(\frac{a}{b}\) × x = \(\frac{a}{b}\) × \(\frac{x}{1}\) = \(\frac{a × x}{b × 1}\) So, \(\frac{1}{5}\) × 10 = \(\frac{1}{5}\) × \(\frac{10}{1}\) = \(\frac{1 × 10}{5 × 1}\) = \(\frac{2}{1}\) = 2 Hence, \(\frac{1}{5}\) × 10 = 2

Question 6. \(\frac{3}{5}\) of 5 Answer: \(\frac{3}{5}\) × 5 = 3

Explanation: The given numbers are: \(\frac{3}{5}\) and 5 We know that, \(\frac{a}{b}\) × x = \(\frac{a}{b}\) × \(\frac{x}{1}\) = \(\frac{a × x}{b × 1}\) So, \(\frac{3}{5}\) × 5 = \(\frac{3}{5}\) × \(\frac{5}{1}\) = \(\frac{3 × 5}{5 × 1}\) = \(\frac{3}{1}\) = 3 Hence, \(\frac{3}{5}\) × 5 = 3

Question 7. \(\frac{1}{6}\) of 3 Answer: \(\frac{1}{6}\) × 3 = \(\frac{1}{2}\)

Explanation: The given numbers are: \(\frac{1}{6}\) and 3 We know that, \(\frac{a}{b}\) × x = \(\frac{a}{b}\) × \(\frac{x}{1}\) = \(\frac{a × x}{b × 1}\) So, \(\frac{1}{6}\) × 3 = \(\frac{1}{6}\) × \(\frac{3}{1}\) = \(\frac{1 × 3}{6 × 1}\) = \(\frac{1}{2}\) Hence, \(\frac{1}{6}\) × 3 = \(\frac{1}{2}\)

Question 8. \(\frac{3}{8}\) × 4 Answer: \(\frac{3}{8}\) × 4 = \(\frac{3}{2}\)

Explanation: The given numbers are: \(\frac{3}{8}\) and 4 We know that, \(\frac{a}{b}\) × x = \(\frac{a}{b}\) × \(\frac{x}{1}\) = \(\frac{a × x}{b × 1}\) So, \(\frac{3}{8}\) × 4 = \(\frac{3}{8}\) × \(\frac{4}{1}\) = \(\frac{3 × 4}{8 × 1}\) = \(\frac{3}{2}\) Hence, \(\frac{3}{8}\) × 4 = \(\frac{3}{2}\)

Question 9. You have 25 beads. You use \(\frac{2}{5}\) of the beads to make a bracelet. How many beads do you use? Answer: The number of beads you used is: 10 beads

Explanation: It is given that you have 25 beads and you use \(\frac{2}{5}\) of the beads to make a bracelet. So, The number of beads you used = ( The fraction of beads used to make 1 bracelet ) × ( The total number of beads ) = \(\frac{2}{5}\) × 25 We know that, \(\frac{a}{b}\) × x = \(\frac{a}{b}\) × \(\frac{x}{1}\) = \(\frac{a × x}{b × 1}\) So, \(\frac{2}{5}\) × 25 = \(\frac{2}{5}\) × \(\frac{25}{1}\) = \(\frac{2 × 25}{5 × 1}\) = \(\frac{10}{1}\) = 10 beads Hence, from the above, We can conclude that the number of beads you used to make bracelets are: 10 beads

Question 10. Writing Write and solve a real-life problem for the expression. \(\frac{3}{4}\) × 20 Answer: Suppose there are 20 passengers in a bus and they occupied the place \(\frac{3}{4}\) of the bus. Hence, The total number of seats in the bus = ( The fraction of the place occupied by the passengers ) × ( The total number of passengers ) = \(\frac{3}{4}\) × 20 Now, We know that, \(\frac{a}{b}\) × x = \(\frac{a}{b}\) × \(\frac{x}{1}\) = \(\frac{a × x}{b × 1}\) So, \(\frac{3}{4}\) × 20 = \(\frac{3}{4}\) × \(\frac{20}{1}\) = \(\frac{3 × 20}{4 × 1}\) = \(\frac{15}{1}\) = 15 seats Hence, from the above, We can conclude that there are 15 seats in the bus.

Explanation: According to Descartes, The given numbers are: \(\frac{2}{3}\) and 6 So, According to Descartes, 6 is divided into 3 equal parts So, 6 ÷ 3 = 2 equal parts So, According to Descartes, these 2 equal parts also divided into 2 parts. So, The total number of parts = 2 equal parts + 2 equal parts = 4 parts So, The product of \(\frac{2}{3}\) and 6 is 4 Now, We know that, \(\frac{a}{b}\) × x = \(\frac{a}{b}\) × \(\frac{x}{1}\) = \(\frac{a × x}{b × 1}\) So, \(\frac{2}{3}\) × 6 = \(\frac{2}{3}\) × \(\frac{6}{1}\) = \(\frac{6 × 2}{3 × 1}\) = \(\frac{4}{1}\) = 4 Hence, from the above, We can conclude that Descartes is not correct.

Question 12. You have 12 tokens. You use \(\frac{3}{4}\) of them to play a pinball game. How many tokens do you have left? Answer: The number of tokens you have left is: 3 tokens

Explanation: It is given that you have 12 tokens and you use \(\frac{3}{4}\) of them to play a pinball game. So, The number of tokens used to play a pinball game = ( The fraction of tokens used to play a pinball game ) × ( The total number of tokens ) = \(\frac{3}{4}\) × 12 = \(\frac{3}{4}\) × \(\frac{12}{1}\) = \(\frac{3 × 12}{4 × 1}\) = \(\frac{9}{1}\) = 9 tokens So, The number of tokens left = ( The total number of tokens ) – ( The number of tokens used to play the pinball ) = 12 – 9 = 3 tokens Hence, from the above, We can conclude that the number of tokens left are: 3 tokens

Explanation: It is given that a male lion sleeps \(\frac{5}{6}\) × 6 hours of each day So, The number of hours the lion sleeps in 1 day = \(\frac{5}{6}\) × 6 hours = \(\frac{5}{6}\) × \(\frac{6}{1}\) = \(\frac{5 × 6}{6 × 1}\) = \(\frac{5}{1}\) = 5 hours We know that, 1 week = 7 days So, The total number of hours the lion sleeps in 1 week = ( The number of hours the lion sleeps in 1 day ) × ( The number od fays in 1 week ) = 5 × 7 = 35 hours Hence, from the above, We can conclude that the lion sleeps for 35 hours in 1 week

Question 14. DIG DEEPER! In a class of 20 students, \(\frac{1}{10}\) of the students are 10 years old, \(\frac{4}{5}\)of the students are 11 years old, and the rest are 12 years old. How many more 11-year-olds than 12-year-olds are in the class? Answer: The number of 11-year-olds more than 12-year-olds is: 14

Explanation: It is given that in a class of 20 students, \(\frac{1}{10}\) of the students are 10 years old, \(\frac{4}{5}\)of the students are 11 years old, and the rest are 12 years old. So, The number of 10-year-olds = ( The fraction of 10-year-olds ) × ( The total number of students ) = \(\frac{1}{10}\) × 20 = \(\frac{1 × 20}{10 × 1}\) = 2 10-year-olds The number of 11-year-olds = ( The fraction of 11-year-olds ) × ( The total number of students ) = \(\frac{4}{5}\) × 20 = \(\frac{4}{5}\) × \(\frac{20}{1}\) = \(\frac{4 × 20}{5 × 1}\) = \(\frac{16}{1}\) = 16 11-year-olds So, The number of 12-year-olds = ( The total number of students ) – ( The number of 10-year-olds + The number of 11-year-olds ) = 20 – ( 16 + 2 ) = 2 12-year-olds So, The number of 11-year-olds more than 12-year-olds = ( The number of 11-year-olds ) – ( The numebr of 12-year-olds ) = 16 – 2 = 14 students Hence, from the above, We can conclude that there are 14 students who are 11-years-old more than 12-years-old.

Multiply. Use a model to help. Question 1. \(\frac{2}{3}\) × 6 Answer: \(\frac{2}{3}\) × 6 = 4

Explanation: The given numbers are: \(\frac{2}{3}\) and 6 We know that, \(\frac{a}{b}\) × x = \(\frac{a}{b}\) × \(\frac{x}{1}\) = \(\frac{a × x}{b × 1}\) So, \(\frac{2}{3}\) × 6 = \(\frac{2}{3}\) × \(\frac{6}{1}\) = \(\frac{2 × 6}{3 × 1}\) = \(\frac{4}{1}\) = 4 Hence, \(\frac{2}{3}\) × 6 = 4

Question 2. \(\frac{3}{5}\) of 10 Answer: \(\frac{3}{5}\) × 10 = 6

Explanation: The given numbers are: \(\frac{3}{5}\) and 10 We know that, \(\frac{a}{b}\) × x = \(\frac{a}{b}\) × \(\frac{x}{1}\) = \(\frac{a × x}{b × 1}\) So, \(\frac{3}{5}\) × 10 = \(\frac{3}{5}\) × \(\frac{10}{1}\) = \(\frac{3 × 10}{5 × 1}\) = \(\frac{6}{1}\) = 6 Hence, \(\frac{3}{5}\) × 10 = 6

Question 3. \(\frac{1}{2}\) of 4 Answer: \(\frac{1}{2}\) × 4 = 2

Explanation: The given numbers are: \(\frac{1}{2}\) and 4 We know that, \(\frac{a}{b}\) × x = \(\frac{a}{b}\) × \(\frac{x}{1}\) = \(\frac{a × x}{b × 1}\) So, \(\frac{1}{2}\) × 4 = \(\frac{1}{2}\) × \(\frac{4}{1}\) = \(\frac{1 × 4}{2 × 1}\) = \(\frac{2}{1}\) = 2 Hence, \(\frac{1}{2}\) × 4 = 2

Question 4. \(\frac{1}{4}\) × 12 Answer: \(\frac{1}{4}\) × 12 = 3

Explanation: The given numbers are: \(\frac{1}{4}\) and 12 We know that, \(\frac{a}{b}\) × x = \(\frac{a}{b}\) × \(\frac{x}{1}\) = \(\frac{a × x}{b × 1}\) So, \(\frac{1}{4}\) × 12 = \(\frac{1}{4}\) × \(\frac{12}{1}\) = \(\frac{1 × 12}{4 × 1}\) = \(\frac{3}{1}\) = 3 Hence, \(\frac{1}{12}\) × 4 = 3

Question 5. \(\frac{5}{6}\) × 3 Answer: \(\frac{5}{6}\) × 3 = \(\frac{5}{2}\)

Explanation: The given numbers are: \(\frac{5}{6}\) and 3 We know that, \(\frac{a}{b}\) × x = \(\frac{a}{b}\) × \(\frac{x}{1}\) = \(\frac{a × x}{b × 1}\) So, \(\frac{5}{6}\) × 3 = \(\frac{5}{6}\) × \(\frac{3}{1}\) = \(\frac{5 × 3}{6 × 1}\) = \(\frac{5}{2}\) Hence, \(\frac{5}{6}\) × 3 = \(\frac{5}{2}\)

Question 6. \(\frac{3}{4}\) of 2 Answer: \(\frac{3}{4}\) × 2 = \(\frac{3}{2}\)

Explanation: The given numbers are: \(\frac{3}{4}\) and 2 We know that, \(\frac{a}{b}\) × x = \(\frac{a}{b}\) × \(\frac{x}{1}\) = \(\frac{a × x}{b × 1}\) So, \(\frac{3}{4}\) × 2 = \(\frac{3}{4}\) × \(\frac{2}{1}\) = \(\frac{3 × 2}{4 × 1}\) = \(\frac{3}{2}\) Hence, \(\frac{3}{4}\) × 2 = \(\frac{3}{2}\)

Explanation: It is given that there are 27 foam balls and you use \(\frac{1}{3}\) of the balls for a model. So, The number of balls you used = ( The fraction of balls you used ) × ( The total number of balls ) = \(\frac{1}{3}\) × 27 Now, We know that, \(\frac{a}{b}\) × x = \(\frac{a}{b}\) × \(\frac{x}{1}\) = \(\frac{a × x}{b × 1}\) So, \(\frac{1}{3}\) × 27 = \(\frac{1}{3}\) × \(\frac{27}{1}\) = \(\frac{1 × 27}{3 × 1}\) = \(\frac{9}{1}\) = 9 balls Hence, from the above, We can conclude that the number of foam balls you used are: 27

Question 8. An object that weighs 1 pound on Earth weighs about \(\frac{1}{15}\) pound on Pluto. A man weighs 240 pounds on Earth. How many pounds does he weigh on Pluto? Answer: The number of pounds the man weigh in Pluto is: 16 pounds

Explanation: It is given that an object that weighs 1 pound on Earth weighs about \(\frac{1}{15}\) pound on Pluto. It is also given that a man weighs 240 pounds on Earth. So, The weight of a man on pluto = ( The fraction of weight of a man on pluto when compared to earth ) × ( The weight of the man on earth ) = \(\frac{1}{15}\) × 240 Now, We know that, \(\frac{a}{b}\) × x = \(\frac{a}{b}\) × \(\frac{x}{1}\) = \(\frac{a × x}{b × 1}\) So, \(\frac{1}{15}\) × 240 = \(\frac{1}{15}\) × \(\frac{240}{1}\) = \(\frac{1 × 240}{15 × 1}\) = \(\frac{16}{1}\) = 16 pounds Hence, from the above, We can conclude that the weight of the man on pluto is: 16 pounds

Explanation: Let the expressions be named as A), B), C), and D) So, A) The given fractions are: \(\frac{1}{3}\) and 4 Let the missing number be X So, X × \(\frac{1}{3}\) = 4 When multiplication goes from left to right or from right to left, it will become the division When division goes from left to right or from right to left, it will become multiplication. So, X = 4 ÷ \(\frac{1}{3}\) We know that, \(\frac{a}{b}\) ÷ \(\frac{x}{y}\) = \(\frac{a}{b}\) × \(\frac{y}{x}\) So, \(\frac{4}{1}\) ÷ \(\frac{1}{3}\) = \(\frac{4}{1}\) × \(\frac{3}{1}\) = \(\frac{4 × 3}{1 × 1}\) = 12 Hence, the missing number is 12 B) The given numbers are: \(\frac{2}{5}\) and 15 We know that, \(\frac{a}{b}\) × x = \(\frac{a}{b}\) × \(\frac{x}{1}\) = \(\frac{a × x}{b × 1}\) So, \(\frac{2}{5}\) × 15 = \(\frac{2}{5}\) × \(\frac{15}{1}\) = \(\frac{2 × 15}{5 × 1}\) = \(\frac{6}{1}\) = 6 Hence, \(\frac{2}{5}\) × 15 = 6 Like the above two expressions, the remaining two expressions can also be solved. hence, from the above, We can conclude that The missing number of A is: 12 The missing number of B is: 6 The missing number of C is: 2 The missing number of D is: 5

Question 11. Modeling Real Life You have 28 craft sticks. You use \(\frac{4}{7}\) of them for a project. How many craft sticks do you have left? Answer: The number of craft sticks you have left is: 12 craft sticks

Explanation: It is given that you have 28 craft sticks and you used \(\frac{4}{7}\) of them for a project. So, The number of craft sticks you used = ( The fraction of craft sticks you used ) × ( The total number of craft sticks ) = \(\frac{4}{7}\) × 28 = \(\frac{4}{7}\) × \(\frac{28}{1}\) = \(\frac{4 × 28}{7 × 1}\) = \(\frac{16}{1}\) = 16 So, The number of craft sticks you left = ( The total number of craft sticks ) – ( The number of craft sticks you used ) = 28 – 16 = 12 craft sticks Hence, from the above, We can conclude that the number of craft sticks you left is: 12 craft sticks

Explanation: It is given that a mother otter spends \(\frac{1}{3}\) of each day feeding her baby. We know that, 1 day = 24 hours So, The number of hours the mother otter spends feeding her baby in 1 day = ( The fraction of time the mother otter spends in feeding milk ) × 24 = \(\frac{1}{3}\) × 24 = \(\frac{1}{3}\) × \(\frac{24}{1}\) = \(\frac{1 × 24}{3 × 1}\) = \(\frac{8}{1}\) = 8 hours Now, We know that, 1 week = 7 days So, The number of hours the mother otter spends feeding her milk in 1 week = ( The number of hours the mother otter feeding her milk in 1 day ) × 7 = 8 × 7 = 56 hours Hence, from the above, We can conclude that the mother otter spends feeding her milk for 56 hours in 1 week

Estimate the quotient. Question 13. 5,692 ÷ 5 Answer: 5,692 ÷ 5 =1,138 R 2

Explanation: By using the partial quotients method, 5,692 ÷ 5 = ( 5,000 + 600 + 90 ) ÷ 5 = ( 5,000 ÷ 5 ) + ( 600 ÷ 5 ) + ( 90 ÷ 5 ) = 1,000 + 120 + 18 = 1,138 R 2 Hence, 5,692 ÷ 5 = 1,138 R 2

Question 14. 309 ÷ 12 Answer: 309 ÷ 12 = 25 R 9

Explanation: By using the partial quotients method, 309 ÷ 12 = ( 240 + 60 ) ÷ 12 = ( 240 ÷ 12 ) + ( 60 ÷ 12 ) = 20 + 5 = 25 R 9 Hence, 309 ÷ 12 = 25 R 2

Question 15. 2,987 ÷ 53 Answer: 2,987 ÷ 53 = 2,987 ÷ ( 50 + 3 ) = ( 2,985 ÷ 50 ) + ( 2,985 ÷ 3 ) = 995 R 2 + 59.7 = 1,054.7 R 2

From the completed table, We can observe the multiplication of the whole number and the fraction. The product of a whole number and a fraction may be a whole number or a fraction.

Construct Arguments Explain how to multiply fractions and whole numbers without using models. Answer: We can multiply the whole numbers and fractions by using the properties of the multiplication. They are: A) \(\frac{a}{b}\) × p = \(\frac{a}{b}\) × \(\frac{p}{1}\) = \(\frac{a × p}{ b × 1}\) B) p = \(\frac{p}{1}\)

Think and Grow: Multiply Fractions and Whole Numbers

Multiply. Question 1. 3 × \(\frac{5}{8}\) = _______ Answer: 3 × \(\frac{5}{8}\) = \(\frac{15}{8}\)

Explanation: The given numbers are: 3 and \(\frac{5}{8}\) We know that, \(\frac{a}{b}\) × p = \(\frac{a}{b}\) × \(\frac{p}{1}\) = \(\frac{a × p}{ b × 1}\) p = \(\frac{p}{1}\) So, 3 × \(\frac{5}{8}\) = \(\frac{3}{1}\) × \(\frac{5}{8}\) = \(\frac{5 × 3}{8 × 1}\) = \(\frac{15}{8}\) Hence, 3 × \(\frac{5}{8}\) = \(\frac{15}{8}\)

Question 2. 6 × \(\frac{4}{9}\) = _______ Answer: 6 × \(\frac{4}{9}\) = \(\frac{8}{3}\)

Explanation: The given numbers are: 6 and \(\frac{4}{9}\) We know that, \(\frac{a}{b}\) × p = \(\frac{a}{b}\) × \(\frac{p}{1}\) = \(\frac{a × p}{ b × 1}\) p = \(\frac{p}{1}\) So, 6 × \(\frac{4}{9}\) = \(\frac{6}{1}\) × \(\frac{4}{9}\) = \(\frac{6 × 4}{9 × 1}\) = \(\frac{24}{9}\) = \(\frac{8}{3}\) Hence, 6 × \(\frac{4}{9}\) = \(\frac{8}{3}\)

Question 3. \(\frac{2}{5}\) × 15 = _______ Answer: 15 × \(\frac{2}{5}\) = 6

Explanation: The given numbers are: 15 and \(\frac{2}{5}\) We know that, \(\frac{a}{b}\) × p = \(\frac{a}{b}\) × \(\frac{p}{1}\) = \(\frac{a × p}{ b × 1}\) p = \(\frac{p}{1}\) So, 15× \(\frac{2}{5}\) = \(\frac{15}{1}\) × \(\frac{2}{5}\) = \(\frac{15 × 2}{5 × 1}\) = \(\frac{6}{1}\) = 6 Hence, 15 × \(\frac{2}{5}\) = 6

Multiply. Question 4. \(\frac{3}{5}\) × 2 = _______ Answer: 2 × \(\frac{3}{5}\) = \(\frac{6}{5}\)

Explanation: The given numbers are: 2 and \(\frac{3}{5}\) We know that, \(\frac{a}{b}\) × p = \(\frac{a}{b}\) × \(\frac{p}{1}\) = \(\frac{a × p}{ b × 1}\) p = \(\frac{p}{1}\) So, 2 × \(\frac{3}{5}\) = \(\frac{2}{1}\) × \(\frac{3}{5}\) = \(\frac{2 × 3}{5 × 1}\) = \(\frac{6}{5}\) Hence, 2 × \(\frac{3}{5}\) = \(\frac{6}{5}\)

Question 5. 5 × \(\frac{2}{9}\) = _______ Answer: 5 × \(\frac{2}{9}\) = \(\frac{10}{9}\)

Explanation: The given numbers are: 5 and \(\frac{2}{9}\) We know that, \(\frac{a}{b}\) × p = \(\frac{a}{b}\) × \(\frac{p}{1}\) = \(\frac{a × p}{ b × 1}\) p = \(\frac{p}{1}\) So, 5 × \(\frac{2}{9}\) = \(\frac{5}{1}\) × \(\frac{2}{9}\) = \(\frac{5 × 2}{9 × 1}\) = \(\frac{10}{9}\) Hence, 5 × \(\frac{2}{9}\) = \(\frac{10}{9}\)

Question 6. \(\frac{5}{6}\) × 4 = _______ Answer: 4 × \(\frac{5}{6}\) = \(\frac{10}{3}\)

Explanation: The given numbers are: 4 and \(\frac{5}{6}\) We know that, \(\frac{a}{b}\) × p = \(\frac{a}{b}\) × \(\frac{p}{1}\) = \(\frac{a × p}{ b × 1}\) p = \(\frac{p}{1}\) So, 4 × \(\frac{5}{6}\) = \(\frac{4}{1}\) × \(\frac{5}{6}\) = \(\frac{5 × 4}{6 × 1}\) = \(\frac{20}{6}\) = \(\frac{10}{3}\) Hence, 4 × \(\frac{5}{6}\) = \(\frac{10}{3}\)

Question 7. 8 × \(\frac{3}{10}\) = _______ Answer: 8 × \(\frac{3}{10}\) = \(\frac{12}{5}\)

Explanation: The given numbers are: 8 and \(\frac{3}{10}\) We know that, \(\frac{a}{b}\) × p = \(\frac{a}{b}\) × \(\frac{p}{1}\) = \(\frac{a × p}{ b × 1}\) p = \(\frac{p}{1}\) So, 8 × \(\frac{3}{10}\) = \(\frac{8}{1}\) × \(\frac{3}{10}\) = \(\frac{8 × 3}{10 × 1}\) = \(\frac{24}{10}\) = \(\frac{12}{5}\) Hence, 8 × \(\frac{3}{10}\) = \(\frac{12}{5}\)

Question 8. \(\frac{1}{5}\) × 7 = _______ Answer: 7 × \(\frac{1}{5}\) = \(\frac{7}{5}\)

Explanation: The given numbers are: 7 and \(\frac{1}{5}\) We know that, \(\frac{a}{b}\) × p = \(\frac{a}{b}\) × \(\frac{p}{1}\) = \(\frac{a × p}{ b × 1}\) p = \(\frac{p}{1}\) So, 7 × \(\frac{1}{5}\) = \(\frac{7}{1}\) × \(\frac{1}{5}\) = \(\frac{7 × 1}{5 × 1}\) = \(\frac{7}{5}\) Hence, 7 × \(\frac{1}{5}\) = \(\frac{7}{5}\)

Question 9. 9 × \(\frac{5}{12}\) = _______ Answer: 9 × \(\frac{5}{12}\) = \(\frac{15}{4}\)

Explanation: The given numbers are: 9 and \(\frac{5}{12}\) We know that, \(\frac{a}{b}\) × p = \(\frac{a}{b}\) × \(\frac{p}{1}\) = \(\frac{a × p}{ b × 1}\) p = \(\frac{p}{1}\) So, 9 × \(\frac{5}{12}\) = \(\frac{9}{1}\) × \(\frac{5}{12}\) = \(\frac{9 × 5}{12 × 1}\) = \(\frac{45}{12}\) = \(\frac{15}{4}\) Hence, 9 × \(\frac{5}{12}\) = \(\frac{15}{4}\)

Question 10. 15 × \(\frac{5}{8}\) = _______ Answer: 15 × \(\frac{5}{8}\) = \(\frac{75}{8}\)

Explanation: The given numbers are: 15 and \(\frac{5}{8}\) We know that, \(\frac{a}{b}\) × p = \(\frac{a}{b}\) × \(\frac{p}{1}\) = \(\frac{a × p}{ b × 1}\) p = \(\frac{p}{1}\) So, 15 × \(\frac{5}{8}\) = \(\frac{15}{1}\) × \(\frac{5}{8}\) = \(\frac{15 × 5}{8 × 1}\) = \(\frac{75}{8}\) Hence, 15 × \(\frac{5}{8}\) = \(\frac{75}{8}\)

Question 11. \(\frac{3}{4}\) × 20 = _______ Answer: 20 × \(\frac{3}{4}\) = 15

Explanation: The given numbers are: 20 and \(\frac{3}{4}\) We know that, \(\frac{a}{b}\) × p = \(\frac{a}{b}\) × \(\frac{p}{1}\) = \(\frac{a × p}{ b × 1}\) p = \(\frac{p}{1}\) So, 20 × \(\frac{3}{4}\) = \(\frac{20}{1}\) × \(\frac{3}{4}\) = \(\frac{20 × 3}{4 × 1}\) = \(\frac{60}{4}\) = \(\frac{15}{1}\) = 15 Hence, 20 × \(\frac{3}{4}\) = 15

Question 12. \(\frac{7}{9}\) × 5 = _______ Answer: 5 × \(\frac{7}{9}\) = \(\frac{35}{9}\)

Explanation: The given numbers are: 5 and \(\frac{7}{9}\) We know that, \(\frac{a}{b}\) × p = \(\frac{a}{b}\) × \(\frac{p}{1}\) = \(\frac{a × p}{ b × 1}\) p = \(\frac{p}{1}\) So, 5 × \(\frac{7}{9}\) = \(\frac{5}{1}\) × \(\frac{7}{9}\) = \(\frac{5 × 7}{9 × 1}\) = \(\frac{35}{9}\) Hence, 5 × \(\frac{7}{9}\) = \(\frac{35}{9}\)

Explanation: It is given that one-tenth of the 50 states in the United States of America have a mockingbird as their state bird So, The number of states that have a mockingbird as their state bird = ( The fraction of the states that have a mockingbird as their state bird ) × ( The total number of states in the United States of America ) = \(\frac{1}{10}\) × 50 Now, We know that, \(\frac{a}{b}\) × p = \(\frac{a}{b}\) × \(\frac{p}{1}\) = \(\frac{a × p}{ b × 1}\) p = \(\frac{p}{1}\) So, 50 × \(\frac{1}{10}\) = \(\frac{50}{1}\) × \(\frac{1}{10}\) = \(\frac{50 × 1}{10 × 1}\) = \(\frac{50}{10}\) = \(\frac{5}{1}\) = 5 states Hence, from the above, We can conclude that there are 5 states that have mockingbird as their state bird

Question 14. Writing Explain why 9 × \(\frac{2}{3}\) is equivalent to \(\frac{2}{3}\) × 9. Answer: By using the Commutative property of multiplication, a × b = b × a So, By using the above property, In 9 × \(\frac{2}{3}\), ‘a’ is: 9 ‘b’ is: \(\frac{2}{3}\) Hence, from the above, We can conclude that  9 × “\(\frac{2}{3}\)” is equivalent to “\(\frac{2}{3}\) × 9″ by using the Commutative property of multiplication.

Question 15. Reasoning Without calculating, determine which product is greater. Explain. \(\frac{1}{8}\) × 24 \(\frac{7}{8}\) × 24 Answer: The product of “\(\frac{7}{8}\) × 24″ is greater than “\(\frac{1}{8}\) × 24″

Explanation: The given products are: \(\frac{1}{8}\) × 24 and \(\frac{7}{8}\) × 24 We know that, \(\frac{a}{b}\) × p = \(\frac{a}{b}\) × \(\frac{p}{1}\) = \(\frac{a × p}{ b × 1}\) p = \(\frac{p}{1}\) So, from the above fractions, We can observe that except numerator, the denominator and the whole number is all the same. So, To find which product is greater, we just have to compare the numerators of the two fractions without calculating the value of the product. In the comparison of the numerators, We can observe that, 1 < 7 So, First fraction numerator < Second fraction numerator Hence, from the above, We can conclude that the product of “\(\frac{7}{8}\) × 24″ is greater than “\(\frac{1}{8}\) × 24″

Question 16. You take 48 pictures on a walking tour. Five-twelfths of them is of buildings. Your friend takes 45 pictures. Six-fifteenths of them are of buildings. Who takes more pictures of buildings? How many more? Answer: You take more pictures You take 2 pictures more than your friend

Explanation: It is given that you take 48 pictures on a walking tour and five-twelfths if them is of buildings. So, The number of buildings taken by you = ( The total number of pictures taken by you ) × ( The fraction of pictures that is of buildings ) = 48 × \(\frac{5}{12}\) = \(\frac{48}{1}\) × \(\frac{5}{12}\) = \(\frac{48 × 5}{1 × 12}\) = 20 pictures It is also given that your friend taken 45 pictures and six-fifteenth of them is buildings So, The number of buildings taken by your friend = ( The total number of pictures taken by your friend ) × ( The fraction of pictures that is of buildings ) = 45 × \(\frac{6}{15}\) = \(\frac{45}{1}\) × \(\frac{6}{15}\) = \(\frac{45 × 6}{1 × 15}\) = 18 pictures So, In the comparison of the pictures of buildings, You take more pictures than your friend. Now, The number of pictures more taken by you than your friend = ( The number of pictures taken by you ) – ( The number of pictures taken by your friend ) = 20 – 18 = 2 pictures Hence, from the above, We can conclude that You take more pictures You take 2 pictures more than your friend

Question 17. You have 72 rocks in your rock collection. Five-eighths of them are sedimentary, one-sixth of them are igneous, and the rest are metamorphic. How many of your rocks are metamorphic? Answer: The number of rocks that are metamorphic is: 15 metamorphous rocks

Explanation: It is given that you have 72 rocks in your rock collection and five-eighths of them are sedimentary, one-sixth of them are igneous, and the rest are metamorphic. So, The number of rocks that are sedimentary = ( The total number of rocks ) × ( The fraction of rocks that are sedimentary ) = 72 × \(\frac{5}{8}\) = \(\frac{72}{1}\) × \(\frac{5}{8}\) = \(\frac{72 × 5}{1 × 8}\) = 45 sedimentary rocks The number of rocks that are igneous = ( The total number of rocks ) × ( The fraction of rocks that are igneous ) = 72 × \(\frac{1}{6}\) = \(\frac{72}{1}\) × \(\frac{1}{6}\) = \(\frac{72 × 1}{1 × 6}\) = 12 igneous rocks So, The number of metamorphous rocks = ( The total number of rocks ) – ( The number of sedimentary rocks + The number of igneous rocks) = 72 – ( 45 + 12 ) = 15 metamorphous rocks Hence, from the above, We can conclude that there are 15 metamorphous rocks

Explanation: It is given that, each day you spent \(\frac{3}{4}\) hour reading and \(\frac{1}{2}\) hour writing in a journal. So, The number of hours you spent on reading and writing a journal in 1 day = ( The number of hours spent on reading ) + ( The number of hours spent on writing ) = \(\frac{3}{4}\) + \(\frac{1}{2}\) Multiply \(\frac{1}{2}\) with \(\frac{2}{2}\) So, \(\frac{3}{4}\) + \(\frac{2}{4}\) = \(\frac{3 + 2}{4}\) = \(\frac{5}{4}\) hours We know that, 1 day = 24 hours 1 week = 7 days So, 1 week = 7 × 24 hours So, The number of hours you spent on reading and writing in 1 week = ( The number of hours you spent on reading and writing in 1 day ) × ( The number of hours in 1 week ) = \(\frac{5}{4}\) × 7 × 24 = \(\frac{5}{4}\) × \(\frac{7}{1}\) × \(\frac{24}{1}\) = \(\frac{5 × 7 × 24}{4 × 1}\) = 5 × 7 × 6 = 210 hours Hence, from the above, We can conclude that the number of hours you spent on reading and writing a journal in 1 week is: 210 hours

Multiply. Question 1. \(\frac{5}{6}\) × 3 = _______ Answer: 3 × \(\frac{5}{6}\) = \(\frac{5}{2}\)

Explanation: The given numbers are: 3 and \(\frac{5}{6}\) We know that, \(\frac{a}{b}\) × p = \(\frac{a}{b}\) × \(\frac{p}{1}\) = \(\frac{a × p}{ b × 1}\) p = \(\frac{p}{1}\) So, 3 × \(\frac{5}{6}\) = \(\frac{3}{1}\) × \(\frac{5}{6}\) = \(\frac{3 × 5}{6 × 1}\) = \(\frac{5}{6}\) = \(\frac{5}{2}\) Hence, 3 × \(\frac{5}{6}\) = \(\frac{5}{2}\)

Question 2. \(\frac{2}{3}\) × 6 = _______ Answer: 6 × \(\frac{2}{3}\) = 4

Explanation: The given numbers are: 6 and \(\frac{2}{3}\) We know that, \(\frac{a}{b}\) × p = \(\frac{a}{b}\) × \(\frac{p}{1}\) = \(\frac{a × p}{ b × 1}\) p = \(\frac{p}{1}\) So, 6 × \(\frac{2}{3}\) = \(\frac{6}{1}\) × \(\frac{2}{3}\) = \(\frac{6 × 2}{3 × 1}\) = \(\frac{12}{3}\) = \(\frac{4}{1}\) = 4 Hence, 6 × \(\frac{2}{3}\) = 4

Question 3. 7 × \(\frac{1}{8}\) = _______ Answer: 7 × \(\frac{1}{8}\) = \(\frac{7}{8}\)

Explanation: The given numbers are: 7 and \(\frac{1}{8}\) We know that, \(\frac{a}{b}\) × p = \(\frac{a}{b}\) × \(\frac{p}{1}\) = \(\frac{a × p}{ b × 1}\) p = \(\frac{p}{1}\) So, 7 × \(\frac{1}{8}\) = \(\frac{7}{1}\) × \(\frac{1}{8}\) = \(\frac{7 × 1}{8 × 1}\) = \(\frac{7}{8}\) Hence, 7 × \(\frac{1}{8}\) = \(\frac{7}{8}\)

Question 4. 2 × \(\frac{1}{2}\) = _______ Answer: 2 × \(\frac{1}{2}\) = 1

Explanation: The given numbers are: 2 and \(\frac{1}{2}\) We know that, \(\frac{a}{b}\) × p = \(\frac{a}{b}\) × \(\frac{p}{1}\) = \(\frac{a × p}{ b × 1}\) p = \(\frac{p}{1}\) So, 2 × \(\frac{1}{2}\) = \(\frac{2}{1}\) × \(\frac{1}{2}\) = \(\frac{2 × 1}{2 × 1}\) = \(\frac{2}{2}\) = 1 Hence, 2 × \(\frac{1}{2}\) = 1

Question 5. \(\frac{4}{5}\) × 9 = _______ Answer: 9 × \(\frac{4}{5}\) = \(\frac{36}{5}\)

Explanation: The given numbers are: 9 and \(\frac{4}{5}\) We know that, \(\frac{a}{b}\) × p = \(\frac{a}{b}\) × \(\frac{p}{1}\) = \(\frac{a × p}{ b × 1}\) p = \(\frac{p}{1}\) So, 9 × \(\frac{4}{5}\) = \(\frac{9}{1}\) × \(\frac{4}{5}\) = \(\frac{9 × 4}{5 × 1}\) = \(\frac{36}{5}\) Hence, 9 × \(\frac{4}{5}\) = \(\frac{36}{5}\)

Question 6. 4 × \(\frac{5}{12}\) = _______ Answer: 4 × \(\frac{5}{12}\) = \(\frac{5}{3}\)

Explanation: The given numbers are: 4 and \(\frac{5}{12}\) We know that, \(\frac{a}{b}\) × p = \(\frac{a}{b}\) × \(\frac{p}{1}\) = \(\frac{a × p}{ b × 1}\) p = \(\frac{p}{1}\) So, 4 × \(\frac{5}{12}\) = \(\frac{4}{1}\) × \(\frac{5}{12}\) = \(\frac{4 × 5}{12 × 1}\) = \(\frac{20}{12}\) = \(\frac{5}{3}\) Hence, 4 × \(\frac{5}{12}\) = \(\frac{5}{3}\)

Question 7. \(\frac{1}{4}\) × 24 = _______ Answer: 24 × \(\frac{1}{24}\) = 6

Explanation: The given numbers are: 24 and \(\frac{1}{4}\) We know that, \(\frac{a}{b}\) × p = \(\frac{a}{b}\) × \(\frac{p}{1}\) = \(\frac{a × p}{ b × 1}\) p = \(\frac{p}{1}\) So, 24 × \(\frac{1}{4}\) = \(\frac{24}{1}\) × \(\frac{1}{4}\) = \(\frac{24 × 1}{4 × 1}\) = \(\frac{24}{4}\) = 6 Hence, 24 × \(\frac{1}{4}\) = 6

Question 8. 16 × \(\frac{3}{8}\) = _______ Answer: 16 × \(\frac{3}{8}\) = 6

Explanation: The given numbers are: 16 and \(\frac{3}{8}\) We know that, \(\frac{a}{b}\) × p = \(\frac{a}{b}\) × \(\frac{p}{1}\) = \(\frac{a × p}{ b × 1}\) p = \(\frac{p}{1}\) So, 16 × \(\frac{3}{8}\) = \(\frac{16}{1}\) × \(\frac{3}{8}\) = \(\frac{16 × 3}{8 × 1}\) = \(\frac{48}{8}\) = 6 Hence, 16 × \(\frac{3}{8}\) = 6

Question 9. \(\frac{7}{10}\) × 25 = _______ Answer: 25 × \(\frac{7}{10}\) = \(\frac{35}{2}\)

Explanation: The given numbers are: 25 and \(\frac{7}{10}\) We know that, \(\frac{a}{b}\) × p = \(\frac{a}{b}\) × \(\frac{p}{1}\) = \(\frac{a × p}{ b × 1}\) p = \(\frac{p}{1}\) So, 25 × \(\frac{7}{10}\) = \(\frac{25}{1}\) × \(\frac{7}{10}\) = \(\frac{7 × 25}{10 × 1}\) = \(\frac{175}{10}\) = \(\frac{35}{2}\) Hence, 25 × \(\frac{7}{10}\) = \(\frac{35}{2}\)

Explanation: It is given that you spent \(\frac{3}{4}\) hour jumping rope every week for 8 weeks. So, The number of hours you spent jumping rope in 8 weeks = ( The number of hours you jump rope in 1 week ) × ( The total number of weeks ) = \(\frac{3}{4}\) × 8 = \(\frac{3}{4}\) × \(\frac{8}{1}\) = \(\frac{3 × 8}{4 × 1}\) = 6 hours hence, from the above, We can conclude that you spent 6 hours on jumping rope in 8 weeks.

Question 11. Logic Your friend finds 25 items that are either insects or flowers. She says that \(\frac{1}{6}\) of the items are insects. Can this be true? Explain. Answer: No, this is not true

Explanation: It is given that your friend finds 25 items that are either insects or flowers and she says that \(\frac{1}{6}\) of the items are insects. But, The number of either insects or flowers will always be a whole number. But, from the given information, 25 will not be divided by \(\frac{1}{6}\) So, The statement “\(\frac{1}{6}\) of the items are insects” is false.

Question 12. Open-Ended Write two different pairs of fractions that could represent the insects and flowers your friend finds in Exercise 11. Answer: The different pairs of fractions that could represent the insects and flowers are: \(\frac{1}{5}\) and \(\frac{1}{25}\)

Explanation: These different pairs of fractions have to divide 25 So, We have to take the fractions the multiples of 5 i.e., 5 and 25 Hence, from the above, We can conclude that The different pairs of fractions that could represent the insects and flowers are: \(\frac{1}{5}\) and \(\frac{1}{25}\)

Explanation: It is given that Newton bakes 56 treats and five-eights of them contains peanut butter So, The number of peanut butter treats made by Newton = \(\frac{5}{8}\) × 56 = \(\frac{5}{8}\) × \(\frac{56}{1}\) = \(\frac{5 × 56}{8 × 1}\) = 35 peanut butter treats It is also given that Descartes bakes 120 treats and five-sixths of them contain peanut butter So, the number of peanut butter treats made by Descartes = \(\frac{5}{6}\) × 120 = \(\frac{5}{6}\) × \(\frac{120}{1}\) = \(\frac{5 × 120}{6 × 1}\) = 100 peanut butter treats So, In the comparison of peanut butter treats, Descartes bakes more Now, The number of peanut butter treats baked by Descartes more than Newton = 100 – 35 = 65 peanut butter treats Hence, from the above, We can conclude that Descartes bakes more peanut butter treats The number of peanut butter treats Descartes made more than Descartes is: 65

Question 14. Modeling Real Life Your class conducts an egg-dropping experiment with 60 eggs. Three-fifths of the eggs break open, one-sixth of the eggs crack, and the rest do not break at all. How many of the eggs do not crack or break open? Answer: The number of eggs that do not break is:

Explanation: It is given that your class conducts an egg-dropping experiment with 60 eggs and three-fifths of the eggs break open, one-sixth of the eggs crack and the rest do not break. So, The number of eggs that break open = 60 × \(\frac{3}{5}\) = \(\frac{60}{1}\) × \(\frac{3}{5}\) = 36 The number of eggs that cracked = 60 × \(\frac{1}{6}\) = \(\frac{60}{1}\) × \(\frac{1}{6}\) = 10 So, The number of eggs that do not breaked = ( The total number of eggs ) – ( The number of eggs that break open + The number of eggs that cracked ) = 60 – ( 36 + 10 ) = 16 Hence, from the above, We can conclude that the number of eggs that do not break is: 16 eggs

Add. Question 15. 5\(\frac{5}{8}\) + 6\(\frac{3}{4}\) = _______ Answer: 5\(\frac{5}{8}\) + 6\(\frac{3}{4}\) = \(\frac{99}{8}\)

Explanation: The give mixed fractions are: 6\(\frac{3}{4}\) and 5\(\frac{5}{8}\) The representation of mixed fractions in the improper form is: \(\frac{45}{8}\) and \(\frac{27}{4}\) In addition, We have to make the denominators equal. So, Multiply \(\frac{27}{4}\) with \(\frac{2}{2}\) So, \(\frac{45}{8}\) + \(\frac{54}{8}\) = \(\frac{45 + 54}{8}\) = \(\frac{99}{8}\) Hence, 5\(\frac{5}{8}\) + 6\(\frac{3}{4}\) = \(\frac{99}{8}\)

Question 16. 1\(\frac{5}{6}\) + 8\(\frac{1}{12}\) = ________ Answer: 1\(\frac{5}{6}\) + 8\(\frac{1}{12}\) = \(\frac{119}{12}\)

Explanation: The give mixed fractions are: 1\(\frac{5}{6}\) and 8\(\frac{1}{12}\) The representation of mixed fractions in the improper form is: \(\frac{11}{6}\) and \(\frac{97}{12}\) In addition, We have to make the denominators equal. So, Multiply \(\frac{11}{6}\) with \(\frac{2}{2}\) So, \(\frac{97}{12}\) + \(\frac{22}{12}\) = \(\frac{97 + 22}{12}\) = \(\frac{119}{12}\) Hence, 1\(\frac{5}{6}\) + 8\(\frac{1}{12}\) = \(\frac{119}{12}\)

Question 17. 3\(\frac{1}{2}\) + \(\frac{3}{5}\) + 2\(\frac{7}{10}\) = _______ Answer: 3\(\frac{1}{2}\) + 2\(\frac{7}{10}\) + \(\frac{3}{5}\)  = \(\frac{68}{10}\)

Explanation: The give mixed fractions are: 3\(\frac{1}{2}\), \(\frac{3}{5}\) and 2\(\frac{7}{10}\) The representation of mixed fractions in the improper form is: \(\frac{7}{2}\) and \(\frac{27}{10}\) In addition, We have to make the denominators equal. So, Multiply \(\frac{7}{2}\) with \(\frac{5}{5}\) Multiply \(\frac{3}{5}\) with \(\frac{2}{2}\) So, \(\frac{27}{10}\) + \(\frac{35}{10}\) + \(\frac{6}{10}\)  = \(\frac{27 + 35 + 6}{10}\) = \(\frac{68}{10}\) Hence, 3\(\frac{1}{2}\) + 2\(\frac{7}{10}\) + \(\frac{3}{5}\)  = \(\frac{68}{10}\)

Fold a sheet of paper in half. Shade \(\frac{1}{4}\) of either half. What fraction of the entire sheet of paper did you shade? Draw a model to support your answer. Answer: The fraction of the entire sheet of paper you shaded is: \(\frac{1}{8}\)

Explanation: Take a full sheet of paper and fold in half So, The number of parts of full sheet of paper = \(\frac{1}{2}\) Now, Shade \(\frac{1}{4}\) of the half of the paper So,, The fraction of the paper you shaded = \(\frac{1}{4}\) × \(\frac{1}{2}\) We know that, \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) a = \(\frac{a}{1}\) So, \(\frac{1}{4}\) × \(\frac{1}{2}\) = \(\frac{1 × 1}{4 × 2}\) = \(\frac{1}{8}\) Hence, from the above, We can conclude that the \(\frac{1}{8}\) part of the full sheet is shaded

Reasoning What multiplication expression does your model represent? Explain your reasoning. Answer: The multiplication expression your model represents is: \(\frac{1}{8}\) We can obtain the multiplication expression by using the following multiplication properties. They are: A) \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) B) a = \(\frac{a}{1}\)

Think and Grow: Use Models to Multiply Fractions

Multiply. Use a model to help. Question 1. \(\frac{1}{3}\) × \(\frac{1}{4}\) = _______ Answer: \(\frac{1}{3}\) × \(\frac{1}{4}\) = \(\frac{1}{12}\)

Explanation: The given fractions are: \(\frac{1}{4}\) and \(\frac{1}{3}\) We know that, \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) a = \(\frac{a}{1}\) So, \(\frac{1}{4}\) × \(\frac{1}{3}\) = \(\frac{1 × 1}{4 × 3}\) = \(\frac{1}{12}\) Hence, \(\frac{1}{4}\) × \(\frac{1}{3}\) = \(\frac{1}{12}\)

Question 2. \(\frac{2}{3}\) × \(\frac{1}{2}\) = _______ Answer: \(\frac{2}{3}\) × \(\frac{1}{2}\) = \(\frac{1}{3}\)

Explanation: The given fractions are: \(\frac{2}{3}\) and \(\frac{1}{2}\) We know that, \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) a = \(\frac{a}{1}\) So, \(\frac{1}{2}\) × \(\frac{2}{3}\) = \(\frac{1 × 2}{2 × 3}\) = \(\frac{1}{3}\) Hence, \(\frac{1}{2}\) × \(\frac{2}{3}\) = \(\frac{1}{3}\)

Multiply. Use a model to help. Question 3. \(\frac{1}{2}\) × \(\frac{1}{6}\) = _______ Answer: \(\frac{1}{2}\) × \(\frac{1}{6}\) = \(\frac{1}{12}\)

Explanation: The given fractions are: \(\frac{1}{2}\) and \(\frac{1}{6}\) We know that, \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) a = \(\frac{a}{1}\) So, \(\frac{1}{2}\) × \(\frac{1}{6}\) = \(\frac{1 × 1}{2 × 6}\) = \(\frac{1}{12}\) Hence, \(\frac{1}{2}\) × \(\frac{1}{6}\) = \(\frac{1}{12}\)

Question 4. \(\frac{1}{5}\) × \(\frac{1}{8}\) = _______ Answer: \(\frac{1}{5}\) × \(\frac{1}{8}\) = \(\frac{1}{40}\)

Explanation: The given fractions are: \(\frac{1}{5}\) and \(\frac{1}{8}\) We know that, \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) a = \(\frac{a}{1}\) So, \(\frac{1}{5}\) × \(\frac{1}{8}\) = \(\frac{1 × 1}{5 × 8}\) = \(\frac{1}{40}\) Hence, \(\frac{1}{5}\) × \(\frac{1}{8}\) = \(\frac{1}{40}\)

Question 5. \(\frac{1}{4}\) × \(\frac{1}{6}\) = _______ Answer: \(\frac{1}{4}\) × \(\frac{1}{6}\) = \(\frac{1}{24}\)

Explanation: The given fractions are: \(\frac{1}{4}\) and \(\frac{1}{6}\) We know that, \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) a = \(\frac{a}{1}\) So, \(\frac{1}{4}\) × \(\frac{1}{6}\) = \(\frac{1 × 1}{4 × 6}\) = \(\frac{1}{24}\) Hence, \(\frac{1}{4}\) × \(\frac{1}{6}\) = \(\frac{1}{24}\)

Question 6. \(\frac{2}{3}\) × \(\frac{1}{3}\) = _______ Answer: \(\frac{2}{3}\) × \(\frac{1}{3}\) = \(\frac{2}{9}\)

Explanation: The given fractions are: \(\frac{2}{3}\) and \(\frac{1}{3}\) We know that, \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) a = \(\frac{a}{1}\) So, \(\frac{2}{3}\) × \(\frac{1}{3}\) = \(\frac{2 × 1}{3 × 3}\) = \(\frac{2}{9}\) Hence, \(\frac{2}{3}\) × \(\frac{1}{3}\) = \(\frac{2}{9}\)

Question 9. One-fifth of the students in your school have tried skating. Of those students, \(\frac{1}{7}\) have tried ice skating. What fraction of students in your school have tried ice skating? Answer: The fraction of students that have tried ice skating is: \(\frac{1}{35}\)

Explanation: It is given that one-fifth of the students in your school have tried skating and of these students, \(\frac{1}{7}\) have tried ice skating So, The fraction of students that have tried ice skating = ( The fraction of students that have tried skating ) × ( The fraction of students that tried ice skating out of the total students ) = \(\frac{1}{5}\) × \(\frac{1}{7}\) We know that, \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) a = \(\frac{a}{1}\) So, \(\frac{1}{5}\) × \(\frac{1}{7}\) = \(\frac{1 × 1}{5 × 7}\) = \(\frac{1}{35}\) Hence, from the above, We can conclude that \(\frac{1}{35}\) of the total students tried ice skating.

Explanation: It is given that the mass of mango is \(\frac{2}{5}\) kilogram and the mass of guava is \(\frac{1}{4}\) as much as the mango. So, The mass of guava = ( The mass of mango ) × ( The fraction of mass of guava in the mass of mango ) = \(\frac{2}{5}\) × \(\frac{1}{4}\) = \(\frac{1 × 2}{4 × 5}\) = \(\frac{2}{20}\) = \(\frac{1}{10}\) kilograms Hnce, from the above, We can conclude that the mass of guava is: \(\frac{1}{10}\) kilogram

Question 12. A giant panda spends \(\frac{2}{3}\) of 1-day eating and foraging. It spends \(\frac{3}{4}\) of that time for eating bamboo. What fraction of 1 day does the panda spend eating bamboo? Answer: The fraction of 1 day the panda spent on eating bamboo is: \(\frac{1}{2}\)

Explanation: It is given that a giant panda spends \(\frac{2}{3}\) of 1-day eating and foraging and it spends \(\frac{3}{4}\) of that time for eating bamboo. So, The fraction of 1 day the panda spent on eating bamboo = ( The fraction of time the panda spent on eating ) × ( The fraction of time the panda spent time on eating bamboo out of the total time ) = \(\frac{2}{3}\) × \(\frac{3}{4}\) = \(\frac{3 × 2}{4 × 3}\) = \(\frac{6}{12}\) = \(\frac{1}{2}\) hour Hence, from the above, We can conclude that the time spent by the panda on eating bamboo is: \(\frac{1}{2}\) hour

Question 13. DIG DEEPER! You have a half-gallon carton of milk that you use only for cereal. You use the same amount each day for 5 days. There is \(\frac{3}{8}\) of the carton left. How many cups of milk do you use each day? Explain. Answer:

Multiply. Use a model to help. Question 1. \(\frac{1}{3}\) × \(\frac{1}{7}\) = _______ Answer: \(\frac{1}{3}\) × \(\frac{1}{7}\) = \(\frac{1}{21}\)

Explanation: The given fractions are: \(\frac{1}{7}\) and \(\frac{1}{3}\) We know that, \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) a = \(\frac{a}{1}\) So, \(\frac{1}{7}\) × \(\frac{1}{3}\) = \(\frac{1 × 1}{7 × 3}\) = \(\frac{1}{21}\) Hence, \(\frac{1}{7}\) × \(\frac{1}{3}\) = \(\frac{1}{21}\)

Question 2. \(\frac{1}{2}\) × \(\frac{1}{9}\) = _______ Answer: \(\frac{1}{2}\) × \(\frac{1}{9}\) = \(\frac{1}{18}\)

Explanation: The given fractions are: \(\frac{1}{2}\) and \(\frac{1}{9}\) We know that, \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) a = \(\frac{a}{1}\) So, \(\frac{1}{2}\) × \(\frac{1}{9}\) = \(\frac{1 × 1}{2 × 9}\) = \(\frac{1}{18}\) Hence, \(\frac{1}{2}\) × \(\frac{1}{9}\) = \(\frac{1}{18}\)

Question 3. \(\frac{2}{5}\) × \(\frac{1}{6}\) = _______ Answer: \(\frac{2}{5}\) × \(\frac{1}{6}\) = \(\frac{1}{15}\)

Explanation: The given fractions are: \(\frac{2}{5}\) and \(\frac{1}{6}\) We know that, \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) a = \(\frac{a}{1}\) So, \(\frac{2}{5}\) × \(\frac{1}{6}\) = \(\frac{2 × 1}{5 × 6}\) = \(\frac{1}{15}\) Hence, \(\frac{2}{5}\) × \(\frac{1}{6}\) = \(\frac{1}{15}\)

Question 4. \(\frac{3}{4}\) × \(\frac{2}{7}\) = _______ Answer: \(\frac{3}{4}\) × \(\frac{2}{7}\) = \(\frac{3}{14}\)

Explanation: The given fractions are: \(\frac{3}{4}\) and \(\frac{2}{7}\) We know that, \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) a = \(\frac{a}{1}\) So, \(\frac{3}{4}\) × \(\frac{2}{7}\) = \(\frac{3 × 2}{4 × 7}\) = \(\frac{3}{14}\) Hence, \(\frac{3}{4}\) × \(\frac{2}{7}\) = \(\frac{3}{14}\)

Question 7. One-sixth of the animals at a zoo are birds. Of the birds, \(\frac{1}{3}\) are female. What fraction of the animals at the zoo are female birds? Answer: The fraction of the animals at the zoo that is female birds is: \(\frac{1}{18}\)

Explanation: It is given that one-sixth of the animals at the zoo are birds and of the birds, \(\frac{1}{3}\) is female. So, The fraction of the animals at the zoo are female birds = ( The fraction of animals in the zoo that is birds ) × ( The fraction of the animals in the zoo that is female birds ) = \(\frac{1}{3}\) × \(\frac{1}{6}\) \(\frac{1 × 1}{3 × 6}\) = \(\frac{1}{18}\) Hence, from the above, We can conclude that the number of animals that are female birds at the zoo is: \(\frac{1}{18}\)

Question 8. Writing Write and solve a real-life problem for the expression. \(\frac{2}{3}\) × \(\frac{3}{7}\) Answer: Suppose in a school, there are boys ad girls. In school, \(\frac{2}{3}\) of the boys and \(\frac{3}{7}\) of the girls passed the PET test So, The fraction of the students that passed the PET test out of the total number of students = ( The fraction of the boys that passed the PET test ) × ( The fraction of the girls that passed the PET test ) = \(\frac{2}{3}\) × \(\frac{3}{7}\) = \(\frac{3 × 2}{7 × 3}\) = \(\frac{6}{21}\) = \(\frac{2}{7}\) Hence, from the above, We can conclude that the number of students who passed the PET test is: \(\frac{2}{7}\)

Explanation: It is given that a Gouldian finch is \(\frac{1}{2}\) the length of the sun conure. From the given figure, The length of the sun conure = \(\frac{11}{12}\) feet So, The length of Gouldian finch = ( The length of Sun Conure ) ÷ 2 = \(\frac{11}{12}\) ÷ 2 = \(\frac{11}{12}\) ÷ \(\frac{2}{1}\) = \(\frac{11}{12}\) × \(\frac{1}{2}\) = \(\frac{11 × 1}{12 × 2}\) = \(\frac{11}{24}\) feet Hence, from the above, We can conclude that the length of the Gouldian finch is: \(\frac{11}{24}\) feet

Question 10. DIG DEEPER! A recipe calls for \(\frac{2}{3}\) cup of chopped walnuts. You chop 4 walnuts and get \(\frac{1}{4}\) of the amount you need. How much more of a cup of chopped walnuts do you need? All of your walnuts are the same size. How many more walnuts should you chop? Explain. Answer: The cup of chopped walnuts more you need is: \(\frac{1}{3}\) The number of walnuts you should chop is: 6

Explanation: It is given that a recipe calls for \(\frac{2}{3}\) cup of chopped walnuts. It is also given that you chop 4 walnuts and get \(\frac{1}{4}\) of the amount you need. So, The cup more of chopped walnuts you need = 1- ( The cup of chopped walnuts you need for recipe ) = 1 – \(\frac{2}{3}\) = \(\frac{3}{3}\) – \(\frac{2}{3}\) = \(\frac{3 – 2}{3}\) = \(\frac{1}{3}\) more cup Now, The number of walnuts you need more = ( The number of walnuts you needed to get \(\frac{1}{4}\) of cup of chopped walnuts ) ÷ ( The amount of chopped walnuts you need for recipe ) = 4 ÷ \(\frac{2}{3}\) = \(\frac{4}{1}\) ÷ \(\frac{2}{3}\) = \(\frac{4}{1}\) × \(\frac{3}{2}\) = \(\frac{4 × 3}{1 × 2}\) = \(\frac{12}{2}\) = 6 Hence, from the above, We can conclude that The cup of chopped walnuts more you need is: \(\frac{1}{3}\) The number of walnuts you should chop is: 6

Subtract. Question 11. 5 – 1\(\frac{3}{4}\) = ______ Answer: 5 – 1\(\frac{3}{4}\) = \(\frac{13}{4}\)

Explanation: The given numbers are: 5 and 1\(\frac{3}{4}\) The representation of 1\(\frac{3}{4}\) in the improper fraction form is: \(\frac{7}{4}\) So, 5 – \(\frac{7}{4}\) = \(\frac{20}{4}\) – \(\frac{7}{4}\) = \(\frac{20 – 7}{4}\) = \(\frac{13}{4}\) Hence, 5 – 1\(\frac{3}{4}\) = \(\frac{13}{4}\)

Question 12. 13\(\frac{1}{4}\) – 7\(\frac{5}{8}\) = ________ Answer: 13\(\frac{1}{4}\) – 7\(\frac{5}{8}\) = \(\frac{45}{8}\)

Explanation: The given mixed fractions are: 13\(\frac{1}{4}\) and 7\(\frac{5}{8}\) The representation of 13\(\frac{1}{4}\) in the improper fraction form is: \(\frac{53}{4}\) The representation of 13\(\frac{1}{4}\) in the improper fraction form is: \(\frac{61}{8}\) To subtract the fractions, we have to make the denominators equal. So, Multiply \(\frac{53}{4}\) by \(\frac{2}{2}\) So, \(\frac{53}{4}\) = \(\frac{106}{8}\) So, \(\frac{106}{8}\) – \(\frac{61}{8}\) = \(\frac{106 – 61}{8}\) = \(\frac{45}{8}\) Hence, 13\(\frac{1}{4}\) – 13\(\frac{1}{4}\) = \(\frac{45}{8}\)

Question 13. 12\(\frac{7}{10}\) – 5\(\frac{3}{10}\) – 1\(\frac{1}{5}\) = ________ Answer: 12\(\frac{7}{10}\) – 5\(\frac{3}{10}\) – 1\(\frac{1}{5}\) = \(\frac{62}{10}\)

Explanation: The given mixed fractions are: 12\(\frac{7}{10}\), 5\(\frac{3}{10}\) and 1\(\frac{1}{5}\) The representation of 12\(\frac{7}{10}\) in the improper fraction form is: \(\frac{127}{10}\) The representation of 5\(\frac{3}{10}\) in the improper fraction form is: \(\frac{53}{10}\) The representation of 1\(\frac{1}{5}\) in the improper fraction form is: \(\frac{6}{5}\) To subtract the fractions, we have to make the denominators equal. So, Multiply \(\frac{6}{5}\) by \(\frac{2}{2}\) So, \(\frac{6}{5}\) = \(\frac{12}{10}\) So, \(\frac{127}{10}\) – \(\frac{53}{10}\) – \(\frac{12}{10}\) = \(\frac{127 – 53 – 12}{8}\) = \(\frac{62}{10}\) Hence, 12\(\frac{7}{10}\) – 5\(\frac{3}{10}\) – 1\(\frac{1}{5}\) = \(\frac{62}{10}\)

Construct Arguments Explain how to multiply two fractions without using a model. Answer: We can multiply the fractions by using the properties of multiplication. They are: A) \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) B) a = \(\frac{a}{1}\)

Think and Grow: Multiply Fractions

Multiply. Question 1. \(\frac{1}{2}\) × \(\frac{4}{3}\) = _______ Answer: \(\frac{1}{2}\) × \(\frac{4}{3}\) = \(\frac{2}{3}\)

Explanation: The given fractions are: \(\frac{1}{2}\) and \(\frac{4}{3}\) We know that, \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) a = \(\frac{a}{1}\) So, \(\frac{1}{2}\) × \(\frac{4}{3}\) = \(\frac{1 × 4}{2 × 3}\) = \(\frac{4}{6}\) = \(\frac{2}{3}\) Hence, \(\frac{1}{2}\) × \(\frac{4}{3}\) = \(\frac{2}{3}\)

Question 2. \(\frac{2}{5}\) × \(\frac{2}{3}\) = _______ Answer: \(\frac{2}{5}\) × \(\frac{2}{3}\) = \(\frac{4}{15}\)

Explanation: The given fractions are: \(\frac{2}{5}\) and \(\frac{2}{3}\) We know that, \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) a = \(\frac{a}{1}\) So, \(\frac{2}{5}\) × \(\frac{2}{3}\) = \(\frac{2 × 2}{5 × 3}\) = \(\frac{4}{15}\) Hence, \(\frac{2}{5}\) × \(\frac{2}{3}\) = \(\frac{4}{15}\)

Question 3. \(\frac{3}{4}\) × \(\frac{5}{8}\) = _______ Answer: \(\frac{3}{4}\) × \(\frac{5}{8}\) = \(\frac{15}{32}\)

Explanation: The given fractions are: \(\frac{3}{4}\) and \(\frac{5}{8}\) We know that, \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) a = \(\frac{a}{1}\) So, \(\frac{3}{4}\) × \(\frac{5}{8}\) = \(\frac{3 × 5}{4 × 8}\) = \(\frac{15}{32}\) Hence, \(\frac{3}{4}\) × \(\frac{5}{8}\) = \(\frac{15}{32}\)

Multiply. Question 4. \(\frac{1}{4}\) × \(\frac{1}{4}\) = _______ Answer: \(\frac{1}{4}\) × \(\frac{1}{4}\) = \(\frac{1}{16}\)

Explanation: The given fractions are: \(\frac{1}{4}\) and \(\frac{1}{4}\) We know that, \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) a = \(\frac{a}{1}\) So, \(\frac{1}{4}\) × \(\frac{1}{4}\) = \(\frac{1 × 1}{4 × 4}\) = \(\frac{1}{16}\) Hence, \(\frac{1}{4}\) × \(\frac{1}{4}\) = \(\frac{1}{16}\)

Question 5. \(\frac{5}{6}\) × \(\frac{7}{10}\) = _______ Answer: \(\frac{5}{6}\) × \(\frac{7}{10}\) = \(\frac{7}{12}\)

Explanation: The given fractions are: \(\frac{5}{6}\) and \(\frac{7}{10}\) We know that, \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) a = \(\frac{a}{1}\) So, \(\frac{5}{6}\) × \(\frac{7}{10}\) = \(\frac{5 × 7}{6 × 10}\) = \(\frac{35}{60}\) = \(\frac{7}{12}\) Hence, \(\frac{5}{6}\) × \(\frac{7}{10}\) = \(\frac{7}{12}\)

Question 6. \(\frac{6}{9}\) × \(\frac{8}{2}\) = _______ Answer: \(\frac{6}{9}\) × \(\frac{8}{2}\) = \(\frac{8}{3}\)

Explanation: The given fractions are: \(\frac{6}{9}\) and \(\frac{8}{2}\) We know that, \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) a = \(\frac{a}{1}\) So, \(\frac{6}{9}\) × \(\frac{8}{2}\) = \(\frac{6 × 8}{2 × 9}\) = \(\frac{48}{18}\) = \(\frac{8}{3}\) Hence, \(\frac{6}{9}\) × \(\frac{8}{2}\) = \(\frac{8}{3}\)

Question 7. \(\frac{21}{100}\) × \(\frac{3}{5}\) = _______ Answer: \(\frac{21}{100}\) × \(\frac{3}{5}\) = \(\frac{63}{500}\)

Explanation: The given fractions are: \(\frac{21}{100}\) and \(\frac{3}{5}\) We know that, \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) a = \(\frac{a}{1}\) So, \(\frac{21}{100}\) × \(\frac{3}{5}\) = \(\frac{21 × 3}{100 × 5}\) = \(\frac{63}{500}\) Hence, \(\frac{21}{100}\) × \(\frac{3}{5}\) = \(\frac{63}{500}\)

Question 8. \(\frac{1}{12}\) × \(\frac{9}{4}\) = _______ Answer: \(\frac{1}{12}\) × \(\frac{9}{4}\) = \(\frac{3}{16}\)

Explanation: The given fractions are: \(\frac{1}{12}\) and \(\frac{9}{4}\) We know that, \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) a = \(\frac{a}{1}\) So, \(\frac{1}{12}\) × \(\frac{9}{4}\) = \(\frac{1 × 9}{12 × 4}\) = \(\frac{9}{48}\) = \(\frac{3}{16}\) Hence, \(\frac{1}{12}\) × \(\frac{9}{4}\) = \(\frac{3}{16}\)

Question 9. \(\frac{4}{7}\) × \(\frac{8}{8}\) = _______ Answer: \(\frac{4}{7}\) × \(\frac{8}{8}\) = \(\frac{4}{7}\)

Explanation: The given fractions are: \(\frac{4}{7}\) and \(\frac{8}{8}\) We know that, \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) a = \(\frac{a}{1}\) So, \(\frac{4}{7}\) × \(\frac{8}{8}\) = \(\frac{8 × 4}{7 × 8}\) = \(\frac{4}{7}\) Hence, \(\frac{4}{7}\) × \(\frac{8}{8}\) = \(\frac{4}{7}\)

Evaluate Question 10. \(\left(\frac{1}{2} \times \frac{7}{8}\right)\) × 2 = _______ Answer: \(\left(\frac{1}{2} \times \frac{7}{8}\right)\) × 2 = \(\frac{7}{8}\)

Explanation: The given fractions are: \(\frac{7}{8}\) and \(\frac{1}{2}\)

We know that, \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) a = \(\frac{a}{1}\) So, ( \(\frac{1}{2}\) × \(\frac{7}{8}\)  ) × 2 = ( \(\frac{1 × 7}{2 × 8}\) ) × \(\frac{2}{1}\) = \(\frac{7}{16}\) × \(\frac{2}{1}\) = \(\frac{7 × 2}{16 × 1}\) = \(\frac{7}{8}\) Hence, \(\frac{4}{7}\) × \(\frac{8}{8}\) = \(\frac{4}{7}\)

Question 11. \(\left(\frac{7}{6}-\frac{5}{6}\right)\) × \(\frac{2}{3}\) = _______ Answer: \(\left(\frac{7}{6}-\frac{5}{6}\right)\) × \(\frac{2}{3}\) = \(\frac{2}{9}\)

Explanation: The given fractions are: \(\frac{7}{6}\) , \(\frac{5}{6}\), and \(\frac{2}{3}\) We know that, \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) a = \(\frac{a}{1}\) So, ( \(\frac{7}{6}\) – \(\frac{5}{6}\)  ) × \(\frac{2}{3}\) = ( \(\frac{7 – 5}{6}\) ) × \(\frac{2}{3}\) = \(\frac{2}{6}\) × \(\frac{2}{3}\) = \(\frac{2 × 2}{6 × 3}\) = \(\frac{4}{18}\) = \(\frac{2}{9}\) Hence, \(\left(\frac{7}{6}-\frac{5}{6}\right)\) × \(\frac{2}{3}\) = \(\frac{2}{9}\)

Question 12. \(\frac{9}{10}\) × \(\left(\frac{4}{9}+\frac{1}{3}\right)\) = _________ Answer: \(\frac{9}{10}\) × \(\left(\frac{4}{9}+\frac{1}{3}\right)\) = \(\frac{7}{10}\)

Explanation: The given fractions are: \(\frac{9}{10}\), \(\frac{4}{9}\) and \(\frac{1}{3}\) We know that, \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) a = \(\frac{a}{1}\) So, \(\frac{9}{10}\) × ( \(\frac{4}{9}\) + \(\frac{1}{3}\)  ) = ( \(\frac{4 + 3}{9}\) ) × \(\frac{9}{10}\) = \(\frac{7}{9}\) × \(\frac{9}{10}\) = \(\frac{7 × 9}{9 × 10}\) = \(\frac{7}{10}\) Hence, \(\frac{9}{10}\) × \(\left(\frac{4}{9}+\frac{1}{3}\right)\) = \(\frac{7}{10}\)

Explanation: It is given that at a school, \(\frac{3}{4}\) of the students play a sport and of the students that play a sport, \(\frac{1}{5}\) play baseball. So, The fraction of the students at the school that play baseall = ( The fraction of the students that play a sport ) × ( The fraction of the students that play baseball out of thetotal number of students ) = \(\frac{3}{4}\) × \(\frac{1}{5}\) = \(\frac{3 × 1}{4 × 5}\) = \(\frac{3}{20}\) Hence, from the above, We can conclude that the fraction of the students that play baseball is: \(\frac{3}{20}\)

Question 14. Reasoning Descartes says he can find the product of a whole number and a fraction the same way he finds the product of two fractions. Explain why his reasoning makes sense. Answer: The product of a whole number and a fraction follows the same procedure as that of the product of the two fractions because of the following properties of multiplication: A) \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) B) a= \(\frac{a}{1}\) Hence, from the above two properties of multiplication, We can conclude that the reasoning of Descartes makes sense

Question 15. Writing Explain how multiplying fractions is different than adding and subtracting fractions. Answer: In the multiplication of the fractions, we multiply numerators and denominators. Example: \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) In addition of the fractions, we add only numerators making the denominators equal Example: \(\frac{a}{b}\) + \(\frac{p}{b}\) = \(\frac{a + p}{b}\) In subtraction of the fractions, we subtract only numerators making the denominators equal Example: \(\frac{a}{b}\) – \(\frac{p}{b}\) = \(\frac{a – p}{b}\)

Explanation: It is given that at the zoo, \(\frac{3}{5}\) of the animals are mammals and of the animals, \(\frac{5}{12}\) are primates So, The fraction of the animals that are primates =( The fraction of the animals that are primates ) × ( The fraction of the animals that are primates ) = \(\frac{3}{5}\) × \(\frac{5}{12}\) = \(\frac{3 × 5}{5 × 12}\) = \(\frac{15}{60}\) = \(\frac{1}{4}\) Now, Let the total number of animals that are mammals =1 So, The fraction of the animals that are not primates = ( The total number of mammals ) – ( The fraction of the animals that are primates ) = 1 – \(\frac{1}{4}\) = \(\frac{4}{4}\) – \(\frac{1}{4}\) = \(\frac{4 – 1}{4}\) = \(\frac{3}{4}\) Hence, from the above, We can conclude that the fraction of the animals that are not primates is: \(\frac{3}{4}\)

Question 17. You have an album of 216 trading cards. One page contains \(\frac{1}{24}\) of the cards. On that page, \(\frac{2}{3}\) of the cards are epic. You only have one page with any epic cards. How many epic cards do you have? Answer: The number of epic cards is: 6 epic cards

Explanation: It is given that you have an album of 216 trading cards It is also given that one page contains \(\frac{1}{24}\) of the cards. On that page, \(\frac{2}{3}\) of the cards are epic. So, The fraction of the epic cards = ( The total number of cards on that page ) × ( The fraction of the epic cards that is on that page ) = \(\frac{1}{24}\) × \(\frac{2}{3}\) = \(\frac{2 × 1}{3 × 24}\) = \(\frac{1}{36}\) So, The number of epic cards = ( The total number of cards ) × ( The fraction of the epic cards ) = 216 × \(\frac{1}{36}\) = \(\frac{216}{1}\) × \(\frac{}{36}\) = \(\frac{216}{36}\) = 6 epic cards Hence, from the above, We can conclude that there are 6 epic cards

Question 18. DIG DEEPER! In a class, \(\frac{2}{5}\) of the students play basketball and \(\frac{7}{10}\) play soccer.Of the students who play basketball, \(\frac{2}{3}\) also play soccer. There are 30 students in the class. How many students play soccer but do not play basketball? Answer: The number of students that play soccer but do not play basketball is: 4 students

Explanation: It is given that in a class, \(\frac{2}{5}\) of the students play basketball and \(\frac{7}{10}\) play soccer.Of the students who play basketball, \(\frac{2}{3}\) also play soccer. So, The number of students who play basketball and soccer = ( The number of students who play basketball ) × ( The number of students playing soccer who are also playing basketball ) = \(\frac{2}{5}\) × \(\frac{2}{3}\) = \(\frac{4}{15}\) Now, The number of students who only play soccer but not basketball = ( The number of students who play soccer only ) – ( The number of students who play both basketball and soccer ) = \(\frac{2}{5}\) – \(\frac{4}{15}\) In subtraction, the denominators must be equal. So, Multiply \(\frac{2}{5}\) by \(\frac{3}{3}\) So, \(\frac{6}{15}\) – \(\frac{4}{15}\) = \(\frac{4}{15}\) It is also given that the total number of students are: 30 So, The number of students who play soccer only = ( The fraction of the students who play soccer only ) × ( The total number of students ) = \(\frac{2}{15}\) × 30 = \(\frac{2}{15}\) × \(\frac{30}{1}\) = 4 students Hence, from the above, We can conclude that there are 4 students who play soccer only.

Multiply. \(\frac{1}{4}\) × \(\frac{1}{5}\) = _______ Answer: \(\frac{1}{4}\) × \(\frac{1}{5}\) = \(\frac{1}{20}\)

Explanation: The given fractions are: \(\frac{1}{4}\) and \(\frac{1}{5}\) We know that, \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) a = \(\frac{a}{1}\) So, \(\frac{1}{4}\) × \(\frac{1}{5}\) = \(\frac{1 × 1}{4 × 5}\) = \(\frac{1}{20}\) Hence, \(\frac{1}{4}\) × \(\frac{1}{5}\) = \(\frac{1}{20}\)

Question 2. \(\frac{2}{7}\) × \(\frac{1}{2}\) = _______ Answer: \(\frac{1}{2}\) × \(\frac{2}{7}\) = \(\frac{1}{7}\)

Explanation: The given fractions are: \(\frac{1}{2}\) and \(\frac{2}{7}\) We know that, \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) a = \(\frac{a}{1}\) So, \(\frac{1}{2}\) × \(\frac{2}{7}\) = \(\frac{1 × 2}{2 × 7}\) = \(\frac{2}{14}\) = \(\frac{1}{7}\) Hence, \(\frac{1}{2}\) × \(\frac{2}{7}\) = \(\frac{1}{7}\)

Question 3. \(\frac{9}{10}\) × \(\frac{2}{3}\) = _______ Answer: \(\frac{9}{10}\) × \(\frac{2}{3}\) = \(\frac{3}{5}\)

Explanation: The given fractions are: \(\frac{9}{10}\) and \(\frac{2}{3}\) We know that, \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) a = \(\frac{a}{1}\) So, \(\frac{9}{10}\) × \(\frac{2}{3}\) = \(\frac{9 × 2}{10 × 3}\) = \(\frac{18}{30}\) = \(\frac{3}{5}\) Hence, \(\frac{9}{10}\) × \(\frac{2}{3}\) = \(\frac{3}{5}\)

Question 4. \(\frac{5}{8}\) × \(\frac{5}{6}\) = _______ Answer: \(\frac{5}{8}\) × \(\frac{5}{6}\) = \(\frac{25}{48}\)

Explanation: The given fractions are: \(\frac{5}{8}\) and \(\frac{5}{6}\) We know that, \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) a = \(\frac{a}{1}\) So, \(\frac{5}{8}\) × \(\frac{5}{6}\) = \(\frac{5 × 5}{8 × 6}\) = \(\frac{25}{48}\) Hence, \(\frac{5}{8}\) × \(\frac{5}{6}\) = \(\frac{25}{48}\)

Question 5. \(\frac{9}{7}\) × \(\frac{3}{4}\) = _______ Answer: \(\frac{9}{7}\) × \(\frac{3}{4}\) = \(\frac{27}{28}\)

Explanation: The given fractions are: \(\frac{9}{7}\) and \(\frac{3}{4}\) We know that, \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) a = \(\frac{a}{1}\) So, \(\frac{9}{7}\) × \(\frac{3}{4}\) = \(\frac{9 × 3}{7 × 4}\) = \(\frac{27}{28}\) Hence, \(\frac{9}{7}\) × \(\frac{3}{4}\) = \(\frac{27}{28}\)

Question 6. \(\frac{11}{100}\) × \(\frac{2}{5}\) = _______ Answer: \(\frac{11}{100}\) × \(\frac{2}{5}\) = \(\frac{11}{250}\)

Explanation: The given fractions are: \(\frac{11}{100}\) and \(\frac{2}{5}\) We know that, \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) a = \(\frac{a}{1}\) So, \(\frac{11}{100}\) × \(\frac{2}{5}\) = \(\frac{11 × 2}{100 × 5}\) = \(\frac{22}{500}\) = \(\frac{11}{250}\) Hence, \(\frac{11}{100}\) × \(\frac{2}{5}\) = \(\frac{11}{250}\)

Question 7. \(\frac{7}{20}\) × \(\frac{6}{2}\) = _______ Answer: \(\frac{7}{20}\) × \(\frac{6}{2}\) = \(\frac{21}{20}\)

Explanation: The given fractions are: \(\frac{7}{20}\) and \(\frac{6}{2}\) We know that, \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) a = \(\frac{a}{1}\) So, \(\frac{7}{20}\) × \(\frac{6}{2}\) = \(\frac{7 × 6}{20 × 2}\) = \(\frac{42}{40}\) = \(\frac{21}{20}\) Hence, \(\frac{7}{20}\) × \(\frac{6}{2}\) = \(\frac{21}{20}\)

Question 8. \(\frac{15}{16}\) × \(\frac{1}{3}\) = _______ Answer: \(\frac{15}{16}\) × \(\frac{1}{3}\) = \(\frac{5}{16}\)

Explanation: The given fractions are: \(\frac{15}{16}\) and \(\frac{1}{3}\) We know that, \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) a = \(\frac{a}{1}\) So, \(\frac{15}{16}\) × \(\frac{1}{3}\) = \(\frac{15 × 1}{16 × 3}\) = \(\frac{15}{48}\) = \(\frac{5}{16}\) Hence, \(\frac{15}{16}\) × \(\frac{1}{3}\) = \(\frac{5}{16}\)

Question 9. \(\frac{5}{12}\) × \(\frac{3}{10}\) = _______ Answer: \(\frac{5}{12}\) × \(\frac{3}{10}\) = \(\frac{1}{8}\)

Explanation: The given fractions are: \(\frac{5}{12}\) and \(\frac{3}{10}\) We know that, \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) a = \(\frac{a}{1}\) So, \(\frac{5}{12}\) × \(\frac{3}{10}\) = \(\frac{5 × 3}{12 × 10}\) = \(\frac{15}{120}\) = \(\frac{1}{8}\) Hence, \(\frac{5}{12}\) × \(\frac{3}{10}\) = \(\frac{1}{8}\)

Evaluate. Question 10. 3 × \(\frac{3}{10}\) = _______ Answer: 3 ×  \(\frac{3}{10}\) = \(\frac{9}{10}\)

Explanation: The given numbers are: 3 and \(\frac{3}{10}\) We know that, \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) a = \(\frac{a}{1}\) So, 3 × \(\frac{3}{10}\) = \(\frac{3}{1}\) × \(\frac{3}{10}\) = \(\frac{3 × 3}{10 × 1}\) = \(\frac{9}{10}\) Hence, 3 × \(\frac{3}{10}\) = \(\frac{9}{10}\)

Question 11. \(\left(\frac{1}{3}+\frac{1}{3}\right)\) × \(\frac{4}{5}\) = _______ Answer: \(\left(\frac{1}{3}+\frac{1}{3}\right)\) × \(\frac{4}{5}\) = \(\frac{8}{15}\)

Explanation: The given fractions are: \(\frac{1}{3}\), \(\frac{1}{3}\) and \(\frac{4}{5}\)

We know that, \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) a = \(\frac{a}{1}\) So, ( \(\frac{1}{3}\) + \(\frac{1}{3}\)  ) × \(\frac{4}{5}\) = ( \(\frac{1 + 1}{3}\) ) × \(\frac{4}{5}\) = \(\frac{2}{3}\) × \(\frac{4}{5}\) = \(\frac{4 × 2}{3 × 5}\) = \(\frac{8}{15}\) Hence, \(\left(\frac{1}{3}+\frac{1}{3}\right)\) × \(\frac{4}{5}\) = \(\frac{8}{15}\)

Question 12. \(\frac{6}{7}\) × \(\left(\frac{3}{4}-\frac{5}{12}\right)\) = _______ Answer: \(\frac{6}{7}\) × \(\left(\frac{3}{4}-\frac{5}{12}\right)\) = \(\frac{2}{7}\)

Explanation: The given fractions are: \(\frac{6}{7}\), \(\frac{3}{4}\) and \(\frac{5}{12}\)

We know that, \(\frac{a}{b}\) × \(\frac{p}{q}\) = \(\frac{a × p}{b × q}\) a = \(\frac{a}{1}\) So, ( \(\frac{3}{4}\) – \(\frac{5}{12}\)  ) × \(\frac{6}{7}\) = ( \(\frac{9}{12}\) – \(\frac{5}{12}\)  ) × \(\frac{6}{7}\) = \(\frac{4}{12}\) × \(\frac{6}{7}\) = \(\frac{4 × 6}{12 × 7}\) = \(\frac{2}{7}\) Hence, \(\frac{6}{7}\) × \(\left(\frac{3}{4}-\frac{5}{12}\right)\) = \(\frac{2}{7}\)

Question 13. A pancake recipe calls for \(\frac{1}{3}\) cup of water. You want to halve the recipe. What fraction of a cup of water do you need? Answer: The fraction of a cup of water you need is: \(\frac{1}{6}\)

Explanation: It is given that a pancake recipe calls for \(\frac{1}{3}\) cup of water and you want to halve the recipe. So, The fraction of a cup of water you need for half of the recipe = ( The cup of water you need for a recipe ) × \(\frac{1}{2}\) = \(\frac{1}{3}\) × \(\frac{1}{2}\) = \(\frac{1 × 1}{3 × 2}\) = \(\frac{1}{6}\) Hence, from the above, We can conclude that the fraction of water you need for half of the recipe is: \(\frac{1}{6}\)

Question 14. Number Sense Which is greater, \(\frac{3}{4}\) × \(\frac{1}{5}\) or \(\frac{3}{4}\) × \(\frac{1}{8}\)? Explain. Answer: \(\frac{3}{4}\) × \(\frac{1}{5}\)” is greater than “\(\frac{3}{4}\) × \(\frac{1}{8}\)

Explanation: The given fractions are: \(\frac{3}{4}\), \(\frac{1}{5}\) and \(\frac{1}{8}\) Now, \(\frac{3}{4}\) × \(\frac{1}{5}\) = \(\frac{3 × 1}{4 × 5}\) = \(\frac{3}{20}\) Now, \(\frac{3}{4}\) × \(\frac{1}{8}\) = \(\frac{3 × 1}{4 × 8}\) = \(\frac{3}{32}\) So, For the comparison of the products, equate the numerators. Multiply \(\frac{3}{20}\) by \(\frac{32}{32}\) Multiply \(\frac{3}{32}\) by \(\frac{20}{20}\) So, \(\frac{3}{20}\) = \(\frac{96}{640}\) \(\frac{3}{32}\) = \(\frac{60}{640}\) So, By comparison of the products, We can observe that 96 > 60 Hence, from the above, We can conclude that “\(\frac{3}{4}\) × \(\frac{1}{5}\)” is greater than “\(\frac{3}{4}\) × \(\frac{1}{8}\)”

Question 15. Reasoning Is \(\frac{17}{24}\) × \(\frac{7}{8}\) equal to \(\frac{17}{8}\) × \(\frac{7}{24}\)? Explain. Answer: \(\frac{17}{24}\) × \(\frac{7}{8}\) is equal to \(\frac{17}{8}\) × \(\frac{7}{24}\)

Explanation: By the commutative property of multiplication, a × b = b × a So, By the above property of multiplication, We can interchange the numbers Hence, from the above, We can conclude that \(\frac{17}{24}\) × \(\frac{7}{8}\) is equal to \(\frac{17}{8}\) × \(\frac{7}{24}\)

Question 17. Modeling Real Life At a town hall meeting, \(\frac{37}{50}\) of the members are present. Of those who are present, \(\frac{1}{2}\) vote in favor of a new park. What fraction of the members do not vote in favor of the new park? Answer: The fraction of the members that do not vote in favor of the new park is: \(\frac{13}{50}\)

Explanation: It is given that at a town hall meeting, \(\frac{37}{50}\) of the members are present. Of those who are present, \(\frac{1}{2}\) vote in favor of a new park. So, The fraction of the members that do not vote in favor of the new park = ( The total number of members at the town hall meeting ) – ( The fraction of the members that vote in the favor of a new park ) = 1 – \(\frac{37}{50}\) = \(\frac{50}{50}\) – \(\frac{37}{50}\) = \(\frac{50 – 37}{50}\) = \(\frac{13}{50}\) Hence, from the above, We can conclude that the number of members that do not vote in the favor of a new park is: \(\frac{13}{50}\)

Question 18. Modeling Real Life There are 50 U.S. states. Seven twenty-fifths of the states share a land border with Canada or Mexico.Of those states, \(\frac{2}{7}\) share a border with Mexico. How many states share a border with Canada? Answer: The number of states that share a border with Canada is: 4 states

Explanation: It is given that there are 50 U.S. states. Seven twenty-fifths of the states share a land border with Canada or Mexico and of those states, \(\frac{2}{7}\) share a border with Mexico. So, The number of states that share a border with Canada or Mexico = \(\frac{7}{25}\) × 50 = \(\frac{7}{25}\) × \(\frac{50}{1}\) = 14 states So, The number of states that share a border with Canada = \(\frac{2}{7}\) × 14 = \(\frac{2}{7}\) × \(\frac{14}{1}\) = 4 states Hence, from the above, We can conclude that there are 4 states that share a border with Canada

Evaluate. Check whether your answer is reasonable. Question 19. 15.67 + 4 + 6.5 = _____ Answer: 15.67 + 4 + 6.5 = 26.17

Explanation: The given numbers are: 15.67, 4, and 6.5 The representation of the numbers in the fraction form is: \(\frac{1567}{100}\), \(\frac{65}{10}\) and 4 As the highest number in the denominator is 100, make all the denominators hundred So, The representation of 4 with 100 in the denominator is: \(\frac{400}{100}\) The representation of 6.5 with 100 in the denominator is: \(\frac{650}{100}\) So, \(\frac{1567}{100}\) + \(\frac{400}{100}\) + \(\frac{650}{100}\) = \(\frac{  1567 + 400 + 650 }{100}\) = \(\frac{2,617}{100}\) = 26.17 Hence, 15.67 + 4 + 6.5 = 26.17

Question 20. 20.7 – 9.54 + 25.81 = _______ Answer: 20.7 – 9.54 + 25.81 = 36.97

Explanation: The given numbers are: 20.7, 9.54, and 25.81 The representation of the numbers in the fraction form is: \(\frac{2581}{100}\), \(\frac{207}{10}\) and \(\frac{954}{100}\) As the highest number in the denominator is 100, make all the denominators hundred So, The representation of 20.7 with 100 in the denominator is: \(\frac{2070}{100}\) So, \(\frac{2581}{100}\) + \(\frac{2070}{100}\) – \(\frac{954}{100}\) = \(\frac{  2581 + 2070 – 954 }{100}\) = \(\frac{3,697}{100}\) = 36.97 Hence, 20.7 – 9.54 + 25.81 = 36.97

Draw and cut out a rectangle that has any two of the side lengths below. \(\frac{1}{2}\) ft \(\frac{1}{3}\) ft \(\frac{1}{4}\) ft Use several copies of your rectangle to create a unit square. What is the area (in square feet) of each small rectangle? Explain your reasoning. Answer: Let there are two small rectangles with the following lengths: \(\frac{1}{2}\) ft and \(\frac{1}{3}\) ft \(\frac{1}{2}\) ft and \(\frac{1}{4}\) ft So, The area of the unit square with lengths \(\frac{1}{2}\) ft and \(\frac{1}{3}\) ft is: \(\frac{1}{2}\) ft × \(\frac{1}{3}\) ft = \(\frac{1 × 1}{2 × 3}\) ft = \(\frac{1}{6}\) ft The area of the unit square with lengths \(\frac{1}{2}\) ft and \(\frac{1}{4}\) ft is: \(\frac{1}{2}\) ft × \(\frac{1}{4}\) ft = \(\frac{1 × 1}{2 × 4}\) ft = \(\frac{1}{8}\) ft Hence, from the above, We can conclude that the areas ofthe smaller rectngles with unit lengths is: \(\frac{1}{6}\) ft and \(\frac{1}{8}\) ft

Think and Grow: Find Areas of Rectangles

Question 6. Find the area of a rectangle with side lengths of \(\frac{5}{8}\) and \(\frac{4}{3}\)? Answer: The area of a rectangle is: \(\frac{20}{24}\)

Explanation: The given side lengths of a rectangle are: \(\frac{5}{8}\) and \(\frac{4}{3}\) So, The area of the rectangle = \(\frac{5}{8}\) × \(\frac{4}{3}\) = \(\frac{5 × 4}{8 × 3}\) = \(\frac{20}{24}\) Hence, from the above, We can conclude that the area of the rectangle is: \(\frac{20}{24}\)

Question 7. Find the area of a rectangle with side lengths of \(\frac{7}{9}\) and \(\frac{1}{2}\)? Answer: The area of a rectangle is: \(\frac{7}{18}\)

Explanation: The given side lengths of a rectangle are: \(\frac{7}{9}\) and \(\frac{1}{2}\) So, The area of the rectangle = \(\frac{7}{9}\) × \(\frac{1}{2}\) = \(\frac{7 × 1}{9 × 2}\) = \(\frac{7}{18}\) Hence, from the above, We can conclude that the area of the rectangle is: \(\frac{7}{18}\)

Question 8. Reasoning Can you find the area of a rectangle with fractional side lengths the same way you find the area of a rectangle with whole-number side lengths? Explain. Answer: Yes, we can find the area of a rectangle with fractional side lengths the same way you find the area of a rectangle with whole-number side lengths. Example: Let the fractional side lengths be: \(\frac{1}{5}\) and \(\frac{1}{9}\) Let the whole number side lengths be: 3 and 4 So, The area of the rectangle with the fractional side lengths = \(\frac{1}{5}\) × \(\frac{1}{9}\) = \(\frac{1 × 1}{9 × 5}\) = \(\frac{1}{45}\) The area of the rectangle with the whole number side lengths = 3 × 4 = \(\frac{3}{1}\) × \(\frac{4}{1}\) = \(\frac{3 × 4}{1 × 1}\) = \(\frac{12}{1}\) = 12 Hence, from the above, We can conclude that we can find the area of a rectangle with fractional side lengths the same way you find the area of a rectangle with whole-number side lengths.

Question 9. YOU BE THE TEACHER Our friend says she can find the area of a square given only one fractional side length. Is your friend correct? Explain. Answer: Yes, she can find the area of a square by only one fractional side length

Explanation: Let the unit fractional side length of the square be: \(\frac{1}{2}\) We know that, The length of all the sides in a square are equal. So, By using this property, we can find the area of the square by taking the fractional unit side length of 1 side as the same for all the sides So, The area of the square = \(\frac{1}{2}\) × \(\frac{1}{2}\) = \(\frac{1 × 1}{2 × 2}\) = \(\frac{1}{4}\) Hence, from the above, We can conclude that she can find the area of a square by only one fractional side length

Explanation: It is given that the area of a square dog kennel is 4 square yards and the side of the square mat is \(\frac{5}{3}\) yd We know that, The length of all the sides in a square are equal. So, By using this property, The area of the square mat = \(\frac{5}{3}\) × \(\frac{5}{3}\) = \(\frac{5 × 5}{3 × 3}\) = \(\frac{25}{9}\) yd Now, The area of a square dog kennel can be written as: \(\frac{36}{9}\) yd So, When we compare the 2 areas, We can get 25 < 36 Hence, from the above, We can conclude that the square mat will fit in the kennel

Explanation: It is given that the side lengths of each chalk art square are \(\frac{11}{4}\) meters. The side lengths of the square zone around each chalk square are an additional \(\frac{3}{4}\) meter. So, The total side length of the chalk walk = \(\frac{11}{4}\) + \(\frac{3}{4}\) = \(\frac{11 + 3}{4}\) = \(\frac{14}{4}\) meters We know that, The length of all sides in the square are equal. So, By using this property, The amount of concrete used for the chalk walk = \(\frac{14}{4}\) × \(\frac{14}{4}\) = \(\frac{14 × 14}{4 × 4}\) = \(\frac{196}{16}\) square meters Hence, from the above, We can conclude that the amount of the concrete used to create the chalk walk is: \(\frac{196}{16}\) square meters

Question 5. Find the area of a rectangle with a side length of \(\frac{3}{4}\) and \(\frac{5}{12}\). Answer: The area of a rectangle is: \(\frac{15}{48}\)

Explanation: The given side lengths of a rectangle are: \(\frac{5}{12}\) and \(\frac{3}{4}\) So, The area of the rectangle = \(\frac{5}{12}\) × \(\frac{3}{4}\) = \(\frac{5 × 3}{12 × 4}\) = \(\frac{15}{48}\) Hence, from the above, We can conclude that the area of the rectangle is: \(\frac{15}{48}\)

Question 6. Find the area of a square with side lengths of \(\frac{9}{16}\) Answer: The area of a square is: \(\frac{81}{256}\)

Explanation: We know that, The length of all the sides of the square is equal. So, The given side lengths of a square are: \(\frac{9}{16}\) and \(\frac{9}{16}\) So, The area of the square = \(\frac{9}{16}\) × \(\frac{9}{16}\) = \(\frac{9 × 9}{16 × 16}\) = \(\frac{81}{256}\) Hence, from the above, We can conclude that the area of the rectangle is: \(\frac{81}{256}\)

Question 7. Open-Ended The area of a rectangle is \(\frac{16}{24}\). What are the possible side lengths of the rectangle? Answer: The possible side lengths for the given area of a rectangle are: A) \(\frac{4}{2}\) and \(\frac{4}{12}\) B) \(\frac{1}{3}\) and \(\frac{4}{8}\)

Explanation: The given area of a rectangle is: \(\frac{16}{24}\) So, To find the possible side lengths of a rectangle, find the factors for the numerator and denominator of \(\frac{16}{24}\) So, The factors of 16 are: 1, 2, 4, 8, 16 The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24 So, The possible side lengths of a rectangle are: A) \(\frac{4}{2}\) and \(\frac{4}{12}\) B) \(\frac{1}{3}\) and \(\frac{4}{8}\) There are so many possible side lengths of a rectangle like above Hence, from the above, We can conclude that the possible side lengths of a rectangle are: A) \(\frac{4}{2}\) and \(\frac{4}{12}\) B) \(\frac{1}{3}\) and \(\frac{4}{8}\)

Divide. Then check your answer. Question 10. 365 ÷ 14 = _______ Answer: 365 ÷ 14 = 26 R 1

Explanation: By using the partial quotients method, 365 ÷ 14 = ( 280 ÷ 84 ) ÷ 14 = ( 280 ÷ 14 ) + ( 84 ÷ 14 ) = 20 + 6 = 26 R 1 Hence, 365 ÷ 14 = 26 R 1

Question 11. 282 ÷ 27 = ______ Answer: 282 ÷ 27 = 10 R 12

Explanation: By using the partial quotients method, 282 ÷ 27 = 270 ÷ 27 = 10 R 12 Hence, 270 ÷ 27 = 10 R 12

Question 12. 601 ÷ 72 = _____ Answer: 601 ÷ 72 = 8 R 25

Explanation: By using the partial quotients method, 601 ÷ 72 = 576 ÷ 72 = 8 R 25 Hence, 601 ÷ 72 = 8 R 25

Structure Find the area using a different method. Explain how you found your answer. Answer: The given mixed fractions of a rectangle are: 2\(\frac{1}{2}\) and 3\(\frac{1}{2}\) To convert a mixed fraction into an improper fraction, we have to add the whole number part and the fractional part in the mixed fraction. So, 2\(\frac{1}{2}\) = 2 + \(\frac{1}{2}\) = \(\frac{4}{2}\) + \(\frac{1}{2}\) = \(\frac{5}{2}\) 3\(\frac{1}{2}\) = 3 + \(\frac{1}{2}\) = \(\frac{6}{2}\) + \(\frac{1}{2}\) = \(\frac{7}{2}\) So, \(\frac{5}{2}\) × \(\frac{7}{2}\) = \(\frac{35}{4}\)

Think and Grow: Multiply Mixed Numbers

Multiply. Question 1. 2\(\frac{1}{2}\) × 1\(\frac{1}{2}\) = ________ Answer: 2\(\frac{1}{2}\) × 1\(\frac{1}{2}\) = \(\frac{15}{4}\)

Explanation: The given fractions are: 2\(\frac{1}{2}\) and 1\(\frac{1}{2}\) The representation of the fractions in the improper form is: \(\frac{5}{2}\) and \(\frac{3}{2}\) So, \(\frac{5}{2}\) × \(\frac{3}{2}\) = \(\frac{5 × 3}{2 × 2}\) = \(\frac{15}{4}\) Hence, 2\(\frac{1}{2}\) × 1\(\frac{1}{2}\) = \(\frac{15}{4}\)

Question 2. 3\(\frac{1}{4}\) × 2\(\frac{2}{3}\) = ______ Answer: 3\(\frac{1}{4}\) × 2\(\frac{2}{3}\) = \(\frac{104}{12}\)

Explanation: The given fractions are: 3\(\frac{1}{4}\) and 2\(\frac{2}{3}\) The representation of the fractions in the improper form is: \(\frac{13}{4}\) and \(\frac{8}{3}\) So, \(\frac{13}{4}\) × \(\frac{8}{3}\) = \(\frac{13 × 8}{4 × 3}\) = \(\frac{104}{12}\) Hence, 3\(\frac{1}{4}\) × 2\(\frac{2}{3}\) = \(\frac{104}{12}\)

Multiply. Question 3. 1\(\frac{3}{4}\) × 2\(\frac{1}{6}\) = ______ Answer: 1\(\frac{3}{4}\) × 2\(\frac{1}{6}\) = \(\frac{91}{24}\)

Explanation: The given fractions are: 1\(\frac{3}{4}\) and 2\(\frac{1}{6}\) The representation of the fractions in the improper form is: \(\frac{7}{4}\) and \(\frac{13}{6}\) So, \(\frac{7}{4}\) × \(\frac{13}{6}\) = \(\frac{13 × 7}{4 × 6}\) = \(\frac{91}{24}\) Hence, 1\(\frac{3}{4}\) × 2\(\frac{1}{6}\) = \(\frac{91}{24}\)

Question 4. 4\(\frac{1}{3}\) × 1\(\frac{5}{6}\) = ______ Answer: 4\(\frac{1}{3}\) × 1\(\frac{5}{6}\) = \(\frac{143}{18}\)

Explanation: The given fractions are: 4\(\frac{1}{3}\) and 1\(\frac{5}{6}\) The representation of the fractions in the improper form is: \(\frac{13}{3}\) and \(\frac{11}{6}\) So, \(\frac{13}{3}\) × \(\frac{11}{6}\) = \(\frac{13 × 11}{6 × 3}\) = \(\frac{143}{18}\) Hence, 4\(\frac{1}{3}\) × 1\(\frac{5}{6}\) = \(\frac{143}{18}\)

Question 5. 3\(\frac{2}{5}\) × 1\(\frac{9}{10}\) = ______ Answer: 3\(\frac{2}{5}\) × 1\(\frac{9}{10}\) = \(\frac{323}{50}\)

Explanation: The given fractions are: 3\(\frac{2}{5}\) and 1\(\frac{9}{10}\) The representation of the fractions in the improper form is: \(\frac{17}{5}\) and \(\frac{19}{10}\) So, \(\frac{17}{5}\) × \(\frac{19}{10}\) = \(\frac{17 × 19}{5 × 10}\) = \(\frac{323}{50}\) Hence, 3\(\frac{2}{5}\) × 1\(\frac{9}{10}\) = \(\frac{323}{50}\)

Question 6. 2\(\frac{3}{8}\) × 3\(\frac{1}{2}\) = ______ Answer: 2\(\frac{3}{8}\) × 3\(\frac{1}{2}\) = \(\frac{133}{16}\)

Explanation: The given fractions are: 2\(\frac{3}{8}\) and 3\(\frac{1}{2}\) The representation of the fractions in the improper form is: \(\frac{19}{8}\) and \(\frac{7}{2}\) So, \(\frac{19}{8}\) × \(\frac{7}{2}\) = \(\frac{19 × 7}{8 × 2}\) = \(\frac{133}{16}\) Hence, 2\(\frac{3}{8}\) × 3\(\frac{1}{2}\) = \(\frac{133}{16}\)

Evaluate. Question 7. 5\(\frac{1}{4}\) × \(\frac{2}{5}\) × 6\(\frac{1}{12}\) = _________ Answer: 5\(\frac{1}{4}\) × \(\frac{2}{5}\) × 6\(\frac{1}{12}\) = \(\frac{3,066}{240}\)

Explanation: The given fractions are: 5\(\frac{1}{4}\), \(\frac{2}{5}\) and 6\(\frac{1}{12}\) The representation of the fractions in the improper form is: \(\frac{21}{4}\) and \(\frac{73}{12}\) So, \(\frac{21}{4}\) × \(\frac{73}{12}\) × \(\frac{2}{5}\) = \(\frac{21 × 73 × 2}{4 × 12 × 5}\) = \(\frac{3,066}{240}\) Hence, 5\(\frac{1}{4}\) × \(\frac{2}{5}\) × 6\(\frac{1}{12}\) = \(\frac{3,066}{240}\)

Question 8. 3\(\frac{2}{3}\) × (10\(\frac{7}{8}\) – 2\(\frac{1}{4}\)) = _________ Answer: 3\(\frac{2}{3}\) × (10\(\frac{7}{8}\) – 2\(\frac{1}{4}\)) = \(\frac{759}{24}\)

Explanation: The given fractions are: 3\(\frac{2}{3}\), 10\(\frac{7}{8}\) and 2\(\frac{1}{4}\) The representation of the fractions in the improper form is: \(\frac{11}{3}\), \(\frac{87}{8}\)  and \(\frac{9}{4}\) So, 3\(\frac{2}{3}\) × (10\(\frac{7}{8}\) – 2\(\frac{1}{4}\)) = \(\frac{11}{3}\) × ( \(\frac{87}{8}\) – \(\frac{9}{4}\) ) = \(\frac{11}{3}\) × ( \(\frac{87 – 18}{8}\) ) = \(\frac{11}{3}\) × \(\frac{69}{8}\) = \(\frac{11 × 69}{8 × 3}\) = \(\frac{759}{24}\) Hence, 3\(\frac{2}{3}\) × (10\(\frac{7}{8}\) – 2\(\frac{1}{4}\)) = \(\frac{759}{24}\)

Explanation: The given fractions are: 4\(\frac{1}{2}\) and 3\(\frac{2}{3}\) The representation of the side lengths in the improper form is: \(\frac{9}{2}\) and \(\frac{11}{3}\) So, The area of rectangle = \(\frac{9}{2}\) × \(\frac{11}{3}\) = \(\frac{9 × 11}{3 × 2}\) = \(\frac{99}{6}\) = \(\frac{33}{2}\) = 15\(\frac{3}{2}\) But according to your friend, 4\(\frac{1}{2}\) × 3\(\frac{2}{3}\) = 12\(\frac{1}{2}\) Hence, from the above, We can conclude that your friend is not correct.

Explanation: According to the separation method, To convert the mixed number into the improper fraction, we can add whole numbers and the fractions separately Now, Let the missing numbers be p and q So, p + 5 = 6 So, p = 6 – 5 = 1 Now, \(\frac{1}{4}\) + \(\frac{1}{q}\) = \(\frac{7}{8}\) So, \(\frac{1}{q}\) = \(\frac{7}{8}\) – \(\frac{1}{4}\) = \(\frac{7}{8}\) – \(\frac{2}{8}\) = \(\frac{7 – 2}{8}\) = \(\frac{5}{8}\) So, q= \(\frac{8}{5}\) Hence, from the above, We can conclude that the missing numbers are: 1 and \(\frac{8}{5}\)

Question 12. On Monday,you roller-skate 6\(\frac{1}{4}\) miles. On Tuesday, you skate 1\(\frac{2}{5}\) times as far as you did on Monday. How many total miles do you roller-skate on Monday and Tuesday combined? Answer: The number of miles you do roller-skate on Monday and Tuesday combined is: 15 miles

Explanation: It is given that on Monday,you roller-skate 6\(\frac{1}{4}\) miles. On Tuesday, you skate 1\(\frac{2}{5}\) times as far as you did on Monday. So, The number of miles you do roller-skate on Tuesday = ( The number of miles you do roller-skate on Monday ) × 1\(\frac{2}{5}\) = 6\(\frac{1}{4}\) × 1\(\frac{2}{5}\) = \(\frac{25}{4}\) × \(\frac{7}{5}\) = \(\frac{25 × 7}{5 × 4}\) = \(\frac{175}{20}\) Now, The number of miles you do roller-skate on Monday and Tuesday combined = 6\(\frac{1}{4}\) + \(\frac{175}{20}\) = \(\frac{25}{4}\) + \(\frac{175}{20}\) = \(\frac{125}{20}\) + \(\frac{175}{20}\) = \(\frac{175 + 125}{20}\) = \(\frac{300}{20}\) = 15 miles Hence, from the above, We can conclude that the number of miles you do roller- skate on Monday and Tuesday combined is: 15 miles

Question 13. DIG DEEPER! An artist paints a rectangular mural. The mural is 4\(\frac{1}{3}\) feet wide. The length is 2\(\frac{1}{4}\) times the width. What is the area of the mural? Answer: The area of the mural is: \(\frac{117}{12}\) square feet

Explanation: It is given that an artist paints a rectangular mural. The mural is 4\(\frac{1}{3}\) feet wide. The length is 2\(\frac{1}{4}\) times the width. So, The area of the mural = ( The width of the mural ) × ( The length of the mural ) = 4\(\frac{1}{3}\) × 2\(\frac{1}{4}\) = \(\frac{13}{3}\) × \(\frac{9}{4}\) = \(\frac{13 × 9}{4 × 3}\) = \(\frac{117}{12}\) Hence, from the above, We can conclude that the area of the mural is: \(\frac{117}{12}\) square feet

Multiply. Question 1. 1\(\frac{1}{2}\) × 1\(\frac{1}{8}\) = ______ Answer: 1\(\frac{1}{2}\) × 1\(\frac{1}{8}\) = \(\frac{18}{16}\)

Explanation: The given fractions are: 1\(\frac{1}{2}\) and 1\(\frac{1}{8}\) The representation of the fractions in the improper form is: \(\frac{3}{2}\) and \(\frac{9}{8}\) So, \(\frac{9}{8}\) × \(\frac{3}{2}\) = \(\frac{9 × 3}{8 × 2}\) = \(\frac{18}{16}\) Hence, 1\(\frac{1}{2}\) × 1\(\frac{1}{8}\) = \(\frac{18}{16}\)

Question 2. 1\(\frac{5}{6}\) × 2\(\frac{1}{4}\) = ______ Answer: 1\(\frac{5}{6}\) × 2\(\frac{1}{4}\) = \(\frac{99}{24}\)

Explanation: The given fractions are: 1\(\frac{5}{6}\) and 2\(\frac{1}{4}\) The representation of the fractions in the improper form is: \(\frac{11}{6}\) and \(\frac{9}{4}\) So, \(\frac{9}{4}\) × \(\frac{11}{6}\) = \(\frac{9 × 11}{4 × 6}\) = \(\frac{99}{24}\) Hence, 1\(\frac{5}{6}\) × 2\(\frac{1}{4}\) = \(\frac{99}{24}\)

Question 3. 2\(\frac{3}{8}\) × 2\(\frac{3}{4}\) = ______ Answer: 2\(\frac{3}{8}\) × 2\(\frac{3}{4}\) = \(\frac{209}{32}\)

Explanation: The given fractions are: 2\(\frac{3}{8}\) and 2\(\frac{3}{4}\) The representation of the fractions in the improper form is: \(\frac{19}{8}\) and \(\frac{11}{4}\) So, \(\frac{19}{8}\) × \(\frac{11}{4}\) = \(\frac{19 × 11}{4 × 8}\) = \(\frac{209}{32}\) Hence, 2\(\frac{3}{8}\) × 2\(\frac{3}{4}\) = \(\frac{209}{32}\)

Question 4. 4\(\frac{1}{6}\) × 3\(\frac{2}{7}\) = ______ Answer: 4\(\frac{1}{6}\) × 3\(\frac{2}{7}\) = \(\frac{575}{42}\)

Explanation: The given fractions are: 4\(\frac{1}{6}\) and 3\(\frac{2}{7}\) The representation of the fractions in the improper form is: \(\frac{25}{6}\) and \(\frac{23}{7}\) So, \(\frac{23}{7}\) × \(\frac{25}{6}\) = \(\frac{23 × 25}{7 × 6}\) = \(\frac{575}{42}\) Hence, 4\(\frac{1}{6}\) × 3\(\frac{2}{7}\) = \(\frac{575}{42}\)

Question 5. 2\(\frac{1}{3}\) × 3\(\frac{9}{10}\) × 5\(\frac{1}{5}\) = ______ Answer: 2\(\frac{1}{3}\)× 3\(\frac{9}{10}\) ×5\(\frac{1}{5}\) = \(\frac{7,098}{150}\)

Explanation: The given fractions are: 2\(\frac{1}{3}\), 3\(\frac{9}{10}\) and 5\(\frac{1}{5}\) The representation of the fractions in the improper form is: \(\frac{7}{3}\), \(\frac{39}{10}\)  and \(\frac{26}{5}\) So, \(\frac{7}{3}\) × \(\frac{39}{10}\) ×\(\frac{26}{5}\) = \(\frac{7 × 39 × 26}{3 × 10 × 5}\) = \(\frac{7,098}{150}\) Hence, 2\(\frac{1}{3}\)× 3\(\frac{9}{10}\) ×5\(\frac{1}{5}\) = \(\frac{7,098}{150}\)

Question 6. (1\(\frac{7}{8}\) + 4\(\frac{4}{5}\)) × 2\(\frac{1}{12}\) = _______ Answer: (1\(\frac{7}{8}\) + 4\(\frac{4}{5}\)) × 2\(\frac{1}{12}\) = \(\frac{6,675}{480}\)

Explanation: The given fractions are: 1\(\frac{7}{8}\), 4\(\frac{4}{5}\) and 2\(\frac{1}{12}\) The representation of the fractions in the improper form is: \(\frac{15}{8}\), \(\frac{24}{5}\) and \(\frac{25}{12}\) So, (1\(\frac{7}{8}\) + 4\(\frac{4}{5}\)) × \(\frac{25}{12}\) = ( \(\frac{15}{8}\) + \(\frac{24}{5}\) ) × \(\frac{25}{12}\) = ( \(\frac{75}{40}\) + \(\frac{192}{40}\) ) × \(\frac{25}{12}\) = \(\frac{75 + 192}{40}\) × \(\frac{25}{12}\) = \(\frac{267}{40}\) × \(\frac{25}{12}\) = \(\frac{267 × 25}{40× 12 }\) = \(\frac{6,675}{480}\) Hence, (1\(\frac{7}{8}\) + 4\(\frac{4}{5}\)) × 2\(\frac{1}{12}\) = \(\frac{6,675}{480}\)

Explanation: Let the missing numbers be p and q By using the partial products method, ( p × 4 ) + ( p × \(\frac{1}{7}\) ) = 20 + \(\frac{5}{7}\) p × ( 4 + \(\frac{1}{7}\) ) = 20 + \(\frac{5}{7}\) p × \(\frac{29}{7}\) = \(\frac{145}{7}\) So, p = \(\frac{145}{7}\) ÷ \(\frac{29}{7}\) = \(\frac{145}{7}\) × \(\frac{7}{29}\) = \(\frac{145 × 7}{7 × 29}\) = 5 Now, (q × 4 ) + ( q × \(\frac{1}{7}\) ) = \(\frac{12}{10}\) + \(\frac{3}{70}\) q × ( 4 + \(\frac{1}{7}\) ) = \(\frac{12}{10}\) + \(\frac{3}{70}\) q × \(\frac{29}{7}\) = \(\frac{84 + 3}{70}\) q × \(\frac{29}{7}\) = \(\frac{87}{70}\) So, q = \(\frac{87}{70}\) ÷ \(\frac{29}{7}\) = \(\frac{87}{70}\) × \(\frac{7}{29}\) = \(\frac{87 × 7}{70 × 29}\) = \(\frac{3}{10}\) Hence, from the above, We can conclude that the missing numbers are: 5 and \(\frac{3}{10}\)

Explanation: The given mixed numbers are: 1\(\frac{11}{12}\) and 2\(\frac{3}{8}\) The representation of the mixed numbers in the fraction form is: \(\frac{23}{12}\) and \(\frac{19}{8}\) So, \(\frac{23}{12}\) × \(\frac{19}{8}\) = \(\frac{23 × 19}{8 × 12}\) = \(\frac{437}{96}\) = 4\(\frac{53}{96}\) Hence, from the above, We can conclude that your friend is correct

Question 9. Modeling Real Life Your friend earns 7\(\frac{1}{2}\) dollars each hour. Will she earn enough money to buy a $35 toy after working 4\(\frac{3}{4}\) hours? Answer: she can’t earn enough money to buy $35 toy after working 4\(\frac{3}{4}\) hours

Explanation: It is given that your friend earns 7\(\frac{1}{2}\) dollars each hour. It is given that will she earn enough money to buy a $35 toy after working 4\(\frac{3}{4}\) hours So, The number of dollars she can earn = 7\(\frac{1}{2}\) × 4\(\frac{3}{4}\) = \(\frac{15}{2}\) × \(\frac{19}{4}\) = \(\frac{15 × 19}{4 × 2}\) = \(\frac{285}{8}\) Now, The representation of $35 as the multiple of 8 is: \(\frac{280}{8}\) So, By comparing the given money with he money she earned, We can say that she can’t buy the toy. Hence, from the above, We can conclude that she can’t earn enough money to buy a $35 toy after working 4\(\frac{3}{4}\) hours

Question 10. Modeling Real Life One class collects 8\(\frac{1}{4}\) pounds of recyclable materials. Another class collects 1\(\frac{1}{2}\) times as many pounds as the first class. How many pounds of recyclable materials do the two classes collect altogether? Answer: The number of pounds of recyclable materials the two classes collects together is: \(\frac{231}{8}\) pounds

Explanation: It is given that one class collects 8\(\frac{1}{4}\) pounds of recyclable materials. Another class collects 1\(\frac{1}{2}\) times as many pounds as the first class. So, The number of pounds of recyclable materials collected by another class = 8\(\frac{1}{4}\) × 1\(\frac{1}{2}\) = \(\frac{33}{4}\) × \(\frac{3}{2}\) = \(\frac{99}{8}\) pounds So, The number of pouds of recyclable materials collected by two classes together = 8\(\frac{1}{4}\) + \(\frac{99}{8}\) = \(\frac{33}{4}\) + \(\frac{99}{8}\) = \(\frac{132 + 99}{8}\) = \(\frac{231}{8}\) pounds Hence, from the above, We can conclude that the number of pounds of recyclable materials the two classes collects together is: \(\frac{231}{8}\) pounds

Find the product. Question 11. 6 × 5.7 = ______ Answer: 6 × 5.7 = 34.2

Explanation: By using the partial products method, 6 × 5.7 = 6 × ( 5 + 0.7 ) = ( 6 × 5 ) + ( 6 × 0.7 ) = 30 + 4.2 = 34.2 Hence, 6 × 5.7 = 34.2

Question 12. 0.84 × 9 = ______ Answer: 0.84 × 9 = 73.17

Explanation: By using the partial quotients method, 0.84 × 9 = ( 0.81 + 0.03 ) × 9 = ( 0.81 × 9 ) + ( 0.03 × 9 ) = 72.9 + 0.27 = 73.17 Hence, 0.84 × 9 = 73.17

Construct Arguments Explain your strategy to your partner. Compare your strategies. Answer: The strategy you followed is: A) Write the side lengths of the rectangles B) Find the areas of the four rectangles C) Make the denominators of all the four areas of rectangles equal D) Compare the numerators of all the four areas of the four rectangles

Think and Grow: Compare Factors and Products

Without calculating, tell whether the product is less than, greater than, or equal to each of its factors. Question 1. 8 × \(\frac{3}{10}\) Answer: The value of 8 × \(\frac{3}{10}\) is less than 8

Explanation: The given numbers are: 8 and \(\frac{3}{10}\) We know that, When you multiply a number by a fraction greater than 1, the product is greater than the number. When you multiply a number by a fraction less than 1, the product is less than the number. So, \(\frac{3}{10}\) is less than 1 So, 8 × \(\frac{3}{10}\) is less than 8 Hence, from the above, We can conclude that the value of 8 × \(\frac{3}{10}\) is less than 8

Question 2. \(\frac{4}{4}\) × 5\(\frac{2}{3}\) Answer: The value of \(\frac{4}{4}\) × 5\(\frac{2}{3}\) is greater than \(\frac{4}{4}\)

Explanation: The given fractions are: \(\frac{4}{4}\) and 5\(\frac{2}{3}\) We know that, When you multiply a number by a fraction greater than 1, the product is greater than the number. When you multiply a number by a fraction less than 1, the product is less than the number. We know that, \(\frac{4}{4}\) is 1 Now, The representation of 5\(\frac{2}{3}\) in the improper fraction form is: \(\frac{17}{3}\) So, \(\frac{17}{3}\) is greater than 1 So, \(\frac{4}{4}\) × 5\(\frac{2}{3}\) is greater than 1 Hence, from the above, We can conclude that the value of \(\frac{4}{4}\) × 5\(\frac{2}{3}\) is greater than \(\frac{4}{4}\)

Question 3. \(\frac{4}{3}\) × \(\frac{1}{6}\) Answer: The value of \(\frac{4}{3}\) × \(\frac{1}{6}\) is greater than \(\frac{1}{6}\) or less than \(\frac{4}{3}\)

Explanation: The given fractions are: \(\frac{4}{3}\) and \(\frac{1}{6}\) We know that, When you multiply a number by a fraction greater than 1, the product is greater than the number. When you multiply a number by a fraction less than 1, the product is less than the number. Now, \(\frac{4}{3}\) is greater than 1 \(\frac{1}{6}\) is less than 1 So, \(\frac{4}{3}\) × \(\frac{1}{6}\) is less than \(\frac{4}{3}\) Hence, from the above, We can conclude that the value of \(\frac{4}{3}\) × \(\frac{1}{6}\) is greater than \(\frac{1}{6}\) or less than \(\frac{4}{3}\)

Apply and Grow

Without calculating, tell whether the product is less than, greater than, or equal to each of its factors. Question 4. \(\frac{1}{4}\) × \(\frac{1}{12}\) Answer: The value of \(\frac{1}{4}\) × \(\frac{1}{12}\) is less than 1

Explanation: The given fractions are: latex]\frac{1}{4}[/latex] and \(\frac{1}{12}\) We know that, When you multiply a number by a fraction greater than 1, the product is greater than the number. When you multiply a number by a fraction less than 1, the product is less than the number. So, \(\frac{1}{4}\) is less than 1 \(\frac{1}{12}\) is less than 1 So, \(\frac{1}{4}\) × \(\frac{1}{12}\) is less than 1 Hence, from the above, We can conclude that the value of \(\frac{1}{4}\) × \(\frac{1}{12}\) is less than 1

Question 5. 3\(\frac{4}{5}\) × 6\(\frac{7}{8}\) Answer: The value of 3\(\frac{4}{5}\) × 6\(\frac{7}{8}\) is greater than 1

Explanation: The given fractions are: 3\(\frac{4}{5}\) and 6\(\frac{7}{8}\) The representation of 3\(\frac{4}{5}\) and 6\(\frac{7}{8}\) in the improper fractions form is: \(\frac{19}{5}\) and \(\frac{55}{8}\) We know that, When you multiply a number by a fraction greater than 1, the product is greater than the number. When you multiply a number by a fraction less than 1, the product is less than the number. So, \(\frac{19}{5}\) is greater than 1 \(\frac{55}{8}\) is greater than 1 So, 3\(\frac{4}{5}\) × 6\(\frac{7}{8}\) is greater than 1 Hence, from the above, We can conclude that the value of 3\(\frac{4}{5}\) × 6\(\frac{7}{8}\) is greater than 1

Question 6. \(\frac{1}{6}\) × \(\frac{10}{10}\) Answer: The value of \(\frac{1}{6}\) × \(\frac{10}{10}\) is less than 1

Explanation: The given fractions are: \(\frac{1}{6}\) and \(\frac{10}{10}\) We know that, When you multiply a number by a fraction greater than 1, the product is greater than the number. When you multiply a number by a fraction less than 1, the product is less than the number. We know that, \(\frac{10}{10}\) is 1 So, \(\frac{1}{6}\) is less than 1 So, \(\frac{1}{6}\) × \(\frac{10}{10}\) is less than 1 Hence, from the above, We can conclude that the value of \(\frac{1}{6}\) × \(\frac{10}{10}\) is less than 1

Question 7. \(\frac{2}{3}\) × 5 Answer: The value of \(\frac{2}{3}\) × 5 is less than 5

Explanation: The given fractions are: \(\frac{2}{3}\) and 5 We know that, When you multiply a number by a fraction greater than 1, the product is greater than the number. When you multiply a number by a fraction less than 1, the product is less than the number. So, \(\frac{2}{3}\) is less than 1 So, \(\frac{2}{3}\) × 5 is less than 5 Hence, from the above, We can conclude that the value of \(\frac{2}{3}\) × 5 is less than 5

Question 8. \(\frac{7}{10}\) × 4\(\frac{8}{9}\) Answer: The value of \(\frac{7}{10}\) × 4\(\frac{8}{9}\) is greater than 1

Explanation: The given fractions are: \(\frac{7}{10}\) and 4\(\frac{8}{9}\) The representation of 4\(\frac{8}{9}\) in the improper fraction form is: \(\frac{44}{9}\) We know that, When you multiply a number by a fraction greater than 1, the product is greater than the number. When you multiply a number by a fraction less than 1, the product is less than the number. So, \(\frac{7}{10}\) is less than 1 \(\frac{44}{9}\) is greater than 1 So, \(\frac{7}{10}\) × 4\(\frac{8}{9}\) is greater than 1 Hence, from the above, We can conclude that the value of \(\frac{7}{10}\) × 4\(\frac{8}{9}\) is greater than 1

Question 9. \(\frac{9}{2}\) × 1\(\frac{3}{4}\) Answer: The value of \(\frac{9}{2}\) × 1\(\frac{3}{4}\) is greater than 1

Explanation: The given fractions are: \(\frac{9}{2}\) × 1\(\frac{3}{4}\) The representation of 1\(\frac{3}{4}\) in the improper fraction form is: \(\frac{7}{4}\) We know that, When you multiply a number by a fraction greater than 1, the product is greater than the number. When you multiply a number by a fraction less than 1, the product is less than the number. So, \(\frac{9}{2}\) is greater than 1 \(\frac{7}{4}\) is greater than 1 So, \(\frac{9}{2}\) × 1\(\frac{3}{4}\) is greater than 1 Hence, from the above, We can conclude that the value of \(\frac{9}{2}\) × 1\(\frac{3}{4}\) is greater than 1

Explanation: The given products are: A) \(\frac{5}{6}\) × \(\frac{1}{3}\) B) \(\frac{5}{6}\) × \(\frac{7}{7}\) C) \(\frac{5}{6}\) × 1\(\frac{8}{9}\) So, In A), \(\frac{5}{6}\) and \(\frac{1}{3}\) are less than 1 So, \(\frac{5}{6}\) × \(\frac{1}{3}\) is less than 1 In B), \(\frac{5}{6}\) is less than 1 \(\frac{7}{7}\) is 1 So, \(\frac{5}{6}\) × \(\frac{7}{7}\) is less than 1 In C), \(\frac{5}{6}\) is less than1 The representation of  1\(\frac{8}{9}\) in the improper fraction form is: \(\frac{17}{9}\) So, \(\frac{17}{9}\) is greater than 1 So, \(\frac{5}{6}\) × 1\(\frac{8}{9}\) is greater than 1 Hence, from the above, We can conclude that the order of the products from the least to the greatest is: \(\frac{5}{6}\) × \(\frac{1}{3}\) < \(\frac{5}{6}\) × \(\frac{7}{7}\) < \(\frac{5}{6}\) × 1\(\frac{8}{9}\)

Explanation: The given products are: A) \(\frac{1}{6}\) × \(\frac{1}{4}\) B) \(\frac{1}{10}\) × \(\frac{1}{4}\) C) \(\frac{1}{4}\) × 5\(\frac{7}{10}\) So, In A), \(\frac{1}{6}\) and \(\frac{1}{4}\) are less than 1 So, \(\frac{1}{6}\) × \(\frac{1}{4}\) is less than 1 In B), \(\frac{1}{10}\) is less than 1 \(\frac{1}{4}\) is less than 1 So, \(\frac{1}{10}\) × \(\frac{1}{4}\) is less than 1 In C), \(\frac{1}{4}\) is less than1 The representation of  5\(\frac{7}{10}\) in the improper fraction form is: \(\frac{57}{10}\) So, \(\frac{57}{10}\) is greater than 1 So, \(\frac{1}{4}\) × 5\(\frac{7}{10}\) is greater than 1 Hence, from the above, We can conclude that the order of the products from the least to the greatest is: \(\frac{1}{6}\) × \(\frac{1}{4}\) < \(\frac{1}{10}\) × \(\frac{1}{4}\) < 5\(\frac{7}{10}\) × \(\frac{1}{4}\)

Question 12. YOU BE THE TEACHER Your friend says that \(\frac{1}{2}\) × 8 is half as much as 8. Is your friend correct? Explain. Answer: Yes, your friend is correct

Explanation: The given numbers are: \(\frac{1}{2}\) and 8 So, \(\frac{1}{2}\) × 8 = \(\frac{1}{2}\) × \(\frac{8}{1}\) = \(\frac{1 × 8}{2 × 1}\) = 4 It is also given that according to your friend, \(\frac{1}{2}\) × 8 is half as much as 8. So, 8 ÷ 2 = 4 Hence, from the above, We can conclude that your friend is correct.

Question 13. DIG DEEPER! Without calculating, tell whether the product is less than, greater than, or equal to 3\(\frac{3}{4}\). Explain. \(\left(\frac{1}{2} \times 3 \frac{3}{4}\right)\) × \(\frac{2}{7}\) Answer: \(\left(\frac{1}{2} \times 3 \frac{3}{4}\right)\) × \(\frac{2}{7}\) is less than  3\(\frac{3}{4}\)

Explanation: The representation of 3\(\frac{3}{4}\) in the improper fraction form is: \(\frac{15}{4}\) Now, \(\left(\frac{1}{2} \times 3 \frac{3}{4}\right)\) × \(\frac{2}{7}\) = ( \(\frac{1}{2}\) × \(\frac{15}{4}\) ) × \(\frac{2}{7}\) = \(\frac{15}{8}\) × \(\frac{2}{7}\) = \(\frac{2 × 15}{7 × 8}\) = \(\frac{30}{56}\) = \(\frac{15}{28}\) Now, \(\frac{15}{4}\) is multiplied by \(\frac{7}{7}\) So, \(\frac{15}{4}\) = \(\frac{105}{28}\) So, When we compare \(\frac{105}{28}\) and \(\frac{15}{28}\) We will get \(\frac{105}{28}\) is greater Hence, from the above, We can conclude that \(\left(\frac{1}{2} \times 3 \frac{3}{4}\right)\) × \(\frac{2}{7}\) is less than  3\(\frac{3}{4}\)

Question 14. You practice playing the keyboard for 3\(\frac{1}{2}\) hours. Your friend practices playing the keyboard for \(\frac{5}{4}\) as many hours as you. Does your friend practice for fewer hours, more hours, or the same number of hours as you? Answer: You practice more hours than your friend

Explanation: It is given that you practice playing the keyboard for 3\(\frac{1}{2}\) hours. Your friend practices playing the keyboard for \(\frac{5}{4}\) as many hours as you. So, Now, The representation of 3\(\frac{1}{2}\) in the improper fraction form is: \(\frac{7}{2}\) So, The number of hours you practice playing the keyboard is: \(\frac{7}{2}\) hours The number of hours your friend practice playing the keyboard is: \(\frac{5}{4}\) hours So, For comparison, make the denominators equal. So, Multiply \(\frac{7}{2}\) with \(\frac{2}{2}\) So, \(\frac{7}{2}\) = \(\frac{14}{4}\) By comparing the timings , We can say that \(\frac{14}{4}\) > \(\frac{5}{4}\) Hence, from the above, We can conclude that you practice more hours than your friend

Question 15. The original price of a telescope is $99. The sale price is \(\frac{4}{5}\) of the original price. An astrologist buys the telescope at its sale price and uses a half-off-coupon. What fraction of the original price does the astrologist pay for the telescope? Answer: The fraction of the original price the astrologist pays for the telescope is: \(\frac{4}{10}\)

Explanation: It is given that the original price of a telescope is $99. The sale price is \(\frac{4}{5}\) of the original price. and an astrologist buys the telescope at its sale price and uses a half-off-coupon. So, The cost of the telescope that an astrologist bought = ( The \(\frac{4}{5}\)th of the original price ) × ( Half-off -coupon on the \(\frac{4}{5}\)th of the original price ) = \(\frac{4}{5}\) × \(\frac{1}{2}\) = \(\frac{4 × 1}{5 × 2}\) = \(\frac{4}{10}\) Hence, from the above, We can conclude that the fraction of the original price the astrologist pays for the telescope is: \(\frac{4}{10}\)

Explanation: It is given that the Abraj Al-Bait Clock Tower is \(\frac{6}{10}\) kilometer tall and Zifeng Tower is \(\frac{3}{4}\) as tall as the clock tower. So, The height of Abraj al-Bait Clock tower is: \(\frac{6}{10}\) kilometer Now, The height of Zifeng tower is = \(\frac{6}{10}\) × \(\frac{3}{4}\) = \(\frac{6 × 3}{10 × 4}\) = \(\frac{18}{40}\) Now, for comparison, make the denomonators of both the towers equal. So, \(\frac{6}{10}\) is multiplied by \(\frac{4}{4}\) So, \(\frac{6}{10}\) = \(\frac{24}{40}\) By comparing the heights of the two towers, We can say that \(\frac{24}{40}\) > \(\frac{18}{40}\) Hence, from the above, We can conclude that The Abraj al-Bait Clock tower is taller than the Zifeng tower The height of Abraj al-Bait Clock tower is: \(\frac{24}{40}\) The height of Zifeng tower is: \(\frac{18}{40}\)

Without calculating, tell whether the product is less than, greater than, or equal to each of its factors. Question 1. 1\(\frac{3}{4}\) × 6 Answer: The value of 1\(\frac{3}{4}\) × 6 is greater than 6

Explanation: The given numbers are: 1\(\frac{3}{4}\) and 6 The representation of 1\(\frac{3}{4}\) in the improper fraction form is: \(\frac{7}{4}\) We know that, When you multiply a number by a fraction greater than 1, the product is greater than the number. When you multiply a number by a fraction less than 1, the product is less than the number. So, \(\frac{7}{4}\) is greater than 1 So, 1\(\frac{3}{4}\) × 6 is greater than 6 Hence, from the above, We can conclude that the value of 1\(\frac{3}{4}\) × 6 is greater than 6

Question 2. \(\frac{5}{12}\) × \(\frac{1}{6}\) Answer: The value of \(\frac{5}{12}\) × \(\frac{1}{6}\) is less than 1

Explanation; The given fractions are: \(\frac{5}{12}\) and \(\frac{1}{6}\) We know that, When you multiply a number by a fraction greater than 1, the product is greater than the number. When you multiply a number by a fraction less than 1, the product is less than the number. So, \(\frac{5}{12}\) is less than 1 \(\frac{1}{6}\) is less than 1 So, \(\frac{5}{12}\) × \(\frac{1}{6}\) is less than 1 Hence, from the above, We can conclude that the value of \(\frac{5}{12}\) × \(\frac{1}{6}\) is less than 1

Question 3. \(\frac{2}{7}\) × \(\frac{5}{5}\) Answer: The value of \(\frac{2}{7}\) × \(\frac{5}{5}\) is less than 1

Explanation: The given fractions are: \(\frac{2}{7}\) and \(\frac{5}{5}\) We know that, When you multiply a number by a fraction greater than 1, the product is greater than the number. When you multiply a number by a fraction less than 1, the product is less than the number. So, \(\frac{2}{7}\) is less than 1 We know that \(\frac{5}{5}\) is 1 So, \(\frac{2}{7}\) × \(\frac{5}{5}\) is less than 1 Hence, from the above, We can conclude that the value of \(\frac{2}{7}\) × \(\frac{5}{5}\) is less than 1

Question 4. 3\(\frac{4}{5}\) × 2\(\frac{9}{10}\) Answer: The value of 3\(\frac{4}{5}\) × 2\(\frac{9}{10}\) is greater than 1

Explanation: the given mixed fractions are: 3\(\frac{4}{5}\) and 2\(\frac{9}{10}\) We know that, When you multiply a number by a fraction greater than 1, the product is greater than the number. When you multiply a number by a fraction less than 1, the product is less than the number. So, The representation of the mixed numbers in the fraction form is: \(\frac{19}{5}\) and \(\frac{29}{10}\) So, \(\frac{19}{5}\) is greater than 1 \(\frac{29}{10}\) is greater than So, 3\(\frac{4}{5}\) × 2\(\frac{9}{10}\) is greater than 1 Hence, from the above, We can conclude that the value of 3\(\frac{4}{5}\) × 2\(\frac{9}{10}\) is greater than 1

Question 5. 8 × \(\frac{2}{3}\) Answer: The value of  8 is greater than \(\frac{2}{3}\)

Explanation: The given numbers are: 8 and \(\frac{2}{3}\) We know that, When you multiply a number by a fraction greater than 1, the product is greater than the number. When you multiply a number by a fraction less than 1, the product is less than the number. So, \(\frac{2}{3}\) is less than 1 So, 8 is greater than \(\frac{2}{3}\) Hence, from the above, We can conclude that the value of 8 is greater than \(\frac{2}{3}\)

Question 6. 1\(\frac{7}{8}\) × \(\frac{1}{4}\) Answer: The value of 1\(\frac{7}{8}\) × \(\frac{1}{4}\) is less than 1

Explanation: The given fractions are: 1\(\frac{7}{8}\) and \(\frac{1}{4}\) The representation of 1\(\frac{7}{8}\) in the fraction form is: \(\frac{15}{8}\) We know that, When you multiply a number by a fraction greater than 1, the product is greater than the number. When you multiply a number by a fraction less than 1, the product is less than the number. So, \(\frac{15}{8}\) is greater than 1 \(\frac{1}{4}\) is less than 1 So, 1\(\frac{7}{8}\) × \(\frac{1}{4}\) is less than 1 Hence, from the above, We can conclude that the value of 1\(\frac{7}{8}\) × \(\frac{1}{4}\) is less than 1

Explanation: The given products are: A) \(\frac{1}{3}\) × \(\frac{4}{5}\) B) \(\frac{1}{3}\) × \(\frac{6}{6}\) C) 8\(\frac{2}{9}\) × \(\frac{1}{3}\) So, In A), \(\frac{1}{3}\) and \(\frac{4}{5}\) are less than 1 So, \(\frac{1}{3}\) × \(\frac{4}{5}\) is less than 1 In B), \(\frac{1}{3}\) is less than 1 \(\frac{6}{6}\) is 1 So, \(\frac{1}{3}\) × \(\frac{6}{6}\) is less than 1 In C), \(\frac{1}{3}\) is less than1 The representation of  8\(\frac{2}{9}\) in the improper fraction form is: \(\frac{74}{9}\) So, \(\frac{74}{9}\) is greater than 1 So, \(\frac{1}{3}\) × 8\(\frac{2}{9}\) is greater than 1 Hence, from the above, We can conclude that the order of the products from the least to the greatest is: \(\frac{1}{3}\) × \(\frac{4}{5}\) < \(\frac{1}{3}\) × \(\frac{6}{6}\) < 8\(\frac{2}{9}\) × \(\frac{1}{3}\)

Explanation: The given products are: A) \(\frac{1}{12}\) × \(\frac{3}{5}\) B) 4 × \(\frac{3}{5}\) C) 2\(\frac{1}{2}\) × \(\frac{3}{5}\) So, In A), \(\frac{1}{12}\) and \(\frac{3}{5}\) are less than 1 So, \(\frac{1}{12}\) × \(\frac{3}{5}\) is less than 1 In B), \(\frac{3}{5}\) is less than 1 So, 4 × \(\frac{3}{5}\) is less than 4 In C), \(\frac{3}{5}\) is less than1 The representation of  2\(\frac{1}{2}\) in the improper fraction form is: \(\frac{5}{2}\) So, \(\frac{5}{2}\) is greater than 1 So, \(\frac{3}{5}\) × 2\(\frac{1}{2}\) is greater than 1 Hence, from the above, We can conclude that the order of the products from the least to the greatest is: \(\frac{1}{12}\) × \(\frac{3}{5}\) < 4 × \(\frac{3}{5}\) < 2\(\frac{1}{2}\) × \(\frac{3}{5}\)

Explanation: The given fractions are: 3\(\frac{1}{3}\), \(\frac{1}{6}\) and 3\(\frac{1}{2}\) The representations of the 3\(\frac{1}{3}\) and  3\(\frac{1}{2}\)  in the fraction form is: \(\frac{7}{2}\) and \(\frac{10}{3}\) So, 3\(\frac{1}{3}\) × \(\frac{1}{6}\) = \(\frac{10}{3}\)  × \(\frac{1}{6}\) = \(\frac{10}{18}\) To compare the fractions, equate the denominators So, Multiply \(\frac{7}{2}\) by \(\frac{9}{9}\) = \(\frac{63}{18}\) So, By comparison, We can say that \(\frac{10}{18}\) < \(\frac{63}{18}\) Hence, from the above, We can conclude that the value of 3\(\frac{1}{3}\) × \(\frac{1}{6}\) is less than 3\(\frac{1}{2}\)

Explanation: The given multiplication equation is: x × 1\(\frac{7}{8}\) is greater than 1\(\frac{7}{8}\) So, To find the value of x, there are 3 options. They are: A) 1 B) \(\frac{1}{2}\) and C) 1\(\frac{4}{5}\) Now, Let x = 1\(\frac{4}{5}\) So, 1\(\frac{4}{5}\) × 1\(\frac{7}{8}\) = \(\frac{9}{5}\) × \(\frac{15}{8}\) = \(\frac{27}{8}\) So, by comparison, We will get, \(\frac{27}{8}\) is greater than \(\frac{15}{8}\) Hence, from the above, We can conclude that the missing number is: 1\(\frac{4}{5}\)

Question 11. Reasoning Why does multiplying by a fraction greater than one result in a product greater than the original number? Answer: In the given fraction, if the numerator is greater than the denominator, then the given fraction is greater than 1 So, When the whole number is multiplied by the fraction which is greater than 1, the result in a product will be greater than the original number i.e, the whole number

Question 12. Modeling Real Life You snowboard 1\(\frac{7}{8}\) miles. Your friend snowboards 3\(\frac{2}{3}\) times as far as you. Does your friend snowboard fewer miles, more miles, or the same number of miles as you? Answer: The number of miles you snowboarded is less than your friend

Explanation: It is given that you snowboard 1\(\frac{7}{8}\) miles. Your friend snowboards 3\(\frac{2}{3}\) times as far as you. So, The number of miles you snowboard is: 1\(\frac{7}{8}\) miles So, The number of miles your friend snowboard is = 1\(\frac{7}{8}\) × 3\(\frac{2}{3}\) = \(\frac{15}{8}\) × \(\frac{11}{3}\) = \(\frac{15 × 11}{3 × 8}\) = \(\frac{165}{24}\) miles So, For comparison we have to make the denominators equal. So, Multiply \(\frac{15}{8}\) by \(\frac{3}{3}\) So, \(\frac{15}{8}\) = \(\frac{45}{24}\) So, By comparing, we will get \(\frac{45}{24}\) is less than \(\frac{165}{24}\) miles Hence, from the above, We can conclude that the number of miles you snowboarded is less than your friend

Answer: The dog that weighs the most is: The oldest dog The dog that weighs the least is: The youngest dog

Explanation: It is given that a pet owner has three dogs. The youngest dog weighs \(\frac{1}{4}\) as much as the second oldest dog. The oldest dog weighs 1\(\frac{1}{4}\) as much as the second oldest. The second oldest weighs 20 pounds. So, The weight of the second oldest dog is: 20 pounds Now, The weight of the oldest dog = 1\(\frac{1}{4}\) × ( The weight of the oldest second dog ) = 1\(\frac{1}{4}\) × 20 = \(\frac{5}{4}\) × 20 = \(\frac{5 × 20}{4 × 1}\) = 25 pounds Now, The weight of the youngest dog = \(\frac{1}{4}\) × ( The weight of the second oldest dog ) = \(\frac{1}{4}\) × 20 = \(\frac{1 × 20}{4}\) = 5 pounds Hence, from the above, We can conclude that The dog that weighs the most is: The oldest dog The dog that weighs the least is: The youngest dog

Explanation: The given decimal numbers are: 40.5 and 40.13 For comparison, compare the place value of the given digits If there are more than 2 digits after the decimal, first compare the tenths position and after that only compare the hundredths position If we can get the result in the tenths position only, then there is no need for further comparison Hence, from the above, We can conclude that 40.5 is greater than 40.13

Explanation: The given decimal numbers are: 13.90 and 13.9 For comparison, compare the place value of the given digits If there are more than 2 digits after the decimal, first compare the tenths position and after that only compare the hundredths position If we can get the result in the tenths position only, then there is no need for further comparison Hence, from the above, We can conclude that 13.90 is equal to 13.9

Explanation: The given decimal numbers are: 32.006 and 32.06 For comparison, compare the place value of the given digits If there are more than 2 digits after the decimal, first compare the tenths position and after that only compare the hundredths position If we can get the result in the tenths position only, then there is no need for further comparison Hence, from the above, We can conclude that 32.006 is less than 32.06

b. What is the combined height of the sedimentary rock layers? Answer: The combined height of the sedimentary layers is: \(\frac{15}{2}\) inches

Explanation: The combined height of the sedimentary rock layers = The height of limestone + The height of sandstone + The height of the shale = \(\frac{9}{4}\) + \(\frac{3}{2}\) + \(\frac{15}{4}\) = \(\frac{24}{4}\) + \(\frac{3}{2}\) = \(\frac{12}{2}\) + \(\frac{3}{2}\) = \(\frac{15}{2}\) inches Hence, from the above, We can conclude that the combined height of the sedimentary layers is: \(\frac{15}{2}\) inches

c. Will you use more foam sheets for the granite layers of the shale layers? Explain. Answer: We use  more foam sheets for the granite layers

Explanation: From the above, The height of the shale layers is: \(\frac{15}{4}\) inches From the table, The height of the granite layers is: 3 inches When we compare the two values, We can observe that the height of shale layers is greater than the granite layers Hence, from the above, We can conclude that we will use more foam sheets for granite layers

d. The height of the topsoil layer is 1\(\frac{1}{4}\) times the height of the sandstone layer. How many foam sheets do you use in the topsoil layer? Answer: The height of the topsoil layer is: \(\frac{15}{8}\) inches

Explanation: From the above, The height of the sandstone layer is: \(\frac{3}{2}\) inches So, The height of topsoil layer = 1\(\frac{1}{4}\) × \(\frac{3}{2}\) = \(\frac{5}{4}\) × \(\frac{3}{2}\) = \(\frac{15}{8}\) inches Hence, from the above, We can conclude that the height of the topsoil layer is: \(\frac{15}{8}\) inches

e. On your model, 1 inch represents 40 feet. What is the actual height of the rock formation? Answer: The actual height of the rock formation is: 495 feet

Explanation: From the above, The height of the sedimentary layers is: \(\frac{15}{2}\) inches The height of the topsoil layer is: \(\frac{15}{8}\) inches The height of the granite layer is: 3 inches So, The combined height of the rock formation = \(\frac{15}{2}\) + \(\frac{15}{8}\) + 3 = \(\frac{99}{8}\) inches But, it is given that, 1 inch = 40 feet So, The combined height of the rock formation in feet = \(\frac{99}{8}\) inches × 40 = 495 feet Hence, from the above, We can conclude that the height of the rock formation in feet is: 495 feet

f. Why do you think the rock formation has layers? Answer: The rock formation has layers because of the tectonic plates.

9.1 Multiply Whole Numbers by Fractions

Multiply. Question 1. 5 × \(\frac{1}{2}\) = _______ Answer: 5 × \(\frac{1}{2}\) = \(\frac{5}{2}\)

Explanation: The given numbers are: 5 and \(\frac{1}{2}\) So, 5 × \(\frac{1}{2}\) = \(\frac{5}{1}\) × \(\frac{1}{2}\) = \(\frac{5 × 1}{1 × 2}\) = \(\frac{5}{2}\) Hence, 5 × \(\frac{1}{2}\) = \(\frac{5}{2}\)

Question 2. 2 × \(\frac{7}{10}\) = _______ Answer: 2 × \(\frac{7}{10}\) = \(\frac{14}{10}\)

Explanation: The given numbers are: 2 and \(\frac{7}{10}\) So, 2 × \(\frac{7}{10}\) = \(\frac{2}{1}\) × \(\frac{7}{10}\) = \(\frac{2 × 7}{1 × 10}\) = \(\frac{14}{10}\) Hence, 2 × \(\frac{7}{10}\) = \(\frac{14}{10}\)

Question 3. 9 × \(\frac{5}{8}\) = _______ Answer: 9 × \(\frac{5}{8}\) = \(\frac{45}{8}\)

Explanation: The given numbers are: 9 and \(\frac{5}{8}\) So, 9 × \(\frac{5}{8}\) = \(\frac{9}{1}\) × \(\frac{5}{8}\) = \(\frac{5 × 9}{1 × 8}\) = \(\frac{45}{8}\) Hence, 9 × \(\frac{5}{8}\) = \(\frac{45}{8}\)

Question 4. 6 × \(\frac{71}{100}\) = _______ Answer: 6 × \(\frac{71}{100}\) = \(\frac{426}{100}\)

Explanation: The given numbers are: 6 and \(\frac{71}{100}\) So, 6 × \(\frac{71}{100}\) = \(\frac{6}{1}\) × \(\frac{71}{100}\) = \(\frac{6 × 71}{1 × 100}\) = \(\frac{426}{100}\) Hence, 6 × \(\frac{71}{100}\) = \(\frac{426}{100}\)

Question 5. 4 × \(\frac{8}{5}\) = _______ Answer: 4 × \(\frac{8}{5}\) = \(\frac{32}{5}\)

Explanation: The given numbers are: 4 and \(\frac{8}{5}\) So, 4 × \(\frac{8}{5}\) = \(\frac{4}{1}\) × \(\frac{8}{5}\) = \(\frac{4 × 8}{1 × 5}\) = \(\frac{32}{5}\) Hence, 4 × \(\frac{8}{5}\) = \(\frac{32}{5}\)

Question 6. 7 × \(\frac{5}{3}\) = _______ Answer: 7 × \(\frac{5}{3}\) = \(\frac{35}{3}\)

Explanation: The given numbers are: 7 and \(\frac{5}{3}\) So, 7 × \(\frac{5}{3}\) = \(\frac{7}{1}\) × \(\frac{5}{3}\) = \(\frac{5 × 7}{1 × 3}\) = \(\frac{35}{3}\) Hence, 7 × \(\frac{5}{3}\) = \(\frac{35}{3}\)

9.2 Use Models to Multiply Fractions by Whole Numbers

Multiply. Use a model to help. Question 7. \(\frac{2}{5}\) of 20 Answer: 20 × \(\frac{2}{5}\) = 8

Explanation: The given numbers are: 20 and \(\frac{2}{5}\) So, 20 × \(\frac{2}{5}\) = \(\frac{20}{1}\) × \(\frac{2}{5}\) = \(\frac{20 × 2}{1 × 5}\) = \(\frac{8}{1}\) = 8 Hence, 20 × \(\frac{2}{5}\) = 8

Question 8. \(\frac{1}{6}\) × 12 Answer: 12 × \(\frac{1}{6}\) = 2

Explanation: The given numbers are: 12 and \(\frac{1}{6}\) So, 12 × \(\frac{1}{6}\) = \(\frac{12}{1}\) × \(\frac{1}{6}\) = \(\frac{12 × 1}{1 × 6}\) = \(\frac{2}{1}\) = 2 Hence, 12 × \(\frac{1}{6}\) = 2

Question 9. \(\frac{1}{3}\) × 6 Answer: 6 × \(\frac{1}{3}\) = 2

Explanation: The given numbers are: 6 and \(\frac{1}{3}\) So, 6 × \(\frac{1}{3}\) = \(\frac{6}{1}\) × \(\frac{1}{3}\) = \(\frac{6 × 1}{1 × 3}\) = \(\frac{2}{1}\) = 2 Hence, 6 × \(\frac{1}{3}\) = 2

Question 10. \(\frac{5}{16}\) of 8 Answer: 8 × \(\frac{5}{16}\) = \(\frac{5}{2}\)

Explanation: The given numbers are: 8 and \(\frac{5}{16}\) So, 8 × \(\frac{5}{16}\) = \(\frac{8}{1}\) × \(\frac{5}{16}\) = \(\frac{5 × 8}{1 × 16}\) = \(\frac{5}{2}\) Hence, 8 × \(\frac{5}{16}\) = \(\frac{5}{2}\)

Explanation: It is given that you have 24 apples and you use \(\frac{1}{4}\) of them to make a single serving of applesauce. So, The number of apples you used = ( The total number of apples ) × ( The fraction of apples you used to make a single serving of applesauce ) = 24 × \(\frac{1}{4}\) = \(\frac{24}{1}\) × \(\frac{1}{4}\) = \(\frac{6}{1}\) = 6 apples Now, The number of apples that do not use = ( The total number of apples ) – ( The number of apples you used ) = 24 – 6 = 18 apples Hence, from the above, We can conclude that the number of apples that do not use is: 18 apples

9.3 Multiply Fractions and Whole Numbers

Multiply. Question 12. \(\frac{3}{5}\) × 15 = _______ Answer: 15 × \(\frac{3}{5}\) = 9

Explanation: The given numbers are: 15 and \(\frac{3}{5}\) So, 15 × \(\frac{3}{5}\) = \(\frac{15}{1}\) × \(\frac{3}{5}\) = \(\frac{15 × 3}{1 × 5}\) = \(\frac{9}{1}\) = 9 Hence, 15 × \(\frac{3}{5}\) = 9

Question 13. \(\frac{9}{10}\) × 30 = _______ Answer: 30 × \(\frac{9}{10}\) = 27

Explanation: The given numbers are: 30 and \(\frac{9}{10}\) So, 30 × \(\frac{9}{10}\) = \(\frac{30}{1}\) × \(\frac{9}{10}\) = \(\frac{30 × 9}{1 × 10}\) = \(\frac{27}{1}\) = 27 Hence, 30 × \(\frac{9}{10}\) = 27

Question 14. 48 × \(\frac{3}{4}\) = _______ Answer: 48 × \(\frac{3}{4}\) = 36

Explanation: The given numbers are: 48 and \(\frac{3}{4}\) So, 48 × \(\frac{3}{4}\) = \(\frac{48}{1}\) × \(\frac{3}{4}\) = \(\frac{48 × 3}{1 × 4}\) = \(\frac{36}{1}\) = 36 Hence, 48 × \(\frac{3}{4}\) = 36

Question 15. 11 × \(\frac{5}{9}\) = _______ Answer: 11 × \(\frac{5}{9}\) = \(\frac{55}{9}\)

Explanation: The given numbers are: 11 and \(\frac{5}{9}\) So, 11 × \(\frac{5}{9}\) = \(\frac{11}{1}\) × \(\frac{5}{9}\) = \(\frac{5 × 11}{1 × 9}\) = \(\frac{55}{9}\) Hence, 11 × \(\frac{5}{9}\) = \(\frac{55}{9}\)

Question 16. \(\frac{1}{6}\) × 19 = _______ Answer: 19 × \(\frac{1}{6}\) = \(\frac{19}{6}\)

Explanation: The given numbers are: 19 and \(\frac{1}{6}\) So, 19 × \(\frac{1}{6}\) = \(\frac{19}{1}\) × \(\frac{1}{6}\) = \(\frac{19 × 1}{1 × 6}\) = \(\frac{19}{6}\) Hence, 19 × \(\frac{1}{6}\) = \(\frac{19}{6}\)

Question 17. 7 × \(\frac{13}{50}\) = _______ Answer: 7 × \(\frac{13}{50}\) = \(\frac{91}{50}\)

Explanation: The given numbers are: 7 and \(\frac{13}{50}\) So, 7 × \(\frac{13}{50}\) = \(\frac{7}{1}\) × \(\frac{13}{50}\) = \(\frac{7 × 13}{1 × 50}\) = \(\frac{91}{50}\) Hence, 7 × \(\frac{13}{50}\) = \(\frac{91}{50}\)

9.4 Use Models to Multiply Fractions

Multiply. Use a model to help. Question 18. \(\frac{1}{2}\) × \(\frac{1}{10}\) = _______ Answer: \(\frac{1}{2}\) × \(\frac{1}{10}\) = \(\frac{1}{20}\)

Explanation: The given numbers are: \(\frac{1}{2}\) and \(\frac{1}{10}\) So, \(\frac{1}{2}\) × \(\frac{1}{10}\) = \(\frac{1 × 1}{2 × 10}\) = \(\frac{1}{20}\) Hence, \(\frac{1}{2}\) × \(\frac{1}{10}\) = \(\frac{1}{20}\)

Question 19. \(\frac{1}{5}\) × \(\frac{1}{9}\) = _______ Answer: \(\frac{1}{5}\) × \(\frac{1}{9}\) = \(\frac{1}{45}\)

Explanation: The given numbers are: \(\frac{1}{5}\) and \(\frac{1}{9}\) So, \(\frac{1}{5}\) × \(\frac{1}{9}\) = \(\frac{1 × 1}{5 × 9}\) = \(\frac{1}{45}\) Hence, \(\frac{1}{5}\) × \(\frac{1}{9}\) = \(\frac{1}{45}\)

Question 20. \(\frac{1}{6}\) × \(\frac{1}{7}\) = _______ Answer: \(\frac{1}{6}\) × \(\frac{1}{7}\) = \(\frac{1}{42}\)

Explanation: The given numbers are: \(\frac{1}{6}\) and \(\frac{1}{7}\) So, \(\frac{1}{6}\) × \(\frac{1}{7}\) = \(\frac{1 × 1}{6 × 7}\) = \(\frac{1}{42}\) Hence, \(\frac{1}{6}\) × \(\frac{1}{7}\) = \(\frac{1}{42}\)

Question 21. \(\frac{1}{3}\) × \(\frac{1}{8}\) = _______ Answer: \(\frac{1}{3}\) × \(\frac{1}{8}\) = \(\frac{1}{24}\)

Explanation: The given numbers are: \(\frac{1}{3}\) and \(\frac{1}{8}\) So, \(\frac{1}{3}\) × \(\frac{1}{8}\) = \(\frac{1 × 1}{3 × 8}\) = \(\frac{1}{24}\) Hence, \(\frac{1}{3}\) × \(\frac{1}{8}\) = \(\frac{1}{24}\)

Question 22. \(\frac{2}{5}\) × \(\frac{1}{3}\) = _______ Answer: \(\frac{2}{5}\) × \(\frac{1}{3}\) = \(\frac{2}{15}\)

Explanation: The given numbers are: \(\frac{2}{5}\) and \(\frac{1}{3}\) So, \(\frac{2}{5}\) × \(\frac{1}{3}\) = \(\frac{2 × 1}{5 × 3}\) = \(\frac{2}{15}\) Hence, \(\frac{2}{5}\) × \(\frac{1}{3}\) = \(\frac{2}{15}\)

Question 23. \(\frac{2}{3}\) × \(\frac{3}{5}\) = _______ Answer: \(\frac{2}{3}\) × \(\frac{3}{5}\) = \(\frac{2}{5}\)

Explanation: The given numbers are: \(\frac{2}{3}\) and \(\frac{3}{5}\) So, \(\frac{2}{3}\) × \(\frac{3}{5}\) = \(\frac{2 × 3}{3 × 5}\) = \(\frac{2}{5}\) Hence, \(\frac{2}{3}\) × \(\frac{3}{5}\) = \(\frac{2}{5}\)

9.5 Multiply Fractions

Evaluate. Question 24. \(\frac{1}{3}\) × \(\frac{1}{8}\) = _______ Answer: \(\frac{1}{3}\) × \(\frac{1}{8}\) = \(\frac{1}{24}\)

Question 25. \(\frac{5}{6}\) × \(\frac{1}{4}\) = _______ Answer: \(\frac{5}{6}\) × \(\frac{1}{4}\) = \(\frac{5}{24}\)

Explanation: The given numbers are: \(\frac{5}{6}\) and \(\frac{1}{4}\) So, \(\frac{5}{6}\) × \(\frac{1}{4}\) = \(\frac{5 × 1}{6 × 4}\) = \(\frac{5}{24}\) Hence, \(\frac{5}{6}\) × \(\frac{1}{4}\) = \(\frac{5}{24}\)

Question 26. \(\frac{7}{2}\) × \(\frac{2}{5}\) = _______ Answer: \(\frac{7}{2}\) × \(\frac{2}{5}\) = \(\frac{7}{5}\)

Explanation: The given numbers are: \(\frac{7}{2}\) and \(\frac{2}{5}\) So, \(\frac{7}{2}\) × \(\frac{2}{5}\) = \(\frac{7 × 2}{2 × 5}\) = \(\frac{7}{5}\) Hence, \(\frac{7}{2}\) × \(\frac{2}{5}\) = \(\frac{7}{5}\)

Question 27. \(\frac{9}{10}\) × \(\frac{3}{7}\) = _______ Answer: \(\frac{9}{10}\) × \(\frac{3}{7}\) = \(\frac{27}{70}\)

Explanation: The given numbers are: \(\frac{9}{10}\) and \(\frac{3}{7}\) So, \(\frac{9}{10}\) × \(\frac{3}{7}\) = \(\frac{9 × 3}{7 × 10}\) = \(\frac{27}{70}\) Hence, \(\frac{9}{10}\) × \(\frac{3}{7}\) = \(\frac{27}{70}\)

Question 28. \(\frac{4}{5}\) × \(\frac{13}{100}\) = _______ Answer: \(\frac{4}{5}\) × \(\frac{13}{100}\) = \(\frac{52}{500}\)

Explanation: The given numbers are: \(\frac{4}{5}\) and \(\frac{13}{100}\) So, \(\frac{4}{5}\) × \(\frac{13}{100}\) = \(\frac{4 × 13}{5 × 100}\) = \(\frac{52}{500}\) Hence, \(\frac{4}{5}\) × \(\frac{13}{100}\) = \(\frac{52}{500}\)

Question 29. \(\frac{11}{25}\) × \(\frac{3}{4}\) = _______ Answer: \(\frac{11}{25}\) × \(\frac{3}{4}\) = \(\frac{33}{100}\)

Explanation: The given numbers are: \(\frac{11}{25}\) and \(\frac{3}{4}\) So, \(\frac{11}{25}\) × \(\frac{3}{4}\) = \(\frac{11 × 3}{25 × 4}\) = \(\frac{33}{100}\) Hence, \(\frac{11}{25}\) × \(\frac{3}{4}\) = \(\frac{33}{100}\)

Question 30. 9 × \(\left(\frac{2}{9} \times \frac{1}{2}\right)\) = _______ Answer: 9 × ( \(\frac{1}{2}\) × \(\frac{2}{9}\) ) = 1

Explanation: The given numbers are: 9, \(\frac{1}{2}\) and \(\frac{2}{9}\) So, 9 × (\(\frac{1}{2}\) × \(\frac{2}{9}\) ) = 9 × (\(\frac{1 × 2}{2 × 9}\) ) =  9 × \(\frac{1}{9}\) = 1 Hence, 9 × ( \(\frac{1}{2}\) × \(\frac{2}{9}\) ) = 1

Question 31. \(\left(\frac{1}{10}+\frac{7}{10}\right)\) × \(\frac{3}{4}\) = _______ Answer: \(\left(\frac{1}{10}+\frac{7}{10}\right)\) × \(\frac{3}{4}\) = \(\frac{3}{5}\)

Explanation: The given fractions are: \(\frac{1}{10}\), \(\frac{7}{10}\) and \(\frac{3}{4}\) So, \(\frac{3}{4}\) × (\(\frac{1}{10}\) + \(\frac{7}{10}\) ) = \(\frac{3}{4}\) × (\(\frac{7 × 1}{10}\) ) =  \(\frac{3}{4}\) × \(\frac{8}{10}\) = \(\frac{3 × 8}{4 × 10}\) = \(\frac{24}{40}\) = \(\frac{3}{5}\) Hence, \(\left(\frac{1}{10}+\frac{7}{10}\right)\) × \(\frac{3}{4}\) = \(\frac{3}{5}\)

Question 32. \(\frac{4}{7}\) × \(\left(\frac{5}{8}-\frac{1}{4}\right)\) = _______ Answer: \(\frac{4}{7}\) × \(\left(\frac{5}{8}-\frac{1}{4}\right)\) = \(\frac{3}{14}\)

Explanation: The given fractions are: \(\frac{4}{7}\), \(\frac{5}{8}\) and \(\frac{1}{4}\) So, \(\frac{4}{7}\) × (\(\frac{5}{8}\) – \(\frac{1}{4}\) ) = \(\frac{4}{7}\) × (\(\frac{5 – 2}{8}\) ) =  \(\frac{4}{7}\) × \(\frac{3}{8}\) = \(\frac{3 × 4}{7 × 8}\) = \(\frac{12}{56}\) = \(\frac{3}{14}\) Hence, \(\frac{4}{7}\) × \(\left(\frac{5}{8}-\frac{1}{4}\right)\) = \(\frac{3}{14}\)

9.6 Find Areas of Rectangles

Question 35. Find the area of a rectangle with side lengths of \(\frac{2}{9}\) and \(\frac{1}{10}\). Answer: The area of the rectangle is: \(\frac{1}{45}\)

Explanation: The given side lengths of a rectangle are: \(\frac{2}{9}\) and \(\frac{1}{10}\) So, The area of the rectangle = \(\frac{2}{9}\) × \(\frac{1}{10}\) = \(\frac{2 × 1}{10 × 9}\) = \(\frac{1}{45}\) Hence, from the above, We can conclude that the area of the rectangle is: \(\frac{1}{45}\)

Question 36. Find the area of a with a side length of \(\frac{3}{4}\). Answer: The area of the square is: \(\frac{9}{16}\)

Explanation: The given side length of a square is: \(\frac{3}{4}\) We know that, The length of all sides of a square are equal So, the area of the square = \(\frac{3}{4}\) × \(\frac{3}{4}\) = \(\frac{3 × 3}{4 × 4}\) = \(\frac{9}{16}\) Hence, from the above, We can conclude that the area of the square is: \(\frac{9}{16}\)

9.7 Multiply Mixed Numbers

Multiply. Question 37. 1\(\frac{1}{4}\) × 2\(\frac{1}{4}\) = _______ Answer: 1\(\frac{1}{4}\) × 2\(\frac{1}{4}\) =\(\frac{45}{16}\)

Explanation: The given mixed fractions are: 1\(\frac{1}{4}\) and 2\(\frac{1}{4}\) The representation of 1\(\frac{1}{4}\) and 2\(\frac{1}{4}\) in the improper fraction form is: \(\frac{5}{4}\) and \(\frac{9}{4}\) So, \(\frac{5}{4}\) × \(\frac{9}{4}\) = \(\frac{5 × 9}{4 × 4}\) = \(\frac{45}{16}\) Hence, 1\(\frac{1}{4}\) × 2\(\frac{1}{4}\) =\(\frac{45}{16}\)

Question 38. 3\(\frac{4}{5}\) × 1\(\frac{1}{2}\) = _______ Answer: 3\(\frac{4}{5}\) × 1\(\frac{1}{2}\) = \(\frac{57}{10}\)

Explanation: The given mixed fractions are: 3\(\frac{4}{5}\) and 1\(\frac{1}{2}\) The representation of 3\(\frac{4}{5}\) and 1\(\frac{1}{2}\) in the improper fraction form is: \(\frac{19}{5}\)  and \(\frac{3}{2}\) So, \(\frac{19}{5}\) × \(\frac{3}{2}\) = \(\frac{19 × 3}{5 × 2}\) = \(\frac{57}{10}\) Hence, 3\(\frac{4}{5}\) × 1\(\frac{1}{2}\) = \(\frac{57}{10}\)

Question 39. 5\(\frac{1}{3}\) × 4\(\frac{7}{8}\) = _______ Answer: 5\(\frac{1}{3}\) × 4\(\frac{7}{8}\) = \(\frac{624}{24}\)

Explanation: The given mixed fractions are: 5\(\frac{1}{3}\) and 4\(\frac{7}{8}\) The representation of 5\(\frac{1}{3}\) and 4\(\frac{7}{8}\) in the improper fraction form is: \(\frac{16}{3}\)  and \(\frac{39}{8}\) So, \(\frac{16}{3}\) × \(\frac{39}{8}\) = \(\frac{39 × 16}{3 × 8}\) = \(\frac{624}{24}\) Hence, 5\(\frac{1}{3}\) × 4\(\frac{7}{8}\) = \(\frac{624}{24}\)

Question 40. 4\(\frac{5}{6}\) × 2\(\frac{9}{10}\) × \(\frac{1}{8}\) = _______ Answer: 4\(\frac{5}{6}\) × 2\(\frac{9}{10}\) × \(\frac{1}{8}\) = \(\frac{841}{480}\)

Explanation: The given mixed fractions are: 4\(\frac{5}{6}\), 2\(\frac{9}{10}\) and \(\frac{1}{8}\) The representation of 4\(\frac{5}{6}\) and 2\(\frac{9}{10}\) in the improper fraction form is: \(\frac{29}{6}\) and \(\frac{29}{10}\) So, \(\frac{29}{6}\) × \(\frac{29}{10}\)  × \(\frac{1}{8}\) = \(\frac{29 × 29 × 1}{8 × 10 × 6}\) = \(\frac{841}{480}\) Hence, 4\(\frac{5}{6}\) × 2\(\frac{9}{10}\) × \(\frac{1}{8}\) = \(\frac{841}{480}\)

Explanation: Let the missing numbers be: p and q So, 4\(\frac{1}{p}\) × 2\(\frac{1}{5}\) = 9\(\frac{q}{10}\) So, To solve the mixed numbers, we can separate the whole numbers and the fraction numbers So, \(\frac{1}{p}\) × 2\(\frac{1}{5}\) = \(\frac{q}{10}\) \(\frac{1}{p}\) × \(\frac{11}{5}\) = \(\frac{q}{10}\) \(\frac{1 × 11}{p × 5}\) = \(\frac{q}{10}\) \(\frac{11}{5p}\) = \(\frac{q}{10}\) By comparing LHS and RHS q = 11 and p = 2 Hence, from the above, We can conclude that the missing numbers are: 2 and 11

9.8 Compare Factors and Products

Without calculating, tell whether the product is less than, greater than, or equal to each of its factors. Question 42. 1\(\frac{1}{2}\) × 3 Answer: The value of 1\(\frac{1}{2}\) × 3 is greater than 3

Explanation: The given numbers are: 3 and 1\(\frac{1}{2}\) The representation of 1\(\frac{1}{2}\) in the mixed form is: \(\frac{3}{2}\) So, \(\frac{3}{2}\) is greater than 1 So, \(\frac{3}{2}\) × 3 is greater than 3 Hence, from the above, We can conclude that the value of 1\(\frac{1}{2}\) × 3 is greater than 3

Question 43. \(\frac{1}{8}\) × \(\frac{1}{6}\) Answer: The value of \(\frac{1}{8}\) × \(\frac{1}{6}\) is less than 1

Explanation: The given numbers are: \(\frac{1}{8}\) and \(\frac{1}{6}\) So, \(\frac{1}{8}\) is less than 1 \(\frac{1}{6}\) is less than 1 So, \(\frac{1}{8}\) × \(\frac{1}{6}\) is less than 1 Hence, from the above, We can conclude that the value of \(\frac{1}{8}\) × \(\frac{1}{6}\) is less than 1

Question 44. \(\frac{5}{5}\) × 2\(\frac{4}{7}\) Answer: The value of \(\frac{5}{5}\) × 2\(\frac{4}{7}\) is greater than 1

Explanation: The given numbers are: \(\frac{5}{5}\) and 2\(\frac{4}{7}\) The representation of 2\(\frac{4}{7}\) in the improper fraction form is: \(\frac{18}{7}\) So, \(\frac{18}{7}\) is greater than 1 \(\frac{5}{5}\) is 1 So, \(\frac{5}{5}\) × 2\(\frac{4}{7}\) is greater than 1 Hence, from the above, We can conclude that the value of \(\frac{5}{5}\) × 2\(\frac{4}{7}\) is greater than 1

Question 45. \(\frac{3}{4}\) × 2\(\frac{11}{12}\) Answer: The value of \(\frac{3}{4}\) × 2\(\frac{11}{12}\) is greater than 1

Explanation: The given numbers are: \(\frac{3}{4}\) and 2\(\frac{11}{12}\) The representation of 2\(\frac{11}{12}\) in the improper fraction form is: \(\frac{35}{12}\) So, \(\frac{35}{12}\) is greater than 1 \(\frac{3}{4}\) is less than 1 So, \(\frac{3}{4}\) × 2\(\frac{11}{12}\) is greater than 1 Hence, from the above, We can conclude that the value of \(\frac{3}{4}\) × 2\(\frac{11}{12}\) is greater than 1

Question 46. 3\(\frac{1}{9}\) × 2\(\frac{7}{8}\) Answer: The value of 3\(\frac{1}{9}\) × 2\(\frac{7}{8}\) is greater than 1

Explanation: The given mixed fractions are: 3\(\frac{1}{9}\) and 2\(\frac{7}{8}\) The representation of 3\(\frac{1}{9}\) and 2\(\frac{7}{8}\) in the improper fractions form is: \(\frac{28}{9}\) and \(\frac{23}{8}\) So, \(\frac{28}{9}\) is greater than 1 \(\frac{23}{8}\) is greater than 1 So, 3\(\frac{1}{9}\) × 2\(\frac{7}{8}\) is greater than 1 Hence, from the above, We can conclude that the value of 3\(\frac{1}{9}\) × 2\(\frac{7}{8}\) is greater than 1

Question 47. \(\frac{9}{8}\) × \(\frac{5}{2}\) Answer: The value of \(\frac{9}{8}\) × \(\frac{5}{2}\) is greater than 1

Explanation: The given fractions are: \(\frac{9}{8}\) and \(\frac{5}{2}\) So, \(\frac{9}{8}\) is greater than 1 So, \(\frac{9}{8}\) × \(\frac{5}{2}\) is greater than 1 Hence, from the above, We can conclude that the value of \(\frac{9}{8}\) × \(\frac{5}{2}\) is greater than 1

Conclusion:

Hope the information provided in the above answer key is beneficial for all the students of grade 5. Follow our Big Ideas Math Answers and make your learning fun and interesting.

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McGraw Hill My Math Grade 4 Chapter 5 Lesson 4 Answer Key Multiply by a Two-Digit Number

All the solutions provided in McGraw Hill Math Grade 4 Answer Key PDF Chapter 5 Lesson 4 Multiply by a Two-Digit Number  will give you a clear idea of the concepts.

McGraw-Hill My Math Grade 4 Answer Key Chapter 5 Lesson 4 Multiply by a Two-Digit Number

Math in My World

Guided Practice

Independent Practice

Multiply. Use the area model to check.

Question 8. 83 × 67 = ____

Question 9. 91 × 78 = ____

Problem Solving

Question 11. The fourth-grade students at Tremont School received a ribbon if they read 50 books during the school year. The school gave out 69 ribbons at the end of the year. How many books did the students read in all?

Question 12. Mathematical PRACTICE 2 Use Symbols Each day, enough paper is recycled in the United States to fill 15 miles of train boxcars. How many miles of boxcars could be filled in 5 weeks? Complete the equation to help you solve the problem. 15 × 5 × ___ = b

Answer: 15 x 5 x 7 = b 15 x 5 x 7 = 525 miles. So, 525 miles of boxcars could be filled in 5 weeks. Explanation: Given that, the data is, Each day, enough paper is recycled in the United States to fill 15 miles of train boxcars. Now, we need to find out how many miles of boxcars could be filled in 5 weeks. First, complete the equation it helps to solve the problem. The week has 7 days. So the equation is 15 x 5 x 7. Now, the product value is 15 x 5 x 7 = 525. Hence, 525 miles of boxcars could be filled in 5 weeks.

HOT Problems

Answer: 37 x 18 does not belong because it is the only problem whose product does not have a zero in the one’s place. 35 x 18 does have a zero in the one’s place. Explanation: Given that, The value is  22 x 15, $45 x 28, 37 x 18, and $66 x 25. Now, we will find the product value and find which value does not belong with the other three. The product of 22 x 15 = 330. The product of $45 x 28 = 1260. The product of 37 x 18 = 666. The product of $66 x 25 = 1650. Based on the product value 37 x 18 does not belong to the other three.

Question 14. ? Building on the Essential Question Why can’t the product of two 2-digit numbers ever be two digits? Explain.

Answer: The least product of two 2-digit factors is 100, which is a result of multiplying 10 x 10, it is a least 2-digit factors. So, the product of two 2-digit numbers is never two digits.

McGraw Hill My Math Grade 4 Chapter 5 Lesson 4 My Homework Answer Key

Question 9. Mrs. Taylor gave each student 75 pieces of paper at the beginning of the school year. If there are 32 students in her class, how many pieces of paper did she give out altogether?

Question 10. Mr. Matthews gives each of his 32 students 15 minutes to present a book report to the class. How many minutes will it take for all his students to present their reports?

Question 11. Mathematical PRACTICE 2 Use Symbols George collects 25 baseball cards each month. How many cards will he have at the end of one year? Complete the equation to help you solve the problem. 25 × ____ = n

Question 12. The office building is 48 floors high. Each floor has 36 windows. How many windows does the building have in all?

Test Practice

Question 13. There are 26 rows of bleachers in the school gym. At the pep rally, there were 17 students sitting in each row. How many students were there in all? A. 43 students B. 182 students C. 208 students D. 442 students

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