{"id":15930,"date":"2022-03-08T09:41:51","date_gmt":"2022-03-08T04:11:51","guid":{"rendered":"https:\/\/envisionmathanswerkey.com\/?p=15930"},"modified":"2022-03-14T17:20:57","modified_gmt":"2022-03-14T11:50:57","slug":"envision-math-common-core-grade-6-answer-key-topic-1","status":"publish","type":"post","link":"https:\/\/envisionmathanswerkey.com\/envision-math-common-core-grade-6-answer-key-topic-1\/","title":{"rendered":"enVision Math Common Core Grade 6 Answer Key Topic 1 Use Positive Rational Numbers"},"content":{"rendered":"

Go through the\u00a0enVision Math Common Core Grade 6 Answer Key<\/a>\u00a0Topic 1 Use Positive Rational Numbers<\/strong>\u00a0regularly and improve your accuracy in solving questions.<\/p>\n

enVision Math Common Core 6th Grade Answers Key Topic 1 Use Positive, Rational Numbers<\/h2>\n

Topic Essential Question <\/strong>How can you fluently add, subtract, multiply, and divide decimals? How can you multiply and divide fractions?
\nAnswer:
\nThe steps to add, subtract, multiply, and divide decimals are:
\na. Write the numbers vertically with the decimals lined up (if the number is a whole number then the decimal goes on the right end of the number)
\nb. Add zeros to make sure there is the same number of digits in each number.
\nc. Add or subtract as normal.
\nd. Multiply just as you would if the numbers were all whole numbers and add at the end.
\ne. After that count how many decimals places the two factors have (start at the far right of each number) and then give the answer that the total number of decimal places.
\nf. To divide decimals, always use long division.
\nThe process to multiply and divide fractions is:
\na. Dividing two fractions is the same as multiplying the first fraction by the reciprocal of the second fraction.
\nb. The first step to dividing fractions is to find the reciprocal (reverse the numerator and denominator) of the second fraction.
\nc. Next, multiply the two numerators. Then, multiply the two denominators.<\/p>\n

enVision STEM Project<\/strong><\/p>\n

VIDEO<\/strong><\/p>\n

Did You Know?<\/strong><\/p>\n

Engineers design equipment to make you safer.<\/p>\n

Engineering is the application of math and science to solve problems.<\/p>\n

Engineers solve problems by designing and building products, materials, machinery, structures, transportation vehicles, and so many other things.
\n\"Envision
\nEngineers work in nearly every area from chemical and electrical engineering to biomedical and oceanographic engineering.
\n\"Envision
\nEngineers help keep you healthy.
\n\"Envision
\nEngineers find ways to improve and enhance performance of all kinds of products.
\n\"Envision<\/p>\n

Your Task: Improve Your School<\/strong><\/p>\n

Think like an engineer! Take a walk around the inside and the outside of your school building. Make a list of specific things or areas that need improvement. Then choose one idea and do some background research to gain an understanding of factors that might impact improvement efforts. In the next topic, you and your classmates will learn about and implement the engineering design process to propose possible ways to make the improvements.
\n\"Envision<\/p>\n

Topic 1 GET READY!<\/h3>\n

Review What You Know!<\/strong><\/p>\n

Vocabulary<\/strong>
\nChoose the best term from the box to complete each definition.<\/strong><\/p>\n

compatible numbers
\ndecimal
\ndivisor
\nestimate
\nquotient<\/p>\n

Question 1.
\nNumbers that are easy to compute mentally are ____
\nAnswer:
\nWe know that,
\nNumbers that are easy to compute mentally are called “Compatible numbers”
\nHence, from the above,
\nWe can conclude that the best term to complete the given definition is: Compatible numbers<\/p>\n

Question 2.
\nThe number used to be divided is the _____
\nAnswer:
\nWe know that,
\nThe number used to be divided is the “Divisor”
\nHence, from the above,
\nWe can conclude that the best term to complete the given definition is: Divisor<\/p>\n

Question 3.
\nA(n) _____ is an approximate answer.
\nAnswer:
\nWe know that,
\nAn “Estimate” is an approximate answer
\nHence, from the above,
\nWe can conclude that the best term to complete the given definition is: Estimate<\/p>\n

Question 4.
\nThe result of a division problem is a(n) ____
\nAnswer:
\nWe know that,
\nThe result of a division problem is a “Quotient”
\nHence, from the above,
\nWe can conclude that the best term to complete the given definition is: Quotient<\/p>\n

Whole Number Operations<\/strong>
\nCalculate each value.<\/strong><\/p>\n

Question 5.
\n\"Envision
\nAnswer:
\nThe given division problem is: 348 \u00f7 4
\nNow,
\nBy using the Long Division method,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe value of the quotient for the given division problem is: 87<\/p>\n

Question 6.
\n9,007 – 3,128
\nAnswer:
\nThe given subtraction problem is: 9,007 – 3,128
\nSo,
\n9,007 – 3,128 = 5,879
\nHence, from the above,
\nWe can conclude that
\nThe result for the given subtraction problem is: 5,879<\/p>\n

Question 7.
\n35 \u00d7 17
\nAnswer:
\nThe given multiplication problem is: 35 \u00d7 17
\nNow,
\nBy using the Partial Products method,
\n35 \u00d7 17
\n= (30 + 5) \u00d7 (10 + 7)
\n= (30 \u00d7 10) + (30 \u00d7 7) + (5 \u00d7 10) + (5 \u00d7 7)
\n= 300 + 210 + 50 + 35
\n= 595
\nHence, from the above,
\nWe can conclude that
\nThe value of product for the given multiplication problem is: 595<\/p>\n

Question 8.
\n7,964 + 3,872
\nAnswer:
\nThe given addition problem is: 7,964 + 3,872
\nSo,
\n7,964 + 3,872 = 11,836
\nHence, from the above,
\nWe can conclude that
\nThe result for the given addition problem is: 11,836<\/p>\n

Question 9.
\n\"Envision
\nAnswer:
\nThe given division problem is: 4,638 \u00f7 22
\nNow,
\nBy using the Long Division method,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe value of quotient for the given division problem is: 210<\/p>\n

Question 10.
\n181 \u00d7 42
\nAnswer:
\nThe given multiplication problem is: 181 \u00d7 42
\nNow,
\nBy using the Partial Products method,
\n181 \u00d7 42
\n= (100 + 80 + 1) \u00d7 (40 + 2)
\n= (100 \u00d740) + (100 \u00d7 2) + (80 \u00d7 40) + (80 \u00d7 2) + (1 \u00d7 40) + (1 \u00d7 2)
\n= 4,000 + 200 + 3,200 + 160 + 40 + 2
\n= 7,602
\nHence, from the above,
\nWe can conclude that
\nThe value of the product for the given multiplication problem is: 7,602<\/p>\n

Mixed Numbers and Fractions<\/strong>
\nWrite each mixed number as a fraction. Write each fraction as a mixed number.<\/strong><\/p>\n

Question 11.
\n8\\(\\frac{1}{3}\\)
\nAnswer:
\nThe given mixed number is: 8\\(\\frac{1}{3}\\)
\nSo,
\nTo convert the given mixed number into a fraction,
\n8\\(\\frac{1}{3}\\)
\n= \\(\\frac{3 \u00d7 8 + 1}{3}\\)
\n= \\(\\frac{24 + 1}{3}\\)
\n= \\(\\frac{25}{3}\\)
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given mixed number into a fraction is: \\(\\frac{25}{3}\\)<\/p>\n

Question 12.
\n5\\(\\frac{3}{5}\\)
\nAnswer:
\nThe given mixed number is: 5\\(\\frac{3}{5}\\)
\nSo,
\nTo convert the given mixed number into a fraction,
\n5\\(\\frac{3}{5}\\)
\n= \\(\\frac{5 \u00d7 5 + 3}{5}\\)
\n= \\(\\frac{25 + 3}{5}\\)
\n= \\(\\frac{28}{5}\\)
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given mixed number into a fraction is: \\(\\frac{28}{5}\\)<\/p>\n

Question 13.
\n2\\(\\frac{5}{8}\\)
\nAnswer:
\nThe given mixed number is: 2\\(\\frac{5}{8}\\)
\nSo,
\nTo convert the given mixed number into a fraction,
\n2\\(\\frac{5}{8}\\)
\n= \\(\\frac{2 \u00d7 8 + 5}{8}\\)
\n= \\(\\frac{16 + 5}{8}\\)
\n= \\(\\frac{21}{8}\\)
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given mixed number into a fraction is: \\(\\frac{21}{8}\\)<\/p>\n

Question 14.
\n3\\(\\frac{4}{9}\\)
\nAnswer:
\nThe given mixed number is: 3\\(\\frac{4}{9}\\)
\nSo,
\nTo convert the given mixed number into a fraction,
\n3\\(\\frac{4}{9}\\)
\n= \\(\\frac{3 \u00d7 9 + 4}{9}\\)
\n= \\(\\frac{27 + 4}{9}\\)
\n= \\(\\frac{31}{9}\\)
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given mixed number into a fraction is: \\(\\frac{31}{9}\\)<\/p>\n

Question 15.
\n\\(\\frac{24}{7}\\)
\nAnswer:
\nThe given fraction is: \\(\\frac{24}{7}\\)
\nSo,
\nTo convert the given fraction into a mixed number,
\nQuotient\\(\\frac{Remainder}{Divisor}\\)
\nSo,
\n\\(\\frac{24}{7}\\) = 3\\(\\frac{3}{7}\\)
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given fraction into a mixed number is: 3\\(\\frac{3}{7}\\)<\/p>\n

Question 16.
\n\\(\\frac{43}{9}\\)
\nAnswer:
\nThe given fraction is: \\(\\frac{43}{9}\\)
\nSo,
\nTo convert the given fraction into a mixed number,
\nQuotient\\(\\frac{Remainder}{Divisor}\\)
\nSo,
\n\\(\\frac{43}{9}\\) = 4\\(\\frac{7}{9}\\)
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given fraction into a mixed number is: 4\\(\\frac{7}{9}\\)<\/p>\n

Question 17.
\n\\(\\frac{59}{8}\\)
\nAnswer:
\nThe given fraction is: \\(\\frac{59}{8}\\)
\nSo,
\nTo convert the given fraction into a mixed number,
\nQuotient\\(\\frac{Remainder}{Divisor}\\)
\nSo,
\n\\(\\frac{59}{8}\\) = 7\\(\\frac{3}{8}\\)
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given fraction into a mixed number is: 7\\(\\frac{3}{8}\\)<\/p>\n

Question 18.
\n\\(\\frac{32}{5}\\)
\nAnswer:
\nThe given fraction is: \\(\\frac{32}{5}\\)
\nSo,
\nTo convert the given fraction into a mixed number,
\nQuotient\\(\\frac{Remainder}{Divisor}\\)
\nSo,
\n\\(\\frac{32}{5}\\) = 6\\(\\frac{2}{5}\\)
\nHence, from the above,
\nWe can conclude that
\nThe representation of the given fraction into a mixed number is: 6\\(\\frac{2}{5}\\)<\/p>\n

Verbal Expressions<\/strong><\/p>\n

Question 19.
\nHow are the expressions “\\(\\frac{1}{4}\\) of 12″ and “12 divided by 4” related?
\nAnswer:
\nThe given expressions are: \\(\\frac{1}{4}\\) \u00d7 12 and 12 \u00f7 4
\nSo,
\nWe can write
\n12 \u00f7 4
\n= \\(\\frac{12}{4}\\)
\n= 12 \u00d7 \\(\\frac{1}{4}\\)
\nHence,f rom the above,
\nWe can conclude that the given expressions 12 \u00d7 \\(\\frac{1}{4}\\) and 12 \u00f7 4 are the same<\/p>\n

Decimals<\/strong><\/p>\n

Question 20.
\nWhat decimal does this model represent? Explain.
\n\"Envision
\nAnswer:
\nThe given model is:
\n\"Envision
\nNow,
\nFrom the given model,
\nWe can observe that
\nThe total number of parts are: 100
\nThe number of shaded parts is: 73
\nSo,
\nThe fraction of the shaded part in the given model = \\(\\frac{Shaded parts}{Total parts}\\)
\n= \\(\\frac{73}{100}\\)
\n= 0.73
\nHence, from the above,
\nWe can conclude that
\nThe decimal number that the given model represents is: 0.73<\/p>\n

Language Development<\/strong><\/p>\n

Fill in the boxes with terms and phrases related to Decimals and Fractions from the given bank. Include illustrations or examples.<\/p>\n

annex zeroes (1-1)
\ncompatible numbers (1-1)
\ndecimal point (1-1)
\nestimate (1-1)
\nhundredths (1-1)
\nline up place value (1-1)
\nrounding (1-1)
\ntenths (1-1)
\nalgorithm (1-2)
\nthe denominator (1-3)
\nmixed number (1-3)
\nthe numerator (1-3)
\nunit fraction (1-3)
\nreciprocal (1-4)
\nrewrite (1-4)
\nmultistep problems (1-7)<\/p>\n

\"Envision<\/p>\n

Topic 1 PICK A PROJECT<\/h3>\n

PROJECT 1A<\/strong>
\nWhat is the most challenging board game you have ever played?
\nPROJECT: MAKE YOUR OWN BOARD GAME
\n\"Envision<\/p>\n

PROJECT 1B<\/strong>
\nWhat is your favorite party food?
\nPROJECT: PLAN THE MENU FOR A SCHOOL FUNDRAISER
\n\"Envision<\/p>\n

PROJECT 1C<\/strong>
\nIf you planted a garden, what would be in it?
\nPROJECT: DESIGN A VEGETABLE AND HERB GARDEN
\n\"Envision<\/p>\n

PROJECT 10D<\/strong>
\nHow much food does a tiger eat?
\nPROJECT: PRESENT A PROPOSAL FOR A TIGER EXHIBIT
\n\"Envision<\/p>\n

Lesson 1.1 Fluently Add, Subtract and Multiply Decimals<\/h3>\n

ACTIVITY<\/strong><\/p>\n

Solve & Discuss It!<\/strong><\/p>\n

Maxine is making a model windmill for a science fair. She is connecting 4 cardboard tubes together vertically. Each tube is 0.28 meters in length. What is the combined measure of the connected tubes?<\/p>\n

Use Appropriate Tools<\/strong> You can use decimal grids to calculate with decimals.
\n\"Envision
\n\"Envision
\nAnswer:
\nIt is given that
\nMaxine is making a model windmill for a science fair. She is connecting 4 cardboard tubes together vertically. Each tube is 0.28 meters in length
\nSo,
\nThe combined measure of the connected tubes = (The number of cardboard tubes) \u00d7 (The length of each tube)
\n= 4 \u00d7 0.28
\n= 4 \u00d7 (0.20 + 0.08)
\n= (4 \u00d7 0.20) + (4 \u00d7 0.08)
\n= 0.8 + 0.32
\n= 1.12 meters
\nSo,
\nThe representation of the length of 4 cardboard tubes in the decimal grid is:
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe combined measure of the 4 connected tubes is: 1.12 meters<\/p>\n

Focus on math practices<\/strong> Look for Relationships<\/strong> Suppose that Maxine made another windmill model by connecting 4 cardboard tubes that are each 2.8 meters long. What is the combined measure of this model? What relationships do you see in the factors you used here and above? Explain how this helps you solve the problem.
\nAnswer:
\nIt is given that
\nMaxine made another windmill model by connecting 4 cardboard tubes that are each 2.8 meters long
\nSo,
\nThe combined measure of another model made by Maxine = (The number of cardboard tubes of another model) \u00d7 (The length of each tube)
\n= 4 \u00d7 2.8
\n= 4 \u00d7 (2 + 0.8)
\n= (4 \u00d7 2) + (4 \u00d7 0.8)
\n= 8 + 3.2
\n= 11.2 meters
\nNow,
\nFrom the above problem,
\nWe can observe that
\nThe combined measure of 4 connected tubes is: 1.12 meters
\nSo,
\nWhen we multiply 1.12 meters by 10,
\nWe will get the combined measure of the other model made by Maxine
\nHence, from the above,
\nWe can conclude that
\nThe relationship do you see in the factors you used in this problem and the above problem is:
\n(The combined measure of 4 connected tubes of the above problem) \u00d7 10 = (The combined measure of the 4 connected tubes of this problem)<\/p>\n

? Essential Question<\/strong> How can you add, subtract, and multiply with decimals?
\nAnswer:
\nThe steps to add, subtract, multiply, and divide decimals are:
\na. Write the numbers vertically with the decimals lined up (if the number is a whole number then the decimal goes on the right end of the number)
\nb. Add zeros to make sure there is the same number of digits in each number.
\nc. Add or subtract as normal.
\nd. Multiply just as you would if the numbers were all whole numbers and add at the end.
\ne. After that count how many decimals places the two factors have (start at the far right of each number) and then give the answer that the total number of decimal places.
\nf. To divide decimals, always use long division.<\/p>\n

Try It!<\/strong>
\nSuppose that Martin finished the race 0.47 seconds after Kim. What was Martin’s time in the race? Use an estimate to check that your answer is reasonable.
\nAnswer:
\nIt is given that
\nMartin finished the race 0.47 seconds after Kim
\nNow,
\nWe know that,
\nThe time Kim took to complete the race is: 50.9 seconds
\nSo,
\nThe time took by Martin in the race = (The time took by Kim to complete the race) + (The more time taken by Martin to finish the race more than Kim)
\n= 50.9 + 0.47
\n= 50.90 + 0.47
\n= 51.37 seconds
\nNow,
\nEstimate:
\nThe time took by Kim to complete the race is: 51 seconds
\nThe more time taken by Martin to finish the race more than Kim is: 0.50 seconds
\nSo,
\nThe time took by Martin to complete the race = (The time took by Kim in the race) + (The more time taken by Martin to finish the race more than Kim)
\n= 51 + 0.50
\n= 51.50 seconds
\nHence, from the above,
\nWe can conclude that
\nThe time took by Martin in the race is: 51.37 seconds
\nSince the estimated time and the exact time is near, your answer is reasonable<\/p>\n

Convince Me!<\/strong> If Martin finished the race 0.267 seconds after Kim, you would need to add 0.267 to 50.9 to solve the problem. How is adding 0.267 to 50.9 different from adding 0.26 to 50.9?
\nAnswer:
\nIt is given that
\nIf Martin finished the race 0.267 seconds after Kim, you would need to add 0.267 to 50.9 to solve the problem
\nSo,
\nThe time took by Martin in the race = (The time took by Kim in the race) + (The more time taken by Martin to finish the race more than Kim)
\n= 50.9 + 0.267
\n= 50.900 + 0.267
\n= 51.167 seconds ——(1)
\nSo,
\nThe time took by Martin in the race = (The time took by Kim in the race) + (The more time taken by Martin to finish the race more than Kim)
\n= 50.9 + 0.26
\n= 50.90 + 0.26
\n= 51.16 seconds —— (2)
\nSo,
\nWhen we compare eq (1) and eq (2),
\nWe can observe that
\neq (1) gives the more accurate time when compared to eq (2)
\nHence, from the above,
\nWe can conclude that
\nThe difference between the results of eq(1) and eq (2) is the accuracy and precision<\/p>\n

Try It!<\/strong>
\nSuppose that Katie finished the race 0.13 seconds before Amy. What was Katie’s time in the race? Use an estimate to check that your answer is reasonable.
\nAnswer:
\nIt is given that
\nKatie finished the race 0.13 seconds before Amy
\nNow,
\nWe know that,
\nThe time Amy took to complete the race is: 20.7 seconds
\nSo,
\nThe time took by Katie in the race = (The time took by Amy to complete the race) – (The time taken by Katie to finish the race before Amy)
\n= 20.7 – 0.13
\n= 20.70 – 0.13
\n= 20.57 seconds
\nNow,
\nEstimate:
\nThe time took by Amy to complete the race is: 21 seconds
\nThe time is taken by Katie to finish the race before Amy is: 0.10 seconds
\nSo,
\nThe time took by Katie to complete the race = (The time took by Amy in the race) + (The time taken by Katie to finish the race before Amy)
\n= 21 – 0.10
\n= 21.00 – 0.10
\n= 20.90 seconds
\nHence, from the above,
\nWe can conclude that
\nThe time took by Martin in the race is: 20.57 seconds
\nSince the estimated time and the exact time is near, your answer is reasonable<\/p>\n

Try It!<\/strong>
\nHow do you determine where to place the decimal point in the product?
\n\"Envision
\nAnswer:
\nWe know that,
\nDecimals are multiplied as if they were whole numbers, and then the decimal point is placed in the product. To find out where the decimal point should be placed,\u00a0count the number of decimal places after the decimal point in each factor.
\nHence,
\nThe representation of the decimal point in the given product is:
\n\"\"<\/p>\n

KEY CONCEPT<\/strong>
\n\"Envision<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n? Essential Question<\/strong> How can you add, subtract, and multiply with decimals?
\nAnswer:
\nThe steps to add, subtract, multiply, and divide decimals are:
\na. Write the numbers vertically with the decimals lined up (if the number is a whole number then the decimal goes on the right end of the number)
\nb. Add zeros to make sure there is the same number of digits in each number.
\nc. Add or subtract as normal.
\nd. Multiply just as you would if the numbers were all whole numbers and add at the end.
\ne. After that count how many decimals places the two factors have (start at the far right of each number) and then give the answer that the total number of decimal places.
\nf. To divide decimals, always use long division.<\/p>\n

Question 2.
\nGeneralize<\/strong> How is adding and subtracting decimals similar to and different from adding and subtracting whole numbers?
\nAnswer:
\nWe know that,
\nAdding and subtracting decimal numbers, numbers with a decimal point is just like adding and subtracting whole numbers.
\nThe only difference is that we line up the numbers according to the decimal point.
\nFor subtraction, the number we are subtracting goes to the bottom.<\/p>\n

Question 3.
\nWhat can you do if a decimal product has final zeros to the right of the decimal point?
\nAnswer:
\nWe can drop the zeros on the right once the decimal point has been placed in the product. If the number of decimal places is greater than the number of digits in the product, you can insert zeros in front of the product.<\/p>\n

Question 4.
\nCritique Reasoning<\/strong> Diego says that the product of 0.51 \u00d7 2.427 will have five decimal places. Is Diego correct? Explain.
\nAnswer:
\nIt is given that
\nDiego says that the product of 0.51 \u00d7 2.427 will have five decimal places
\nNow,
\nWe know that,
\nDecimals are multiplied as if they were whole numbers, and then the decimal point is placed in the product. To find out where the decimal point should be placed,\u00a0count the number of decimal places after the decimal point in each factor.
\nSo,
\nIn
\n0.51 \u00d7 2.427,
\nWe can observe that
\nThere are 2 decimal places in 0.51
\nThere are 3 decimal places in 2.427
\nSo,
\nIn 0.51 \u00d7 2,427,
\nThere are 5 decimal places
\nHence, from the above,
\nWe can conclude that Diego is correct<\/p>\n

Do You Know How?<\/strong><\/p>\n

In 5-10, find each sum or difference.<\/strong><\/p>\n

Question 5.
\n5.9 + 2.7
\nAnswer:
\nThe given addition problem is: 5.9 + 2.7
\nSo,
\nBy using the Long Addition,
\n\"\"
\nHence, from the above,
\nWe can conclude that the value of the given addition problem is: 8.6<\/p>\n

Question 6.
\n4.01 \u2013 2.95
\nAnswer:
\nThe given subtraction problem is: 4.01 – 2.95
\nSo,
\nBy using the Long Subtraction,
\n\"\"
\nHence, from the above,
\nWe can conclude that the value of the given subtraction problem is: 1.06<\/p>\n

Question 7.
\n6.8 \u2013 1.45
\nAnswer:
\nThe given subtraction problem is: 6.8 – 1.45
\nSo,
\nBy using the Long Subtraction,
\n\"\"
\nHence, from the above,
\nWe can conclude that the value of the given subtraction problem is: 5.35<\/p>\n

Question 8.
\n9.62 – 0.3
\nAnswer:
\nThe given subtraction problem is: 9.62 – 0.3
\nSo,
\nBy using the Long Subtraction,
\n\"\"
\nHence, from the above,
\nWe can conclude that the value of the given subtraction problem is: 9.32<\/p>\n

Question 9.
\n2.57 + 7.706
\nAnswer:
\nThe given addition problem is: 2.57 + 7.706
\nSo,
\nBy using the Long Addition,
\n\"\"
\nHence, from the above,
\nWe can conclude that the value of the given addition problem is: 10.276<\/p>\n

Question 10.
\n15 – 6.108
\nAnswer:
\nThe given subtraction problem is: 15 – 6.108
\nSo,
\nBy using the Long Subtraction,
\n\"\"
\nHence, from the above,
\nWe can conclude that the value of the given subtraction problem is: 8.892<\/p>\n

In 11-16, place the decimal point in the correct location in the product.<\/strong><\/p>\n

Question 11.
\n4 \u00d7 0.94 = 376
\nAnswer:
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n4 \u00d7 0.94 = 3.76<\/p>\n

Question 12.
\n5 \u00d7 0.487 = 2435
\nAnswer:
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n5 \u00d7 0.487 = 2.435<\/p>\n

Question 13.
\n3.4 \u00d7 6.8 = 2312
\nAnswer:
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n3.4 \u00d7 6.8 = 23.12<\/p>\n

Question 14.
\n3.9 \u00d7 0.08 = 312
\nAnswer:
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n3.9 \u00d7 0.08 = 0.312<\/p>\n

Question 15.
\n0.9 \u00d7 0.22 = 198
\nAnswer:
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n0.9 \u00d7 0.22 = 0.198<\/p>\n

Question 16.
\n9 \u00d7 1.2 = 108
\nAnswer:
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n9 \u00d7 1.2 = 10.8<\/p>\n

In 17 and 18, find each product.<\/strong><\/p>\n

Question 17.
\n5.3 \u00d7 2.7
\nAnswer:
\nThe given multiplication equation is: 5.3 \u00d7 2.7
\nSo,
\nBy using the Long Multiplication method,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe value of the product for the given multiplication equation is: 14.31<\/p>\n

Question 18.
\n8 \u00d7 4.09
\nAnswer:
\nThe given multiplication is: 8 \u00d7 4.09
\nSo,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe value of the product for the given multiplication equation is: 32.72<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Scan for Multimedia<\/strong><\/p>\n

In 19-27, find each sum or difference.<\/strong><\/p>\n

Question 19.
\n2.17 \u2013 0.8
\nAnswer:
\nThe given subtraction equation is: 2.17 – 0.8
\nSo,
\nBy using the Long subtraction,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n2.17 – 0.8 = 1.37<\/p>\n

Question 20.
\n4.3 + 4.16
\nAnswer:
\nThe given addition equation is: 4.3 + 4.16
\nSo,
\nBy using the Long Addition,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n4.3 + 4.16 = 8.46<\/p>\n

Question 21.
\n46.91 – 28.7
\nAnswer:
\nThe given subtraction equation is: 46.91 – 28.7
\nSo,
\nBy using the Long subtraction,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n46.91 – 28.7 = 18.21<\/p>\n

Question 22.
\n4.815 + 2.17
\nAnswer:
\nThe given addition equation is: 4.815 + 2.17
\nSo,
\nBy using the Long Addition,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n4.815 + 2.17 = 6.985<\/p>\n

Question 23.
\n5.1 – 0.48
\nAnswer:
\nThe given subtraction equation is: 5.1 – 0.48
\nSo,
\nBy using the Long subtraction,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n5.1 – 0.48 = 4.62<\/p>\n

Question 24.
\n27 + 0.185
\nAnswer:
\nThe given addition equation is: 27 + 0.185
\nSo,
\nBy using the Long Addition,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n27 + 0.185 = 27.185<\/p>\n

Question 25.
\n9.501 – 9.45
\nAnswer:
\nThe given subtraction equation is: 9.501 – 9.45
\nSo,
\nBy using the Long subtraction,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n9.501 – 9.45 = 0.051<\/p>\n

Question 26.
\n14 + 9.8
\nAnswer:
\nThe given addition equation is: 14 + 9.8
\nSo,
\nBy using the Long Addition,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n14 + 9.8 = 23.8<\/p>\n

Question 27.
\n12.65 + 14.24
\nAnswer:
\nThe given addition equation is: 12.65 + 14.24
\nSo,
\nBy using the Long Addition,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n12.65 + 14.24 = 26.89<\/p>\n

In 28-33, find each product.<\/strong><\/p>\n

Question 28.
\n7 \u00d7 0.5
\nAnswer:
\nThe given multiplication equation is: 7 \u00d7 0.5
\nSo,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n7 \u00d7 0.5 = 3.5<\/p>\n

Question 29.
\n12 \u00d7 0.08
\nAnswer:
\nThe given multiplication equation is: 12 \u00d7 0.08
\nSo,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n12 \u00d7 0.08 = 0.96<\/p>\n

Question 30.
\n24 \u00d7 0.17
\nAnswer:
\nThe given multiplication equation is: 24 \u00d7 0.17
\nSo,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n24 \u00d7 0.17 = 4.08<\/p>\n

Question 31.
\n0.4 \u00d7 0.17
\nAnswer:
\nThe given multiplication equation is: 0.4 \u00d7 0.17
\nSo,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n0.4 \u00d7 0.17 = 0.068<\/p>\n

Question 32.
\n1.9 \u00d7 0.46
\nAnswer:
\nThe given multiplication is: 1.9 \u00d7 0.46
\nSo,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n1.9 \u00d7 0.46 = 0.874<\/p>\n

Question 33.
\n3.42 \u00d7 5.15
\nAnswer:
\nThe given multiplication is: 3.42 \u00d7 5.15
\nSo,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n3.42 \u00d7 5.15 = 17.6130<\/p>\n

Question 34.
\nWrite an equation that illustrates the following: A number with two decimal places multiplied by a number with one decimal place. The product has only two nonzero digits.
\nAnswer:
\nIt is given that
\nA number with two decimal places is multiplied by a number with one decimal place. The product has only two nonzero digits.
\nNow,
\nLet the number with two decimal places be: 2.55
\nLet the number with one decimal be: 1.6
\nSo,
\nWe have to find the product of 2.55 \u00d7 1.6
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe multiplication equation with the product value that has only two non-zero digits is: 2.55 \u00d7 1.6<\/p>\n

Question 35.
\nThe Bright-o Shampoo Factory includes 1.078 ounces of vanilla oil in a 6.35-ounce bottle of shampoo. How much of the bottle of shampoo is NOT vanilla oil?
\nAnswer:
\nIt is given that
\nThe Bright-o Shampoo Factory includes 1.078 ounces of vanilla oil in a 6.35-ounce bottle of shampoo.
\nSo,
\nThe total content present in Bright-o-Shampoo = (The amount of vanilla oil) + (The amount of other content)
\nNow,
\nLet the amount of other content be: x
\nSo,
\n6.35 = 1.078 + x
\nSo,
\nx = 6.35 – 1.078
\nNow,
\nBy using the Long Subtraction,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe amount of the bottle of a shampoo that is not vanilla oil is: 5.272 ounces<\/p>\n

In 36-38, use the graph to solve.<\/strong><\/p>\n

\"Envision<\/p>\n

Question 36.
\nThe fastest speed a table tennis ball has been hit is about 13.07 times as fast as the speed for the fastest swimming. What is the speed of the table tennis ball?
\nAnswer:
\nIt is given that
\nThe fastest speed a table tennis ball has been hit is about 13.07 times as fast as the speed for the fastest swimming.
\nNow,
\nThe given graph is:
\n\"Envision
\nNow,
\nThe speed of the table tennis ball = 13.07 \u00d7 (The speed for the fastest swimming)
\n= 13.07 \u00d7 5.35
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that the speed of the tennis ball is about 70 miles per hour<\/p>\n

Question 37.
\nLook for Relationships<\/strong> How fast would 1.5 times the fastest rowing speed be? Before you solve, tell the number of decimal places in your answer.
\nAnswer:
\nThe given graph is:
\n\"Envision
\nNow,
\nFrom the given graph,
\nWe can observe that
\nThe fastest rowing speed is: 13.99 miles per hour
\nSo,
\nThe speed of 1.5 times the fastest rowing speed = 1.5 \u00d7 13.99
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n1.5 times of the fastest rowing speed is: 209.85 miles per hour<\/p>\n

Question 38.
\nWhich activity has a recorded speed about 7 times as fast as the fastest rowing speed?
\nAnswer:
\nThe given graph is:
\n\"Envision
\nNow,
\nFrom the given graph,
\nWe can observe that
\nThe fastest rowing speed is: 13.99 miles per hour
\nNow,
\n7 times of the fastest rowing speed is: 7 \u00d7 13.99
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nSo,
\nThe fastest rowing speed is about: 98 miles per hour
\nNow,
\nFrom the given graph,
\nWe can observe that
\nThe fastest Luge is: 95.69 miles per hour
\nHence, from the above,
\nWe can conclude that
\n“Fastest Luge” has a recorded speed about 7 times as fast as the fastest rowing speed<\/p>\n

Question 39.
\nMatthew a pennant, and a hat. He paid with a $50 bill and some money he borrowed from his friend. If Matthew got $6.01 in change from the cashier, how much did he borrow from his friend to pay for all the items?
\n\"Envision
\nAnswer:
\nIt is given that
\nMatthew a pennant, and a hat. He paid with a $50 bill and some money he borrowed from his friend. If Matthew got $6.01 in change from the cashier
\nNow,
\nLet the amount Matthew borrowed from his friend be: x
\nSo,
\nAccording to the given information,
\nThe amount Matthew borrowed from his friend = (The amount of bill) – (The amount Matthew got in change from the cashier)
\nx = $50 – $6.01
\nNow,
\nBy using the Long Subtraction,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe amount Matthew borrowed from his friend is: $43.99<\/p>\n

Question 40.
\nAnna’s running time for a race was 23.1 seconds. Another runner’s time was 5.86 seconds faster. Find the other runner’s time.
\nAnswer:
\nIt is given that
\nAnna’s running time for a race was 23.1 seconds. Another runner’s time was 5.86 seconds faster
\nSo,
\nThe time took by another runner = 5.86 + (Anna’s running time)
\n= 5.86 + 23.1
\nNow,
\nBy using the Long Addition,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe time took by another runner is: 28.96 seconds<\/p>\n

Question 41.
\nHigher-Order Thinking<\/strong> Explain why 0.25 \u00d7 0.4 has only one decimal place in the product.
\nAnswer:
\nThe given multiplication equation is: 0.25 \u00d7 0.4
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nNow,
\nWe know that,
\nIn the decimal numbers,
\nIf there are zeros on the right side, then we can neglect those zeros
\nSo,
\n0.100 can become 0.1
\nHence, from the above,
\nWe can conclude that
\n0.25 \u00d7 0.4 has only one decimal place in the product because of the following property:
\nIn the decimal numbers,
\nIf there are zeros on the right side, then we can neglect those zeros<\/p>\n

Question 42.
\nThe wings of some hummingbirds beat 52 times per second when hovering. If a hummingbird hovers for 35.5 seconds, how many times do its wings beat?
\nAnswer:
\nIt is given that
\nThe wings of some hummingbirds beat 52 times per second when hovering and a hummingbird hovers for 35.5 seconds
\nSo,
\nThe number of times the wings of a hummingbird beat in 35.5 seconds = (The time a hummingbird hovers) \u00d7 (The number of times the wings of a hummingbird beat)
\n= 35.5 \u00d7 52
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe number of times the wings of a hummingbird beat in 35.5 seconds is: 1,846 times<\/p>\n

Question 43.
\nThe students at Walden Middle School are selling tins of popcorn to raise money for new uniforms. They sold 42 tins in the first week. How much money did they make in the first week?
\n\"Envision
\nAnswer:
\nIt is given that
\nThe students at Walden Middle School are selling tins of popcorn to raise money for new uniforms. They sold 42 tins in the first week
\nNow,
\nThe given figure is:
\n\"\"
\nSo,
\nThe money did the students at Walden Middle School made in the first week = (The number of tins) \u00d7 (The cost of each tin)
\n= 42 \u00d7 $9.25
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe money did the students at Walden Middle School made in the first week is: $388.50<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 44.
\nUse the information in the table to solve each problem.
\n\"Envision
\nPART A<\/strong>
\nWhat is the combined length in kilometers of the Bayshore Loop trail and the Rowdy Bend trail?
\nAnswer:
\nThe given information is:
\n\"Envision
\nSo,
\nThe combined length of the Bayshore Loop trail and the Rowdy Bend trail = 3.2 + 4.2
\nNow,
\nBy using the Long Addition,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe combined length of the Bayshore Loop trail and the Rowdy Bend trail is: 7.4 Kilometers<\/p>\n

PART B<\/strong>
\nHow many kilometers longer is the Coastal Prairie trail than the Snake Bight trail?
\nAnswer:
\nThe given information is:
\n\"Envision
\nSo,
\nThe difference between the length of the Coastal Prairie trail and the Snake Bight trail = 12.1 – 2.6
\nNow,
\nBy using the Long Subtraction,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe Coastal Prairie trail is 9.5 Kilometers longer than the Snake Bight trail<\/p>\n

Lesson 1.2 Fluently Divide Whole Numbers and Decimals<\/h3>\n

Solve & Discuss It!<\/strong>
\nSome friends went to lunch and split the bill equally. If each person paid $6.75, how many people went to lunch? Use a diagram or equation to explain your thinking.
\n\"Envision
\nAnswer:
\nSome friends went to lunch and split the bill equally and each person paid $6.75
\nNow,
\nThe given figure is:
\n\"Envision<\/p>\n

So,
\nThe number of people that went for lunch – (The total bill) \u00f7 (The amount paid by each person)
\n= \\(\\frac{$27}{$6.75}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe number of people that went for lunch is: 4 people<\/p>\n

Reasoning<\/strong> How can you use reasoning to create a representation of the problem?
\nAnswer:
\nReasoning in maths is the process of applying logical thinking to a situation to derive the correct problem-solving strategy for a given question and using this method to develop and describe a solution. Put more simply, mathematical reasoning is the bridge between fluency and problem-solving.<\/p>\n

\n
\n
\n
\n
\"\"<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

Focus on math practices<\/strong>
\nReasoning<\/strong> Suppose $7.00 was added to the bill for a dessert that everyone shared. How much more does each person have to pay?
\nAnswer:
\nIt is given that
\nSuppose $7.00 was added to the bill for a dessert that everyone shared
\nSo,
\nThe total bill = $27 + $7
\n= $34
\nSo,
\nThe amount each person has to pay = (The total bill) \u00f7 (The number of people that the bill was split)
\n= \\(\\frac{$34}{4}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe amount that each person has to pay is about $8<\/p>\n

? Essential Question<\/strong> How can you divide whole numbers and decimals?
\nAnswer:
\nThe steps to divide decimals by Whole Numbers are:
\na. Bring down the next digit from the dividend. Continue dividing.
\nb. Place the decimal point in the quotient.
\nc. Check your answer: Multiply the divisor by the quotient to see if you get the dividend.<\/p>\n

Scan for Multimedia<\/strong><\/p>\n

Try It!<\/strong>
\nWorkers at an electronics company pack 2,610 smartphones in the boxes. Each box holds 9 smartphones. How many boxes do they fill?
\nAnswer:
\nIt is given that
\nWorkers at an electronics company pack 2,610 smartphones in the boxes. Each box holds 9 smartphones
\nSo,
\nThe number of boxes does the workers fill = (The total number of smartphones in the box) \u00f7 (The number of smartphones each box holds)
\n= \\(\\frac{2,610}{9}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe number of boxes does the workers fill are: 290 boxes<\/p>\n

Convince Me!<\/strong> Why is the first digit of the quotient in the Try It! not in the same place as the first digit of the quotient in Example 1?
\n\"Envision
\nAnswer:
\nWe know that,
\nUse place value\u00a0to place the first digit. Look at the first digit. If the first digit is less than the divisor, then the first digit of the quotient will be in the hundreds place. If the first digit is greater than or equal to the divisor, then the first digit of the quotient will be in the thousands place
\nHence, from the above,
\nWe can conclude that the first digit of the quotient in the Try It! is not in the same place as the first digit of the quotient in Example 1<\/p>\n

Try It!<\/strong><\/p>\n

Divide.<\/strong><\/p>\n

a. 65 \u00f7 8
\nAnswer:
\nThe given division equation is: 65 \u00f7 8
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n65 \u00f7 8 = 8.12<\/p>\n

b. 14.4 \u00f7 8
\nAnswer:
\nThe given division equation is: 14.4 \u00f7 8
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n14.4 \u00f7 8 = 1.8<\/p>\n

c. 128.8 \u00f7 1.4
\nAnswer:
\nThe given division equation is: 128.8 \u00f7 1.4
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n128.8 \u00f7 1.4 = 92<\/p>\n

KEY CONCEPT<\/strong>
\nTo divide by a decimal, rewrite the decimal so that you are dividing by a whole number. Multiply both the divisor and the dividend by the same power of 10. Then divide as you would with whole numbers.
\n\"Envision<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n? Essential Question<\/strong> How can you divide whole numbers and decimals?
\nAnswer:
\nThe steps to divide decimals by Whole Numbers are:
\na. Bring down the next digit from the dividend. Continue dividing.
\nb. Place the decimal point in the quotient.
\nc. Check your answer: Multiply the divisor by the quotient to see if you get the dividend.<\/p>\n

Question 2.
\nWhen dividing with decimals, why is it necessary to multiply both the divisor and the dividend by the same power of 10?
\nAnswer:
\nIn cases like this, you can use powers of 10 to help create an easier problem to solve.
\nExample:
\nIn this case, you can multiply the divisor, 0.3, by 10 to move the decimal point 1 place to the right. If you multiply the divisor by 10, then you also have to multiply the dividend by 10 to keep the quotient the same.<\/p>\n

Question 3.
\nUse Structure<\/strong> Explain how you can decide where to place the first digit of the quotient for 6,139 \u00f7 153.
\nAnswer:
\nWe know that,
\nUse place value\u00a0to place the first digit. Look at the first digit. If the first digit is less than the divisor, then the first digit of the quotient will be in the hundreds place. If the first digit is greater than or equal to the divisor, then the first digit of the quotient will be in the thousands place
\nHence, from the above,
\nWe can conclude that
\nFor 6,139 \u00f7 153,
\nThe first digit of the quotient will be in the hundred’s place<\/p>\n

Question 4.
\nUse Structure<\/strong> How do you know where to place the decimal point in the quotient when dividing a decimal by a whole number?
\nAnswer:
\nWe know that,
\nThe decimal point in the quotient is directly above the decimal point in the dividend. To divide a decimal by a whole number, we place the decimal point in the quotient above the decimal point in the dividend and then divide as usual.<\/p>\n

Do You Know How?<\/strong><\/p>\n

Question 5.
\n\"Envision
\nAnswer:
\nThe given division equation is: 9,853 \u00f7 48
\nSo,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n9,853 \u00f7 48 = 205 R 13<\/p>\n

In 6 and 7, divide. Record remainders.<\/strong><\/p>\n

Question 6.
\n2,789 \u00f7 36
\nAnswer:
\nThe given division equation is: 2,789 \u00f7 36
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n2,879 \u00f7 36 = 77 R 17
\nThe remainder for the given division equation is: 17<\/p>\n

Question 7.
\n\"Envision
\nAnswer:
\nThe given division equation is: 153 \u00f7 18
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n153 \u00f7 18 = 8 R 8
\nThe remainder for the given division equation is: 8<\/p>\n

In 8 and 9, divide. Write remainders as decimals.<\/strong><\/p>\n

Question 8.
\n\"Envision
\nAnswer:
\nThe given division equation is: 139 \u00f7 4
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n139 \u00f7 4 = 34 R 3
\nThe remainder for the given division equation is: 3<\/p>\n

Question 9.
\n215 \u00f7 2
\nAnswer:
\nThe given division equation is: 215 \u00f7 2
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n215 \u00f7 2 = 107 R 1
\nThe remainder for the given division equation is: 1<\/p>\n

In 10 and 11, divide.<\/strong><\/p>\n

Question 10.
\n\"Envision
\nAnswer:
\nThe given division equation is: 34.75 \u00f7 5
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n34.75 \u00f7 5 = 6.95<\/p>\n

Question 11.
\n215.25 \u00f7 5
\nAnswer:
\nThe given division equation is: 215.25 \u00f7 5
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n215.25 \u00f7 5 = 43.05<\/p>\n

In 12 and 13, divide. Annex zeros if needed to write remainders as decimals.<\/strong><\/p>\n

Question 12.
\n5.3 \u00f7 0.2
\nAnswer:
\nThe given division equation is: 5.3 \u00f7 0.2
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n5.3 \u00f7 0.2 = 26.5<\/p>\n

Question 13.
\n\"Envision
\nAnswer:
\nThe given division equation is: 8.9 \u00f7 0.4
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n8.9 \u00f7 0.4 = 22.2<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Scan for Multimedia<\/strong><\/p>\n

Leveled Practice<\/strong>
\nIn 14 and 15, divide.<\/strong><\/p>\n

Question 14.
\n\"Envision
\nAnswer:
\nThe given division equation is: 5,841 \u00f7 62
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n5,841 \u00f7 62 = 94 R 13<\/p>\n

Question 15.
\n\"Envision
\nAnswer:
\nThe given division equation is: 350 \u00f7 4
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n350 \u00f7 4 = 87.5<\/p>\n

In 16-19, divide. Record remainders<\/strong>.<\/p>\n

Question 16.
\n2,593 \u00f7 21
\nAnswer:
\nThe given division equation is: 2,593 \u00f7 21
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n2,593 \u00f7 21 = 123 R 10
\nThe remainder for the given division equation is: 10<\/p>\n

Question 17.
\n\"Envision
\nAnswer:
\nThe given division equation is: 6,927 \u00f7 19
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n6,927 \u00f7 19 = 364 R 11
\nThe remainder for the given division equation is: 11<\/p>\n

Question 18.
\n\"Envision
\nAnswer:
\nThe given division equation is: 2,483 \u00f7 9
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n2,483 \u00f7 9 = 276 R 8
\nThe remainder for the given division equation is: 8<\/p>\n

Question 19.
\n968 \u00f7 38
\nAnswer:
\nThe given division equation is: 968 \u00f7 38
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n968 \u00f7 38 = 25 R 18
\nThe remainder for the given division equation is: 25 R 18<\/p>\n

In 20-23, divide. Write remainders as decimals.<\/strong><\/p>\n

Question 20.
\n\"Envision
\nAnswer:
\nThe given division equation is: 56 \u00f7 5
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n56 \u00f7 5 = 11.2<\/p>\n

Question 21.
\n232 \u00f7 40
\nAnswer:
\nThe given division equation is: 232 \u00f7 40
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n232 \u00f7 40 = 5.8<\/p>\n

Question 22.
\n44 \u00f7 10
\nAnswer:
\nThe given division equation is: 44 \u00f7 10
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n44 \u00f7 10 = 4.4<\/p>\n

Question 23.
\n\"Envision
\nAnswer:
\nThe given division equation is: 2,626 \u00f7 4
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n2,626 \u00f7 4 = 656.5<\/p>\n

In 24-27, divide.<\/strong><\/p>\n

Question 24.
\n\"Envision
\nAnswer:
\nThe given division equation is: $54.18 \u00f7 6
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n$54.18 \u00f7 6 = $9.03<\/p>\n

Question 25.
\n187.2 \u00f7 8
\nAnswer:
\nThe given division equation is: 187.2 \u00f7 8
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n187.2 \u00f7 8 = 23.4<\/p>\n

Question 26.
\n\"Envision
\nAnswer:
\nThe given division equation is: 6.3 \u00f7 7
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n6.3 \u00f7 7 = 0.9<\/p>\n

Question 27.
\n137.5 \u00f7 5
\nAnswer:
\nThe given division equation is: 137.5 \u00f7 5
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n137.5 \u00f7 5 = 27.5<\/p>\n

In 28-31, divide. Annex zeros if needed to write remainders as decimals.<\/strong><\/p>\n

Question 28.
\n6.4 \u00f7 0.8
\nAnswer:
\nThe given division equation is: 6.4 \u00f7 0.8
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n6.4 \u00f7 0.8 = 8<\/p>\n

Question 29.
\n\"Envision
\nAnswer:
\nThe given division equation is: 0.2430 \u00f7 0.6
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n0.2430 \u00f7 0.6 = 0.405<\/p>\n

Question 30.
\n52.056 \u00f7 7.23
\nAnswer:
\nThe given division equation is: 52.056 \u00f7 7.23
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n52.056 \u00f7 7.23 = 7.2<\/p>\n

Question 31.
\n\"Envision
\nAnswer:
\nThe given division equation is: 9.089 \u00f7 0.745
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n9.089 \u00f7 0.745 = 12.2<\/p>\n

Question 32.
\nAnts are one of the Thorny Devil lizard’s favorite foods. It can eat 45 ants per minute. How long would it take this lizard to eat 1,080 ants? Express your answer in minutes.
\n\"Envision
\nAnswer:
\nIt is given that
\nAnts are one of the Thorny Devil lizard’s favorite foods. It can eat 45 ants per minute.
\nSo,
\nThe time it will take for Thorny Devil lizard to eat 1,080 ants = (The total number of ants) \u00f7 (The number of ants Thorny Devil lizard eat per minute)
\n= 1,080 \u00f7 45
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe time it will take for Thorny Devil lizard to eat 1,080 ants is: 24 minutes<\/p>\n

Question 33.
\nCritique Reasoning<\/strong> Henrieta divided 0.80 by 20 as shown. Is her work correct? If not, explain why and give a correct response.
\n\"Envision
\nAnswer:
\nIt is given that
\nHenrieta divided 0.80 by 20 as shown
\nSo,
\nThe representation of the division equation Henrieta divided is: 0.80 \u00f7 20
\nNow,
\nBy using the Long Division,
\n\"\"
\nSo,
\nFrom the above,
\nWe can observe that
\n0.80 \u00f7 20 = 0.04
\nHence, from the above,
\nWe can conclude that Henrieta’s work is not correct<\/p>\n

Question 34.
\nWhich brand of fruit snacks costs less per pound? How much less?
\n\"Envision
\nAnswer:
\nThe given figure is:
\n\"Envision
\nNow,
\nThe cost per pound of Brand A’s fruit snacks = (The cost of Brand A fruit snacks) \u00f7 (The amount of Brand A fruit snacks)
\n= $16.20 \u00f7 15
\nNow,
\nBy using the Long Division,
\n\"\"
\nSo,
\nThe cost per pound of Brand A’s fruit snacks is: $1.08
\nNow,
\nThe cost per pound of Brand B’s fruit snacks = (The cost of Brand B fruit snacks) \u00f7 (The amount of Brand B fruit snacks)
\n= $22.25 \u00f7 25
\nNow,
\nBy using the Long Division,
\n\"\"
\nSo,
\nThe cost per pound of Brand B’s fruit snacks is: $0.89
\nSo,
\nThe difference between the cost per pound of two brands = $1.08 – $0.89
\nNow,
\nBy using the Long Subtraction,
\n\"\"
\nSo,
\nThe difference between the cost per pound of two brands is: $0.19
\nHence, from the above,
\nWe can conclude that
\nThe cost per pound of Brand B’s fruit snacks is: $0.89
\nThe difference between the cost per pound of two brands is: $0.19<\/p>\n

Question 35.
\nBe Precise<\/strong> How many times as much does each item cost in 2010 as in 1960?
\n\"Envision
\nAnswer:
\nThe given data is:
\n\"\"
\nNow,
\nLet the number of times as much does each item cost in 2010 as in 1960 be x
\nSo,
\nFor the Movie Ticket,
\nx = $9.75 \u00f7 $0.75
\nNow,
\nBy using the Long Division,
\n\"\"
\nSo,
\nFor Regular popcorn,
\nx = $4.10 \u00f7 $0.25
\nNow,
\nBy using the Long Division,
\n\"\"
\nSo,
\nFor Regular Drink,
\nx = $3.08 \u00f7 $0.35
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe number of times as much does the movie ticket cost in 2010 as in 1960 is: 13 times
\nThe number of times as much does the regular ticket cost in 2010 as in 1960 is: 16.40 times
\nThe number of times as much does the regular drink cost in 2010 as in 1960 is: 8.80 times<\/p>\n

Question 36.
\nHigher-Order Thinking<\/strong> Kendra has 5.5 pounds of popcorn and wants to package it equally in 50 bags. How can she use place-value reasoning to find the amount of popcorn to put in each bag?
\nAnswer:
\nIt is given that
\nKendra has 5.5 pounds of popcorn and wants to package it equally in 50 bags.
\nNow,
\nWe know that,
\nUse place value\u00a0to place the first digit. Look at the first digit. If the first digit is less than the divisor, then the first digit of the quotient will be in the hundreds place. If the first digit is greater than or equal to the divisor, then the first digit of the quotient will be in the thousands place
\nSo,
\nThe amount of popcorn Kendra put in each bag = (The total number of pounds of popcorn Kendra has) \u00f7 (The number of bags Kendra wants to package the popcorn equally)
\n= 5.5 \u00f7 50
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe amount of popcorn Kendra put in each bag is: 0.11 pounds<\/p>\n

Question 37.
\nYou and a friend are paid $38.25 for doing yard work. You worked 2.5 hours and your friend worked 2 hours. You split the money according to the amount of time each of you worked. How much is your share of the money? Explain.
\nAnswer:
\nIt is given that
\nYou and a friend are paid $38.25 for doing yard work. You worked 2.5 hours and your friend worked 2 hours. You split the money according to the amount of time each of you worked
\nSo,
\nThe amount of money that is your share = (The amount of money you and your friend are paid for doing yard work) \u00f7 (The number of hours you worked)
\n= $38.25 \u00f7 2.5
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe amount of money that is your share is: $15.30<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 38.
\nWhat is the value of the expression 1,248 \u00f7 25?
\nA. 49
\nB. 49 R9
\nC. 49.9
\nD. 49 R 23
\nAnswer:
\nThe given division equation is: 1,248 \u00f7 25
\nNow,
\nBy using the Long Division,
\n\"\" \"\"
\nHence, from the above,
\nWe can conclude that
\nThe value of the expression 1,248 \u00f7 25 is:
\n\"\"<\/p>\n

Question 39.
\nWhich expression has the same solution as 3,157 \u00f7 41?
\nA. 1,852 \u00f7 24
\nB. 1,928 \u00f7 25
\nC. 2,079 \u00f7 27
\nD. 2,184 \u00f7 28
\nAnswer:
\nThe given division equation is: 3,157 \u00f7 41
\nNow,
\nBy using the Long Division,
\n\"\"
\nNow,
\nThe expressions that have the same value as the given expression by using the Long Division are:
\nA. 1,852 \u00f7 24
\n\"\"
\nB. 1,928 \u00f7 25
\n\"\"
\nC. 2,079 \u00f7 27
\n\"\"
\nD. 2,184 \u00f7 28
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe expressions that have the same value as the given expression are:
\n\"\"<\/p>\n

Lesson 1-3 Multiply Fractions<\/h3>\n

Solve & Discuss It!<\/strong><\/p>\n

ACTIVITY<\/strong><\/p>\n

The art teacher gave each student half of a sheet of paper. Then she asked the students to color one-fourth of their pieces of paper. What part of the original sheet did the students color?
\nAnswer:
\nIt is given that
\nThe art teacher gave each student half of a sheet of paper. Then she asked the students to color one-fourth of their pieces of paper
\nSo,
\nThe part of the original sheet did the students color = (The piece of paper the art teacher gave to each student) \u00d7 (The part of the paper the are teacher asked the students to color)
\n= \\(\\frac{1}{2}\\) \u00d7 \\(\\frac{1}{4}\\)
\n= \\(\\frac{1 \u00d7 1}{2 \u00d7 4}\\)
\n= \\(\\frac{1}{8}\\) of the piece of paper
\nHence, from the above,
\nWe can conclude that
\nThe part of the original sheet did the students color is: \\(\\frac{1}{8}\\) of the piece of paper<\/p>\n

A model with Math<\/strong>
\nHow can you use a picture to represent the problem?
\n\"Envision
\nAnswer:
\nBy representing units of measurement and other objects visually, students can begin to think about the problem mathematically. Pictures and diagrams are\u00a0also good ways of describing solutions to problems; therefore they are an important part of mathematical communication.<\/p>\n

Focus on math practices
\nReasoning<\/strong> Should your answer be less than or greater than 1? Explain.
\nAnswer:
\nFrom the above problem,
\nWe can observe that
\nThe part of the original sheet did the students color is: \\(\\frac{1}{8}\\) of the piece of paper
\nNow,
\nBy using the Long Division,
\nThe value of \\(\\frac{1}{8}\\) is:
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nYour answer is less than 1<\/p>\n

? Essential Question<\/strong> How can you multiply fractions and mixed numbers?
\nAnswer:
\nThe steps that are used to multiply mixed numbers and fractions are:
\na. Convert the given mixed fraction to an improper fraction.
\nb. Write the whole number in the form of a fraction with denominator 1.
\nc. Multiply the numerators of both the fractions.
\nd. Multiply the denominators of both the fractions.
\ne. Combine them and simplify the fraction to its lowest form.<\/p>\n

Try It!<\/strong><\/p>\n

Find \\(\\frac{1}{4}\\) \u00d7 \\(\\frac{1}{5}\\) using the area model. Explain.
\n\"Envision
\nAnswer:
\nThe given multiplication equation is:
\n\\(\\frac{1}{4}\\) \u00d7 \\(\\frac{1}{5}\\)
\nNow,
\nBy using the Area model,
\n\"\"
\nSo,
\n\"\"
\nSo,
\n\\(\\frac{1}{4}\\) \u00d7 \\(\\frac{1}{5}\\)
\n= \\(\\frac{1 \u00d7 1}{4 \u00d7 5}\\)
\n= \\(\\frac{1}{20}\\)
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{1}{4}\\) \u00d7 \\(\\frac{1}{5}\\) = \\(\\frac{1}{20}\\)<\/p>\n

Convince Me!<\/strong> Why is the product of \\(\\frac{1}{4}\\) \u00d7 \\(\\frac{1}{5}\\) less than each factor?
\n\"Envision
\nAnswer:
\nThe given multiplication equation is:
\n\\(\\frac{1}{4}\\) \u00d7 \\(\\frac{1}{5}\\)
\nSo,
\n\\(\\frac{1}{4}\\) \u00d7 \\(\\frac{1}{5}\\)
\n= \\(\\frac{1}{20}\\)
\nNow,
\nWhen we compare the denominators,
\n4 < 5 and 5 < 20
\nSo,
\n20\u00a0 > 5 > 4 —- (1)
\nBut,
\nWhen we compare the denominators,
\nThe same relationship of eq (1) will be reversed
\nSo,
\n20 < 5 < 4
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{1}{4}\\) \u00d7 \\(\\frac{1}{5}\\) is less than each factor<\/p>\n

Try It!<\/strong>
\nFind \\(\\frac{3}{4}\\) \u00d7 \\(\\frac{4}{6}\\) using the number line. Explain.
\n\"Envision
\nAnswer:
\nThe given multiplication equation is:
\n\\(\\frac{3}{4}\\) \u00d7 \\(\\frac{4}{6}\\)
\nSo,
\n\\(\\frac{3}{4}\\) \u00d7 \\(\\frac{4}{6}\\)
\n= \\(\\frac{3 \u00d7 4}{4 \u00d7 6}\\)
\n= \\(\\frac{12}{24}\\)
\n= \\(\\frac{3}{6}\\)
\nSo,
\nThe representation of the product of \\(\\frac{3}{4}\\) \u00d7 \\(\\frac{4}{6}\\) in the number line is:
\n\"\"
\nHence, from the above,
\nWe can concldue that
\n\\(\\frac{3}{4}\\) \u00d7 \\(\\frac{4}{6}\\) = \\(\\frac{3}{6}\\)<\/p>\n

Try It!<\/strong>
\nA clothing factory makes T-shirts. If each machine makes 3\\(\\frac{1}{3}\\) T-shirts per hour, how many T-shirts does one machine make in 4\\(\\frac{1}{2}\\) hours? Write and solve an equation.
\n\"Envision
\nAnswer:
\nIt is given that
\nA clothing factory makes T-shirts. If each machine makes 3\\(\\frac{1}{3}\\) T-shirts per hour
\nSo,
\nThe number of T-shirts does 1 machine makes in 4\\(\\frac{1}{2}\\) hours = (The number of T-shirts made by each machine in a hour) \u00d7 4\\(\\frac{1}{2}\\) hours
\nNow,
\nThe representation of 3\\(\\frac{1}{3}\\) into a fraction is: \\(\\frac{10}{3}\\)
\nThe representation of 4\\(\\frac{1}{2}\\) into a fraction is: \\(\\frac{9}{2}\\)
\nSo,
\nThe number of T-shirts does 1 machine makes in 4\\(\\frac{1}{2}\\) hours = \\(\\frac{10}{3}\\) \u00d7 \\(\\frac{9}{2}\\)
\n= \\(\\frac{10 \u00d79}{3 \u00d72}\\)
\n= \\(\\frac{90}{6}\\)
\n= 15 T-shirts
\nHence, from the above,
\nWe can conclude that
\nThe number of T-shirts does 1 machine makes in 4\\(\\frac{1}{2}\\) hours is: 15 T-shirts<\/p>\n

KEY CONCEPT<\/strong>
\nYou can find the product of fractions or mixed numbers.
\n\"Envision<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n? Essential Question<\/strong> How can you multiply fractions and mixed numbers?
\nAnswer:
\nThe steps that are used to multiply mixed numbers and fractions are:
\na. Convert the given mixed fraction to an improper fraction.
\nb. Write the whole number in the form of a fraction with denominator 1.
\nc. Multiply the numerators of both the fractions.
\nd. Multiply the denominators of both the fractions.
\ne. Combine them and simplify the fraction to its lowest form.<\/p>\n

Question 2.
\nReasoning<\/strong> Is the product of \\(\\frac{3}{6}\\) \u00d7 \\(\\frac{5}{4}\\) equal to the product of \\(\\frac{3}{6}\\) \u00d7 \\(\\frac{5}{6}\\)? Explain.
\nAnswer:
\nThe given multiplication equations are:
\n\\(\\frac{3}{6}\\) \u00d7 \\(\\frac{5}{4}\\)
\n\\(\\frac{3}{6}\\) \u00d7 \\(\\frac{5}{6}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe product of \\(\\frac{3}{6}\\) \u00d7 \\(\\frac{5}{4}\\)\u00a0 is not equal to the product of \\(\\frac{3}{6}\\) \u00d7 \\(\\frac{5}{6}\\)<\/p>\n

Question 3.
\nConstruct Arguments<\/strong> Why is adding \\(\\frac{3}{9}\\) and \\(\\frac{6}{9}\\) different from multiplying the two fractions?
\nAnswer:<\/p>\n

Question 4.
\nTina has \\(\\frac{1}{2}\\) of a pan of cornbread left from a dinner party. She eats \\(\\frac{1}{2}\\) of the leftover part the next night. How much of the whole pan does Tina eat? Write and solve an equation.
\nAnswer:
\nIt is given that
\nTina has \\(\\frac{1}{2}\\) of a pan of cornbread left from a dinner party. She eats \\(\\frac{1}{2}\\) of the leftover part the next night
\nSo,
\nThe amount of the whole pan of cornbread does Tina eat = (The amount of a pan of cornbread left from a dinner party) \u00d7 (The amount of the leftover from a dinner party that Tine eat the next night)
\n= \\(\\frac{1}{2}\\) \u00d7 \\(\\frac{1}{2}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe amount of the whole pan of cornbread does Tina eat is: \\(\\frac{1}{4}\\) of a pan of cornbread<\/p>\n

Question 5.
\nConstruct Arguments<\/strong> Explain how you would multiply 5 \u00d7 2\\(\\frac{1}{2}\\).
\nAnswer:
\nThe given multiplication is: 5 \u00d7 2\\(\\frac{1}{2}\\)
\nSo,
\nThe representation of 2\\(\\frac{1}{2}\\) into a fraction is:
\n\"\"
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n5 \u00d7 2\\(\\frac{1}{2}\\) = \\(\\frac{25}{2}\\)<\/p>\n

Question 6.
\nIn Example 1, find the fraction of a whole pan of lasagna that Tom ate if he started with \\(\\frac{7}{8}\\) of a pan.
\nAnswer:
\nIt is given that
\nTom has a \\(\\frac{7}{8}\\) of a pan of lasagna left and Tom ate \\(\\frac{1}{3}\\) of this amount
\nSo,
\nThe fraction of a whole pan of lasagna that Tom ate = (The amount of a pan of lasagna left) \u00d7 (The amount that Tom ate from the amount of lasagna that is left)
\n= \\(\\frac{7}{8}\\) \u00d7 \\(\\frac{1}{3}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe fraction of a whole pan of lasagna that Tom ate is: \\(\\frac{7}{24}\\) of a pan of lasagna<\/p>\n

Do You Know How?<\/strong><\/p>\n

Question 7.
\nFind \\(\\frac{5}{6}\\) \u00d7 \\(\\frac{1}{2}\\). Use the model to help solve.
\n\"Envision
\nAnswer:
\nThe given multiplication equation is:
\n\\(\\frac{5}{6}\\) \u00d7 \\(\\frac{1}{2}\\)
\nNow,
\nBy using the Area model,
\n\"\"
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{5}{6}\\) \u00d7 \\(\\frac{1}{2}\\) = \\(\\frac{5}{12}\\)<\/p>\n

Question 8.
\nFind \\(\\frac{3}{4}\\) \u00d7 \\(\\frac{4}{9}\\).
\nAnswer:
\nThe given multiplication equation is:
\n\\(\\frac{3}{4}\\) \u00d7 \\(\\frac{4}{9}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{3}{4}\\) \u00d7 \\(\\frac{4}{9}\\) = \\(\\frac{1}{3}\\)<\/p>\n

In 9-16, find each product.<\/strong><\/p>\n

Question 9.
\n\\(\\frac{2}{3}\\) \u00d7 \\(\\frac{1}{2}\\)
\nAnswer:
\nThe given multiplication equation is:
\n\\(\\frac{2}{3}\\) \u00d7 \\(\\frac{1}{2}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{2}{3}\\) \u00d7 \\(\\frac{1}{2}\\) = \\(\\frac{1}{3}\\)<\/p>\n

Question 10.
\n\\(\\frac{5}{9}\\) \u00d7 \\(\\frac{1}{9}\\)
\nAnswer:
\nThe given multiplication equation is:
\n\\(\\frac{5}{9}\\) \u00d7 \\(\\frac{1}{9}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{5}{9}\\) \u00d7 \\(\\frac{1}{9}\\) = \\(\\frac{5}{81}\\)<\/p>\n

Question 11.
\n\\(\\frac{7}{10}\\) \u00d7 \\(\\frac{3}{4}\\)
\nAnswer:
\nThe given multiplication equation is:
\n\\(\\frac{7}{10}\\) \u00d7 \\(\\frac{3}{4}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{7}{10}\\) \u00d7 \\(\\frac{3}{4}\\) = \\(\\frac{21}{40}\\)<\/p>\n

Question 12.
\n\\(\\frac{1}{3}\\) \u00d7 \\(\\frac{1}{4}\\)
\nAnswer:
\nThe given multiplication equation is:
\n\\(\\frac{1}{3}\\) \u00d7 \\(\\frac{1}{4}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{1}{3}\\) \u00d7 \\(\\frac{1}{4}\\) = \\(\\frac{1}{12}\\)<\/p>\n

Question 13.
\n\\(\\frac{5}{6}\\) \u00d7 \\(\\frac{3}{7}\\)
\nAnswer:
\nThe given multiplication equation is:
\n\\(\\frac{5}{6}\\) \u00d7 \\(\\frac{3}{7}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{5}{6}\\) \u00d7 \\(\\frac{3}{7}\\) = \\(\\frac{5}{14}\\)<\/p>\n

Question 14.
\n\\(\\frac{3}{5}\\) \u00d7 \\(\\frac{11}{12}\\)
\nAnswer:
\nThe give multiplication equation is:
\n\\(\\frac{3}{5}\\) \u00d7 \\(\\frac{11}{12}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, fro the above,
\nWe can conclude that
\n\\(\\frac{3}{5}\\) \u00d7 \\(\\frac{11}{12}\\) = \\(\\frac{11}{20}\\)<\/p>\n

Question 15.
\n\\(\\frac{4}{10}\\) \u00d7 \\(\\frac{2}{5}\\)
\nAnswer:
\nThe given multiplication equation is:
\n\\(\\frac{4}{10}\\) \u00d7 \\(\\frac{2}{5}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{4}{10}\\) \u00d7 \\(\\frac{2}{5}\\) = \\(\\frac{4}{25}\\)<\/p>\n

Question 16.
\n\\(\\frac{3}{4}\\) \u00d7 \\(\\frac{2}{9}\\)
\nAnswer:
\nThe given multiplication equation is:
\n\\(\\frac{3}{4}\\) \u00d7 \\(\\frac{2}{9}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{3}{4}\\) \u00d7 \\(\\frac{2}{9}\\) = \\(\\frac{1}{6}\\)<\/p>\n

In 17 and 18, estimate the product. Then complete the multiplication.<\/strong><\/p>\n

Question 17.
\n\"Envision
\nAnswer:
\nThe given multiplication equation is:
\n2\\(\\frac{3}{4}\\) \u00d7 8
\nNow,
\nThe representation of 2\\(\\frac{3}{4}\\) into a fraction is:
\n\"\"
\nSo,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\"\"<\/p>\n

Question 18.
\n\"Envision
\nAnswer:
\nThe given multiplication equation is:
\n4\\(\\frac{1}{2}\\) \u00d7 1\\(\\frac{1}{4}\\)
\nNow,
\nThe representation of 4\\(\\frac{1}{2}\\) into a fraction is:
\n\"\"
\nThe representation of 1\\(\\frac{1}{4}\\) into a fraction is:
\n\"\"
\nSo,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\"\"<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Scan for Multimedia<\/strong><\/p>\n

In 19 and 20, find each product. Shade the model to help solve.<\/strong><\/p>\n

Question 19.
\n\\(\\frac{1}{3}\\) \u00d7 \\(\\frac{5}{6}\\)
\n\"Envision
\nAnswer:
\nThe given multiplication equation is:
\n\\(\\frac{1}{3}\\) \u00d7 \\(\\frac{5}{6}\\)
\nNow,
\nThe representation of the given multiplication equation in the form of a model is:
\n\"\"
\nSo,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{1}{3}\\) \u00d7 \\(\\frac{5}{6}\\) = \\(\\frac{5}{18}\\)<\/p>\n

Question 20.
\n\\(\\frac{2}{3}\\) \u00d7 \\(\\frac{1}{12}\\)
\n\"Envision
\nAnswer:
\nThe given multiplication equation is:
\n\\(\\frac{2}{3}\\) \u00d7 \\(\\frac{1}{12}\\)
\nSo,
\nThe representation of the given multiplication equation in the form of a model is:
\n\"\"
\nSo,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{2}{3}\\) \u00d7 \\(\\frac{1}{12}\\) = \\(\\frac{1}{18}\\)<\/p>\n

In 21-28, find each product.<\/strong><\/p>\n

Question 21.
\n\\(\\frac{7}{8}\\) \u00d7 \\(\\frac{1}{2}\\)
\nAnswer:
\nThe given multiplication equation is:
\n\\(\\frac{7}{8}\\) \u00d7 \\(\\frac{1}{2}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{7}{8}\\) \u00d7 \\(\\frac{1}{2}\\) = \\(\\frac{7}{16}\\)<\/p>\n

Question 22.
\n\\(\\frac{2}{5}\\) \u00d7 \\(\\frac{1}{12}\\)
\nAnswer:
\nThe given multiplication equation is:
\n\\(\\frac{2}{5}\\) \u00d7 \\(\\frac{1}{12}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{2}{5}\\) \u00d7 \\(\\frac{1}{12}\\) = \\(\\frac{1}{30}\\)<\/p>\n

Question 23.
\n\\(\\frac{5}{7}\\) \u00d7 \\(\\frac{7}{9}\\)
\nAnswer:
\nThe given multiplication equation is:
\n\\(\\frac{5}{7}\\) \u00d7 \\(\\frac{7}{9}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{5}{7}\\) \u00d7 \\(\\frac{7}{9}\\) = \\(\\frac{5}{9}\\)<\/p>\n

Question 24.
\n\\(\\frac{1}{2}\\) \u00d7 \\(\\frac{3}{4}\\)
\nAnswer:
\nThe given multiplication equation is:
\n\\(\\frac{1}{2}\\) \u00d7 \\(\\frac{3}{4}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{1}{2}\\) \u00d7 \\(\\frac{3}{4}\\) = \\(\\frac{3}{8}\\)<\/p>\n

Question 25.
\n\\(\\frac{1}{4}\\) \u00d7 \\(\\frac{7}{8}\\)
\nAnswer:
\nThe given multiplication equation is:
\n\\(\\frac{1}{4}\\) \u00d7 \\(\\frac{7}{8}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{1}{4}\\) \u00d7 \\(\\frac{7}{8}\\) = \\(\\frac{7}{32}\\)<\/p>\n

Question 26.
\n\\(\\frac{5}{6}\\) \u00d7 \\(\\frac{9}{10}\\)
\nAnswer:
\nThe given multiplication equation is:
\n\\(\\frac{5}{6}\\) \u00d7 \\(\\frac{9}{10}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{5}{6}\\) \u00d7 \\(\\frac{9}{10}\\) = \\(\\frac{3}{4}\\)<\/p>\n

Question 27.
\n\\(\\frac{1}{4}\\) \u00d7 \\(\\frac{1}{8}\\)
\nAnswer:
\nThe given multiplication equation is:
\n\\(\\frac{1}{4}\\) \u00d7 \\(\\frac{1}{8}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{1}{4}\\) \u00d7 \\(\\frac{1}{8}\\) = \\(\\frac{1}{32}\\)<\/p>\n

Question 28.
\n\\(\\frac{1}{3}\\) \u00d7 \\(\\frac{3}{7}\\)
\nAnswer:
\nThe given multiplication equation is:
\n\\(\\frac{1}{3}\\) \u00d7 \\(\\frac{3}{7}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{1}{3}\\) \u00d7 \\(\\frac{3}{7}\\) = \\(\\frac{1}{7}\\)<\/p>\n

In 29-36, estimate the product. Then find each product.<\/strong><\/p>\n

Question 29.
\n2\\(\\frac{1}{6}\\) \u00d7 4\\(\\frac{2}{9}\\)
\nAnswer:
\nThe given multiplication equation is:
\n2\\(\\frac{1}{6}\\) \u00d7 4\\(\\frac{2}{9}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n2\\(\\frac{1}{6}\\) \u00d7 4\\(\\frac{2}{9}\\) = \\(\\frac{247}{27}\\)<\/p>\n

Question 30.
\n\\(\\frac{3}{4}\\) \u00d7 8\\(\\frac{1}{2}\\)
\nAnswer:
\nThe given multiplication equation is:
\n\\(\\frac{3}{4}\\) \u00d7 8\\(\\frac{1}{2}\\)
\nNow,
\nThe representation of 8\\(\\frac{1}{2}\\) into a fraction is:
\n\"\"
\nSo,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{3}{4}\\) \u00d7 8\\(\\frac{1}{2}\\) = \\(\\frac{51}{8}\\)<\/p>\n

Question 31.
\n1\\(\\frac{1}{8}\\) \u00d7 3\\(\\frac{1}{3}\\)
\nAnswer:
\nThe given multiplication equation is:
\n1\\(\\frac{1}{8}\\) \u00d7 3\\(\\frac{1}{3}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n1\\(\\frac{1}{8}\\) \u00d7 3\\(\\frac{1}{3}\\) = \\(\\frac{15}{4}\\)<\/p>\n

Question 32.
\n3\\(\\frac{1}{2}\\) \u00d7 \\(\\frac{2}{3}\\)
\nAnswer:
\nThe given multiplication equation is:
\n3\\(\\frac{1}{2}\\) \u00d7 \\(\\frac{2}{3}\\)
\nNow,
\nThe representation of 3\\(\\frac{1}{2}\\) into a fraction is:
\n\"\"
\nSo,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n3\\(\\frac{1}{2}\\) \u00d7 \\(\\frac{2}{3}\\) = \\(\\frac{7}{3}\\)<\/p>\n

Question 33.
\n3\\(\\frac{1}{4}\\) \u00d7 6
\nAnswer:
\nThe given multiplication equation is:
\n3\\(\\frac{1}{4}\\) \u00d7 6
\nNow,
\nThe representation of 3\\(\\frac{1}{4}\\) into a fraction is:
\n\"\"
\nSo,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n3\\(\\frac{1}{4}\\) \u00d7 6 = \\(\\frac{39}{2}\\)<\/p>\n

Question 34.
\n5\\(\\frac{1}{3}\\) \u00d7 3
\nAnswer:
\nThe given multiplication equation is:
\n5\\(\\frac{1}{3}\\) \u00d7 3
\nNow,
\nThe representation of 5\\(\\frac{1}{3}\\) into a fraction is:
\n\"\"
\nSo,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n5\\(\\frac{1}{3}\\) \u00d7 3 = 16<\/p>\n

Question 35.
\n2\\(\\frac{3}{8}\\) \u00d7 4
\nAnswer:
\nThe given multiplication equation is:
\n2\\(\\frac{3}{8}\\) \u00d7 4
\nNow,
\nThe representation of 2\\(\\frac{3}{8}\\) into a fraction is:
\n\"\"
\nSo,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n2\\(\\frac{3}{8}\\) \u00d7 4 = \\(\\frac{19}{2}\\)<\/p>\n

Question 36.
\n4\\(\\frac{1}{8}\\) \u00d7 5\\(\\frac{1}{2}\\)
\nAnswer:
\nThe given multiplication equation is:
\n4\\(\\frac{1}{8}\\) \u00d7 5\\(\\frac{1}{2}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n4\\(\\frac{1}{8}\\) \u00d7 5\\(\\frac{1}{2}\\) = \\(\\frac{363}{16}\\)<\/p>\n

In 37 and 38, use the diagram at the right.<\/strong><\/p>\n

Question 37.
\nLinda walked \\(\\frac{3}{4}\\) of the length of the Tremont Trail before stopping for a rest. How far had Linda walked on the trail?
\n\"Envision
\nAnswer:
\nIt is given that
\nLinda walked \\(\\frac{3}{4}\\) of the length of the Tremont Trail before stopping for a rest
\nNow,
\nThe given information is:
\n\"Envision
\nSo,
\nThe distance covered by Linda on the trail = (The distance Linda walked of the Tremont trail) \u00d7 (The length of the Tremont trail)
\n= \\(\\frac{3}{4}\\) \u00d7 3\\(\\frac{1}{2}\\)
\nNow,
\nThe representation of 3\\(\\frac{1}{2}\\) into a fraction is:
\n\"\"
\nSo,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe distance covered by Linda on the trail is: \\(\\frac{25}{8}\\) miles<\/p>\n

Question 38.
\nThe city plans to extend the Wildflower Trail to make it 2\\(\\frac{1}{2}\\) times its current length in the next 5 years. How long will the Wildflower Trail be at the end of 5 years?
\nAnswer:
\nIt is given that
\nThe city plans to extend the Wildflower Trail to make it 2\\(\\frac{1}{2}\\) times its current length in the next 5 years
\nNow,
\nThe given information is:
\n\"Envision
\nSo,
\nThe length of the Wildflower trail at the end of 5 years = (The extended length of the Wildflower trail at the end of 5 years) \u00d7 (The current length of the Wildflower trail)
\n= 2\\(\\frac{1}{2}\\) \u00d7 2\\(\\frac{3}{8}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe length of the Wildflower trail at the end of 5 years will be: \\(\\frac{95}{16}\\) miles<\/p>\n

Question 39.
\nThe world’s smallest gecko is \\(\\frac{3}{4}\\) inch long. An adult male Western Banded Gecko is 7\\(\\frac{1}{3}\\) times as long. How long is an adult male Western Banded Gecko?
\n\"Envision
\nAnswer:
\nIt is given that
\nThe world’s smallest gecko is \\(\\frac{3}{4}\\) inch long. An adult male Western Banded Gecko is 7\\(\\frac{1}{3}\\) times as long
\nSo,
\nThe length of an adult male Western Banded Gecko = (The length of the world’s small gecko) \u00d7 (The number of times an adult male western banded gecko is as larger than the world’s smallest gecko)
\n= \\(\\frac{3}{4}\\) \u00d7 7\\(\\frac{1}{3}\\)
\nNow,
\nThe representation of 7\\(\\frac{1}{3}\\) into a fraction is:
\n\"\"
\nSo,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe length of an adult male Western Banded Gecko is: \\(\\frac{11}{2}\\) inches<\/p>\n

Question 40.
\nHigher-Order Thinking<\/strong> In Ms. Barclay’s classroom, \\(\\frac{2}{5}\\) of the students play chess. Of the students who play chess, \\(\\frac{5}{6}\\) also play sudoku. If there are 30 students in Ms. Barclay’s class, how many play chess and sudoku?
\nAnswer:
\nIt is given that
\nIn Ms. Barclay’s classroom, \\(\\frac{2}{5}\\) of the students play chess. Of the students who play chess, \\(\\frac{5}{6}\\) also play sudoku. If there are 30 students in Ms. Barclay’s class
\nSo,
\nThe number of students who play chess = (The total number of students) \u00d7 (The fracti of the students that play chess)
\n= 30 \u00d7 \\(\\frac{2}{5}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nSo,
\nThe number of students who play chess are: 12 students
\nNow,
\nThe number of students who play Sudoku = (The number of students who play chess) \u00d7 (The fraction of the students who play Sudoku)
\n= 12 \u00d7 \\(\\frac{5}{6}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nSo,
\nThe number of students who play sudoku are: 10 students
\nNow,
\nThe number of students who play chess and sudoku = (The number of students who play chess) – (The number of students who play sudoku)
\n= 12 – 10
\n= 2 students
\nHence, from the above,
\nWe can conclude that
\nThe number of students who play chess are: 12 students
\nThe number of students who play sudoku are: 10 students
\nThe number of students who play chess and sudoku are: 2 students<\/p>\n

Question 41.
\nThe Boca Grande Causeway in Florida is about 1\\(\\frac{4}{9}\\) times as long as the Golden Gate Bridge in San Francisco. The Golden Gate Bridge is about 9,000 feet long. About how long is the Boca Grande Causeway?
\nAnswer:
\nIt is given that
\nThe Boca Grande Causeway in Florida is about 1\\(\\frac{4}{9}\\) times as long as the Golden Gate Bridge in San Francisco. The Golden Gate Bridge is about 9,000 feet long
\nSo,
\nThe length of the Boca Grande causeway = (The number of times the Boca Grande causeway as long as the Golden Gate Bridge) \u00d7 (The length of the Golden Gate Bridge)
\n= 1\\(\\frac{4}{9}\\) \u00d7 9,000
\nNow,
\nThe representation of 1\\(\\frac{4}{9}\\) into a fraction is:
\n\"\"
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe acn conclude that
\nThe length of the Boca Grande causeway is: 13,000 feet<\/p>\n

Question 42.
\nIf \\(\\frac{7}{8}\\) is multiplied by \\(\\frac{4}{5}\\), will the product be greater than either of the two factors? Explain.
\nAnswer:
\nThe given multiplication equation is:
\n\\(\\frac{7}{8}\\) \u00d7 \\(\\frac{4}{5}\\)
\nNow,
\nThe representation of the given multiplication equation in the form of a decimal numbers is:
\n0.875 \u00d7 0.8
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nNow,
\nThe representation of the value of the product in the form of a decimal number is: 0.7
\nHence, from the above,
\nWe can conclude that
\nThe product would be smaller than either of the two factors<\/p>\n

Question 43.
\nBe Precise<\/strong> To amend the U.S. Constitution, \\(\\frac{3}{4}\\) of the 50 states must approve the amendment. If 35 states approve an amendment, will the Constitution be amended?
\nAnswer:
\nIt is given that
\nTo amend the U.S. Constitution, \\(\\frac{3}{4}\\) of the 50 states must approve the amendment
\nSo,
\nThe number of states that approved the amendment = (The total number of states) \u00d7 (The number of states who approved the amendment)
\n= 50 \u00d7 \\(\\frac{3}{4}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nNow,
\nWe can observe that about 37 states approved the amendment from the above Long Multiplication
\nHence, from the above,
\nWe can conclude that
\nIf 35 states approve an amendment, the Constitution will be amended<\/p>\n

Question 44.
\nA scientist had \\(\\frac{3}{4}\\) of a bottle of a solution. She used \\(\\frac{1}{6}\\) of the solution in an experiment. How much of the bottle did she use?
\n\"Envision
\nAnswer:
\nIt is given that
\nA scientist had \\(\\frac{3}{4}\\) of a bottle of a solution. She used \\(\\frac{1}{6}\\) of the solution in an experiment
\nSo,
\nThe amount of the solution in the bottle did she used = (The total amount of solution in the bottle) \u00d7 (the amount of solution did she used in an experiment)
\n= \\(\\frac{3}{4}\\) \u00d7 \\(\\frac{1}{6}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe amount of the solution in the bottle did she used is: \\(\\frac{1}{8}\\) of the total amount of solution<\/p>\n

Question 45.
\nIn the voting for City Council Precinct 5, only \\(\\frac{1}{2}\\) of all eligible voters cast votes. What fraction of all eligible voters voted for Shelley? Morgan? Who received the most votes?
\n\"Envision
\nAnswer:
\nIt is given that
\nIn the voting for City Council Precinct 5, only \\(\\frac{1}{2}\\) of all eligible voters cast votes
\nNow,
\nThe given table is:
\n\"Envision
\nSo,
\nThe fraction of the eligible voters voted for Shelly = (The fraction of the eligible voters voted) \u00d7 (The fraction of votes received by Shelly)
\n= \\(\\frac{1}{2}\\) \u00d7 \\(\\frac{3}{10}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nSo,
\nThe fraction of the eligible voters voted for Shelly is: \\(\\frac{3}{20}\\) votes
\nSo,
\nThe fraction of the eligible voters voted for Morgan = (The fraction of the eligible voters voted) \u00d7 (The fraction of votes received by Morgan)
\n= \\(\\frac{1}{2}\\) \u00d7 \\(\\frac{5}{8}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nSo,
\nThe fraction of the eligible voters voted for Morgan is: \\(\\frac{5}{16}\\) votes
\nNow,
\nThe representation of \\(\\frac{3}{20}\\) into a decimal number is: 0.15
\nThe representation of \\(\\frac{5}{16}\\) into a decimal number is: 0.3125
\nHence, from the above,
\nWe can conclude that
\nMorgan received the most votes<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 46.
\nWhich of these equations is equivalent to 1\\(\\frac{1}{2}\\) \u00d7 3\\(\\frac{1}{5}\\) = 4\\(\\frac{1}{2}\\)?
\nA. 4\\(\\frac{1}{2}\\) \u00f7 3\\(\\frac{1}{5}\\) = 1\\(\\frac{1}{2}\\)
\nB. 1\\(\\frac{1}{2}\\) \u00f7 4\\(\\frac{1}{2}\\) = 3\\(\\frac{1}{5}\\)
\nC. 1\\(\\frac{1}{2}\\) \u00f7 3\\(\\frac{1}{5}\\) = 4\\(\\frac{1}{2}\\)
\nD. 3\\(\\frac{1}{5}\\) \u00f7 4\\(\\frac{1}{2}\\) = 1\\(\\frac{1}{2}\\)
\nAnswer:
\nThe given multiplication equation is:
\n1\\(\\frac{1}{2}\\) \u00d7 3\\(\\frac{1}{5}\\) = 4\\(\\frac{1}{2}\\)
\nNow,
\nDivide the above equation with 3\\(\\frac{1}{5}\\)
\nSo,
\nWe will get
\n4\\(\\frac{1}{2}\\) \u00f7 3\\(\\frac{1}{5}\\) = 1\\(\\frac{1}{2}\\)
\nHence, from the above,
\nWe can conclude that
\nThe equations that are equivalent to 1\\(\\frac{1}{2}\\) \u00d7 3\\(\\frac{1}{5}\\) = 4\\(\\frac{1}{2}\\) are:
\n\"\"<\/p>\n

Question 47.
\nWhich of these equations is equivalent to \\(\\frac{3}{4}\\) \u00d7 8\\(\\frac{1}{5}\\) = 6\\(\\frac{3}{20}\\)? Select all that apply.
\n\"Envision \\(\\frac{3}{4}\\) \u00f7 8\\(\\frac{1}{5}\\) = 6\\(\\frac{3}{20}\\)
\n\"Envision 6\\(\\frac{3}{20}\\) \u00f7 \\(\\frac{3}{4}\\) = 8\\(\\frac{1}{5}\\)
\n\"Envision 6\\(\\frac{3}{20}\\) \u00f7 8\\(\\frac{1}{5}\\) = \\(\\frac{3}{4}\\)
\n\"Envision \\(\\frac{3}{4}\\) \u00f7 6\\(\\frac{3}{20}\\) = 8\\(\\frac{1}{5}\\)
\n\"Envision 8\\(\\frac{1}{5}\\) \u00f7 6\\(\\frac{3}{20}\\) = \\(\\frac{3}{4}\\)
\nAnswer:
\nThe given multiplication equation is:
\n\\(\\frac{3}{4}\\) \u00d7 8\\(\\frac{1}{5}\\) = 6\\(\\frac{3}{20}\\)
\nNow,
\nDivide the above equation with 8\\(\\frac{1}{5}\\)
\nSo,
\nWe will get
\n6\\(\\frac{3}{20}\\) \u00f7 8\\(\\frac{1}{5}\\) = \\(\\frac{3}{4}\\)
\nHence, from the above,
\nWe can conclude that
\nThe equations that are equivalent to \\(\\frac{3}{4}\\) \u00d7 8\\(\\frac{1}{5}\\) = 6\\(\\frac{3}{20}\\) are:
\n\"\"<\/p>\n

Topic 1 MID-TOPIC CHECKPOINT<\/h3>\n

Question 1.
\nVocabulary<\/strong> How can you use a compatible number to estimate a quotient when dividing a decimal by a whole number? Lesson 1.2
\nAnswer:
\nWhen dividing a decimal by a whole number, we use the following procedure:
\na. Estimate the quotient.
\nb. Perform the division. Remember to place a zero in the quotient when the divisor is larger than the dividend.
\nc. Compare your estimate with your quotient to verify that the answer makes sense.<\/p>\n

Question 2.
\nKeaton is building a rectangular tabletop and wants to put a metal border around the edge. The length of the tabletop is 1.83 meters and the width is 0.74 meters. Use the formula P = 2l + 2w to find the perimeter of the tabletop. Lesson 1.1
\nAnswer:
\nIt is given that
\nKeaton is building a rectangular tabletop and wants to put a metal border around the edge. The length of the tabletop is 1.83 meters and the width is 0.74 meters
\nSo,
\nTo find the length of the metal border around the edge,
\nWe have to find the perimeter of the tabletop
\nNow,
\nFrom the give information,
\nWe can observe that
\nThe perimeter of the tabletop (P) = 2l + 2w
\nWhere,
\n‘l’ is the length of the tabletop
\n‘w’ is the width of the tabletop
\nSo,
\nThe perimeter of the tabletop (P) = (2 \u00d7 1.83) + (2 \u00d7 0.74)
\n= 3.66 + 1.48
\n= 5.14 meters
\nHence, from the above,
\nWe can conclude that
\nThe perimeter of thetabletop is: 5.14 meters<\/p>\n

Question 3.
\nNorbert’s Nursery is having a sale. Flats of flowers are priced as marked, including tax. Jake buys 2 flats of petunias, 3 flats of daisies, and 1 flat of begonias. If he pays with a $50 bill, how much change should Jake receive? Lesson 1.1
\n\"Envision
\nAnswer:
\nIt is given that
\nNorbert’s Nursery is having a sale. Flats of flowers are priced as marked, including tax. Jake buys 2 flats of petunias, 3 flats of daisies, and 1 flat of begonias. If he pays with a $50 bill
\nNow,
\nThe given table is:
\n\"Envision
\nNow,
\nThe total price of flats of flowers Jack bought = (2 \u00d7 The price of each flat of Petunia) + (3 \u00d7 The price of each flat of daisies) + (1 \u00d7 The price of each flat of begonias)
\n= (2 \u00d7 $5.25) + (3 \u00d7 $7.65) + (1 \u00d7 $8.40)
\n= $10.5 + $22.95 + $8.40
\n= $41.85
\nNow,
\nThe amount of change should Jack receive = (The total amount of money Jack gave) – (The total price of the flowers Jack bought)
\n= $50 – $41.85
\n= $8.15
\nHence, from the above,
\nWe can conclude that
\nThe amount of change should Jack receive is: $8.15<\/p>\n

Question 4.
\nMarguerite is selling space in an advertisement book for a community fund-raising event. Each \\(\\frac{1}{4}\\) page in the book costs $15.50. What is the cost for \\(\\frac{3}{4}\\) page? Lesson 1.1
\nA. $62.00
\nB. $46.50
\nC. $20.67
\nD. $11.63
\nAnswer:
\nIt is given that
\nMarguerite is selling space in an advertisement book for a community fund-raising event. Each \\(\\frac{1}{4}\\) page in the book costs $15.50
\nSo,
\nThe cost for \\(\\frac{3}{4}\\) page = 3 \u00d7 (The cost of \\(\\frac{1}{4}\\) page)
\n= 3 \u00d7 $15.50
\n= $46.50
\nHence, from the above,
\nWe can conclude that
\nThe cost for \\(\\frac{3}{4}\\) page is:
\n\"\"<\/p>\n

Question 5.
\nWhat is the value of 170 \u00f7 (4 \u00d7 5)? Lesson 1.2
\nAnswer:
\nThe given division expression is: 170 \u00f7 (4 \u00d7 5)
\nSo,
\n170 \u00f7 (4 \u00d7 5)
\n= 170 \u00f7 20
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n170 \u00f7 20 = 8.5<\/p>\n

Question 6.
\nLucia walks 2\\(\\frac{3}{4}\\) miles on Monday. On Monday, she walks 1\\(\\frac{1}{2}\\) times farther than on Tuesday. Which equation can be used to find how far Lucia walks on Tuesday? Lesson 1-3
\nA. 2\\(\\frac{3}{4}\\) \u00d7 1\\(\\frac{1}{2}\\) = 4\\(\\frac{1}{8}\\)
\nB. 2\\(\\frac{3}{4}\\) \u00d7 1\\(\\frac{1}{2}\\) = 4\\(\\frac{1}{4}\\)
\nC. 2\\(\\frac{3}{4}\\) \u00f7 1\\(\\frac{1}{2}\\) = 1\\(\\frac{5}{6}\\)
\nD. 1\\(\\frac{1}{2}\\) \u00f7 2\\(\\frac{3}{4}\\) = \\(\\frac{6}{11}\\)
\nAnswer:
\nIt is given that
\nLucia walks 2\\(\\frac{3}{4}\\) miles on Monday. On Monday, she walks 1\\(\\frac{1}{2}\\) times farther than on Tuesday
\nSo,
\nThe distance Lucia walked on Tuesday = (The distance Lucia walked on Monday) + 1\\(\\frac{1}{2}\\)
\n= 2\\(\\frac{3}{4}\\) + 1\\(\\frac{1}{2}\\)
\nNow,
\nBy using the Long Addition,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe equation that can be used to find how far Lucia walks on Tuesday is:
\n\"\"
\nHow well did you do on the mid-topic checkpoint? Fill in the stars. \"Envision<\/p>\n

Topic 1 MID-TOPIC PERFORMANCE TASK<\/strong><\/p>\n

Nyan Robotics Team received their challenge for the year and has to buy parts to build their robot for competitions.<\/p>\n

\"Envision<\/p>\n

PART A<\/strong>
\nTeam members Eric and Natalia secure a grant for $75.00 to buy beams and channels. If the team needs 3 beams and 6 channels, will the grant cover the cost? If so, how much of the grant will remain?
\nAnswer:
\nIt is given that
\nTeam members Eric and Natalia secure a grant for $75.00 to buy beams and channels and the team needs 3 beams and 6 channels
\nNow,
\nThe given table is;
\n\"Envision
\nSo,
\nThe cost of 3 beams and 6 channels = (3 \u00d7 The cost of each Beam) + (6 \u00d7 The cost of each Channel)
\n= (3 \u00d7 $5.95) + (6 \u00d7 $8.50)
\n= $17.85 + $51
\n= $68.85
\nSo,
\nThe grant received > The cost of 3 beams and 6 channels
\nNow,
\nThe amount of the grant that will remain = (The total grant granted) – (The cost of 3 beams and 6 channels)
\n= $75 – $68.85
\n= $6.15
\nHence, from the above,
\nWe can conclude that
\nThe amount of grant that will remain is: $6.15<\/p>\n

PART B<\/strong>
\nTeam members Corinne, Kevin, and Tomas decide to share the cost of 2 motor controllers and 4 wheels equally. How much does each member need to contribute?
\nAnswer:
\nIt is given that
\nTeam members Corinne, Kevin, and Tomas decide to share the cost of 2 motor controllers and 4 wheels equally
\nNow,
\nThe given table is:
\n\"Envision
\nSo,
\nThe total cost of 2 motor controllers and 4 wheels = (2 \u00d7 The cost of each motor controller) + (4 \u00d7 The cost of each wheel)
\n= (2 \u00d7 $99.75) + (4 \u00d7 $18.90)
\n= $199.5 + $75.6
\n= $275.1
\nNow,
\nThe amount of money each member needs to contribute to buy 2 motor controllers and 4 wheels = $275.1 \u00f7 3
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe amount of money each member needs to contribute to buy 2 motor controllers and 4 wheels is: $91.70<\/p>\n

PART C<\/strong>
\nNyan Robotics has a budget of $99 to buy sprockets, axles, and gears. If they spend \\(\\frac{2}{3}\\) of the budget on sprockets, how much money from the budget remains to buy axles and gears?
\nAnswer:
\nIt is given that
\nNyan Robotics has a budget of $99 to buy sprockets, axles, and gears and they spend \\(\\frac{2}{3}\\) of the budget on sprockets
\nNow,
\nThe given table is:
\n\"Envision
\nNow,
\nThe amount of budget remaining after they spent on sprockets to buy axles and gears = \\(\\frac{2}{3}\\) \u00d7 $99
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe amount of budget remaining after they spent on sprockets to buy axles and gears is: $66<\/p>\n

3-Act Mathematical Modeling: Stocking Up<\/strong><\/p>\n

3-ACT MATH<\/strong><\/p>\n

Stocking UP<\/strong><\/p>\n

ACT 1<\/strong><\/p>\n

Question 1.
\nAfter watching the video, what is the first question that comes to mind?
\nAnswer:<\/p>\n

Question 2.
\nWrite the Main Question you will answer.
\nAnswer:<\/p>\n

Question 3.
\nConstruct Arguments<\/strong> Predict an answer to this Main Question. Explain your prediction.
\n\"Envision
\nAnswer:<\/p>\n

Question 4.
\nOn the number line below, write a number that is too small to be the answer. Write a number that is too large.
\n\"Envision
\nAnswer:<\/p>\n

Question 5.
\nPlot your prediction on the same number line.
\nAnswer:<\/p>\n

ACT 2<\/strong><\/p>\n

Question 6.
\nWhat information in this situation would be helpful to know? How would you use that information?
\n\"Envision
\nAnswer:<\/p>\n

Question 7.
\nUse Appropriate Tools<\/strong> What tools can you use to solve the problem? Explain how you would use them strategically.
\nAnswer:<\/p>\n

Question 8.
\nModel with Math<\/strong> Represent the situation using mathematics. Use your representation to answer the Main Question.
\nAnswer:<\/p>\n

Question 9.
\nWhat is your answer to the Main Question? Is it higher or lower than your prediction? Explain why.
\n\"Envision
\nAnswer:<\/p>\n

ACT 3<\/strong><\/p>\n

Question 10.
\nWrite the answer you saw in the video.
\nAnswer:<\/p>\n

Question 11.
\nReasoning<\/strong> Does your answer match the answer in the video? If not, what are some reasons that would explain the difference?
\n\"Envision
\nAnswer:<\/p>\n

Question 12.
\nMake Sense and Persevere<\/strong> Would you change your model now that you know the answer? Explain.
\n\"Envision
\nAnswer:<\/p>\n

Reflect<\/strong><\/p>\n

Question 13.
\nModel with Math<\/strong> Explain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question?
\n\"Envision
\nAnswer:<\/p>\n

Question 14.
\nReasoning<\/strong> How did you represent the situation using symbols? How did you use those symbols to solve the problem?
\nAnswer:<\/p>\n

SEQUEL<\/strong><\/p>\n

Question 15.
\nModel with Math<\/strong>
\nThe store purchases boxes of pasta for $0.82 and cans of sauce for $1.62. How much profit does the store make from this purchase?
\nAnswer:<\/p>\n

Lesson 1.4 Understand Division with Fractions<\/h3>\n

ACTIVITY<\/strong><\/p>\n

Explore It!<\/strong>
\nStudents are competing in a 4-kilometer relay race. There are 10 runners.
\n\"Envision<\/p>\n

A. Use the number line to represent the data for the race.
\n\"Envision
\nAnswer:
\nIt is given that
\nStudents are competing in a 4-kilometer relay race. There are 10 runners.
\nNow,
\nThe given data is:
\n\"Envision
\nSo,
\nFrom the given number line,
\nWe can observe that
\nThere are 10 runners and each runner runs \\(\\frac{2}{5}\\) km and the finish line is at 4 km
\nHence,
\nThe representation of the given data in a number line is:
\n\"\"
\nB. Use multiplication or division to describe your work on the number line.
\nAnswer:
\nIt is given that
\nStudents are competing in a 4-kilometer relay race. There are 10 runners.
\nNow,
\nThe given data is:
\n\"Envision
\nNow,
\nUsing Multiplication,
\nThe total distance covered by 10 runners = 10 \u00d7 (The distance each runner runs)
\n= 10 \u00d7 \\(\\frac{2}{5}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nSo,
\nThe total distance covered by 10 runners is: 4 km<\/p>\n

Focus on math practices<\/strong>
\nModel with Math<\/strong>
\nDescribe what a number line would look like if there were 10 runners each running; kilometer in a 5-kilometer
\nrace.
\nAnswer:
\nIt is given that
\nThere were 10 runners each running; kilometer in a 5-kilometer race
\nSo,
\nThe total distance covered by 10 runners = 10 \u00d7 (The distance each runner runs)
\n= 10 \u00d7 \\(\\frac{1}{2}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nSo,
\nThe total distance covered by 10 runners is: 5 km
\nHence,
\nThe representation of the given information in a number line is:
\n\"\"<\/p>\n

? Essential Question<\/strong> How can you represent the division of fractions?
\nAnswer:
\nInterpret a fraction as a division of the numerator by the denominator (\\(\\frac{a}{b}\\) = a \u00f7 b). Solve word problems involving the division of whole numbers leading to answers in the form of fractions or mixed numbers,
\nEx: By using visual fraction models or equations to represent the problem.<\/p>\n

Try It!<\/strong>
\nA board is 6 feet long. How many \\(\\frac{2}{3}\\)-foot-long pieces can be cut from the board? Use the number line to show your work.
\n\"Envision
\nAnswer:
\nIt is given that
\nA board is 6 feet long
\nSo,
\nThe number of \\(\\frac{2}{3}\\)-foot-long pieces can be cut from the board = (The length of the board) \u00f7 \\(\\frac{2}{3}\\)
\n= 6 \u00f7 \\(\\frac{2}{3}\\)
\n= 6 \u00d7 \\(\\frac{3}{2}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe number of \\(\\frac{2}{3}\\)-foot-long pieces that can be cut from the board are: 9 pieces<\/p>\n

Convince Me!<\/strong> Why is the number of pieces that can be cut from the board greater than the number of feet in the length of the board?
\nAnswer:<\/p>\n

Try It!<\/strong>
\nMake a diagram to find \\(\\frac{2}{3}\\) \u00f7 4.
\nAnswer:
\nThe given division expression is: \\(\\frac{2}{3}\\) \u00f7 4
\nHence,
\nThe representation of the given division expression in the form of an area model is:
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{2}{3}\\) \u00f7 4 = \\(\\frac{1}{6}\\)<\/p>\n

Try It!<\/strong>
\nUse the pattern in the table above to find 8 \u00f7 \\(\\frac{3}{4}\\).
\n\"Envision
\nAnswer:
\nThe given division expression is: 8 \u00f7 \\(\\frac{3}{4}\\)
\nSo,
\n8 \u00f7 \\(\\frac{3}{4}\\)
\n= 8 \u00d7 \\(\\frac{4}{3}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\"\"<\/p>\n

KEY CONCEPT<\/strong><\/p>\n

To divide a whole number by a fraction:
\n\"Envision
\nTo divide a fraction by a whole number:
\n\"Envision<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n? Essential Question<\/strong> How can you represent the division of fractions?
\nAnswer:
\nInterpret a fraction as a division of the numerator by the denominator (\\(\\frac{a}{b}\\) = a \u00f7 b). Solve word problems involving the division of whole numbers leading to answers in the form of fractions or mixed numbers,
\nEx: By using visual fraction models or equations to represent the problem.<\/p>\n

Question 2.
\nReasoning<\/strong> Draw a diagram to represent 8 \u00f7 \\(\\frac{2}{3}\\). Then write an equation to show the solution.
\nAnswer:
\n\"\"<\/p>\n

Question 3.
\nReasoning<\/strong> is 4 \u00f7 \\(\\frac{3}{2}\\) the same as 4 \u00f7 \\(\\frac{2}{3}\\)? Explain.
\nAnswer:
\nThe given division expressions are:
\n4 \u00f7 \\(\\frac{3}{2}\\) and 4 \u00f7 \\(\\frac{2}{3}\\)
\nSo,
\n4 \u00f7 \\(\\frac{3}{2}\\)
\n= 4 \u00d7 \\(\\frac{2}{3}\\)
\nSo,
\n4 \u00f7 \\(\\frac{2}{3}\\)
\n= 4 \u00d7 \\(\\frac{3}{2}\\)
\nHence, from the above,
\nWe can conclude that
\n4 \u00f7 \\(\\frac{3}{2}\\) and 4 \u00f7 \\(\\frac{3}{2}\\) are not the same<\/p>\n

Question 4.
\nHow can you write any nonzero whole number as a fraction?
\nAnswer:
\nYou can multiply the numerator and the denominator of a fraction\u00a0by any nonzero whole number, as long as you multiply both by the same whole number<\/p>\n

Question 5.
\nLook for Relationships<\/strong> How does the quotient compare to the dividend when the divisor is a fraction less than 1?
\nAnswer:
\nWhen the divisor is less than 1, the quotient is larger than the dividend. Decreasing the divisor to \\(\\frac{1}{2}\\) increases the quotient to \\(\\frac{1}{2}\\). When the divisor is smaller than the dividend, the quotient is more than 1.<\/p>\n

Question 6.
\nWhat division equation is represented by the diagram?
\n\"Envision
\nAnswer:
\nThe given diagram is:
\n\"Envision
\nNow,
\nFrom the given diagram,
\nWe can observe that
\nThe total number of equal parts are: 8
\nThe number of shaded parts are: 6
\nSo,
\nThe division equation that is represented by the given diagram = (The number of shaded parts) \u00f7 (The total number of equal parts)
\n= \\(\\frac{6}{8}\\)
\n= \\(\\frac{6 \u00f7 2}{8 \u00f7 2}\\)
\n= \\(\\frac{3}{4}\\)
\nHence, from the above,
\nWe can conclude that
\nThe division equation that is represented by the given diagram is: \\(\\frac{3}{4}\\)<\/p>\n

Do You Know How?<\/strong><\/p>\n

In 7-14, find each reciprocal.<\/strong><\/p>\n

Question 7.
\n\\(\\frac{3}{5}\\)
\nAnswer:
\nWe know that,
\nTwo numbers whose product is “1” are called “Reciprocals” of each other. If a nonzero number is named as a fraction \\(\\frac{a}{b}\\), then its reciprocal is \\(\\frac{b}{a}\\)
\nHence, from the above,
\nWe can conclude that
\nThe reciprocal of \\(\\frac{3}{5}\\) is: \\(\\frac{5}{3}\\)<\/p>\n

Question 8.
\n\\(\\frac{1}{6}\\)
\nAnswer:
\nWe know that,
\nTwo numbers whose product is “1” are called “Reciprocals” of each other. If a nonzero number is named as a fraction \\(\\frac{a}{b}\\), then its reciprocal is \\(\\frac{b}{a}\\)
\nHence, from the above,
\nWe can conclude that
\nThe reciprocal of \\(\\frac{1}{6}\\) is: \\(\\frac{6}{1}\\)<\/p>\n

Question 9.
\n9
\nAnswer:
\nWe know that,
\nTwo numbers whose product is “1” are called “Reciprocals” of each other. If a nonzero number is named as a fraction \\(\\frac{a}{b}\\), then its reciprocal is \\(\\frac{b}{a}\\)
\nHence, from the above,
\nWe can conclude that
\nThe reciprocal of 9 is: \\(\\frac{1}{9}\\)<\/p>\n

Question 10.
\n\\(\\frac{7}{4}\\)
\nAnswer:
\nWe know that,
\nTwo numbers whose product is “1” are called “Reciprocals” of each other. If a nonzero number is named as a fraction \\(\\frac{a}{b}\\), then its reciprocal is \\(\\frac{b}{a}\\)
\nHence, from the above,
\nWe can conclude that
\nThe reciprocal of \\(\\frac{7}{4}\\) is: \\(\\frac{4}{7}\\)<\/p>\n

Question 11.
\n\\(\\frac{5}{8}\\)
\nAnswer:
\nWe know that,
\nTwo numbers whose product is “1” are called “Reciprocals” of each other. If a nonzero number is named as a fraction \\(\\frac{a}{b}\\), then its reciprocal is \\(\\frac{b}{a}\\)
\nHence, from the above,
\nWe can conclude that
\nThe reciprocal of \\(\\frac{5}{8}\\) is: \\(\\frac{8}{5}\\)<\/p>\n

Question 12.
\n16
\nAnswer:
\nWe know that,
\nTwo numbers whose product is “1” are called “Reciprocals” of each other. If a nonzero number is named as a fraction \\(\\frac{a}{b}\\), then its reciprocal is \\(\\frac{b}{a}\\)
\nHence, from the above,
\nWe can conclude that
\nThe reciprocal of 16 is: \\(\\frac{1}{16}\\)<\/p>\n

Question 13.
\n\\(\\frac{7}{12}\\)
\nAnswer:
\nWe know that,
\nTwo numbers whose product is “1” are called “Reciprocals” of each other. If a nonzero number is named as a fraction \\(\\frac{a}{b}\\), then its reciprocal is \\(\\frac{b}{a}\\)
\nHence, from the above,
\nWe can conclude that
\nThe reciprocal of \\(\\frac{7}{12}\\) is: \\(\\frac{12}{7}\\)<\/p>\n

Question 14.
\n\\(\\frac{11}{5}\\)
\nAnswer:
\nWe know that,
\nTwo numbers whose product is “1” are called “Reciprocals” of each other. If a nonzero number is named as a fraction \\(\\frac{a}{b}\\), then its reciprocal is \\(\\frac{b}{a}\\)
\nHence, from the above,
\nWe can conclude that
\nThe reciprocal of \\(\\frac{11}{5}\\) is: \\(\\frac{5}{11}\\)<\/p>\n

In 15-22, find each quotient.<\/strong><\/p>\n

Question 15.
\n6 \u00f7 \\(\\frac{2}{3}\\)
\nAnswer:
\nThe given division equation is: 6 \u00f7 \\(\\frac{2}{3}\\)
\nSo,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n6 \u00f7 \\(\\frac{2}{3}\\) = 9
\nThe quotient for the given division equation is: 9<\/p>\n

Question 16.
\n12 \u00f7 \\(\\frac{3}{8}\\)
\nAnswer:
\nThe given division equation is: 12 \u00f7 \\(\\frac{3}{8}\\)
\nSo,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n12 \u00f7 \\(\\frac{3}{8}\\) = 32
\nThe quotient for the given division equation is: 32<\/p>\n

Question 17.
\n\\(\\frac{1}{4}\\) \u00f7 3
\nAnswer:
\nThe given division equation is: \\(\\frac{1}{4}\\) \u00f7 3
\nSo,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{1}{4}\\) \u00f7 3 = \\(\\frac{1}{12}\\)
\nThe quotient for the given division equation is: \\(\\frac{1}{12}\\)<\/p>\n

Question 18.
\n\\(\\frac{2}{5}\\) \u00f7 2
\nAnswer:
\nThe given division equation is: \\(\\frac{2}{5}\\) \u00f7 2
\nSo,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{2}{5}\\) \u00f7 2 = \\(\\frac{1}{5}\\)
\nThe quotient for the given division equation is: \\(\\frac{1}{5}\\)<\/p>\n

Question 19.
\n2 \u00f7 \\(\\frac{1}{2}\\)
\nAnswer:
\nThe given division equation is: 2 \u00f7 \\(\\frac{1}{2}\\)
\nSo,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n2 \u00f7 \\(\\frac{1}{2}\\) = 4
\nThe quotient for the given division equation is: 4<\/p>\n

Question 20.
\n3 \u00f7 \\(\\frac{1}{4}\\)
\nAnswer:
\nThe given division equation is: 2 \u00f7 \\(\\frac{1}{2}\\)
\nSo,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n2 \u00f7 \\(\\frac{1}{2}\\) = 4
\nThe quotient for the given division equation is: 4<\/p>\n

Question 21.
\n9 \u00f7 \\(\\frac{3}{5}\\)
\nAnswer:
\nThe given division equation is: 9 \u00f7 \\(\\frac{3}{5}\\)
\nSo,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n9 \u00f7 \\(\\frac{3}{5}\\) = 15
\nThe quotient for the given division equation is: 15<\/p>\n

Question 22.
\n5 \u00f7 \\(\\frac{2}{7}\\)
\nAnswer:
\nThe given division equation is: 5 \u00f7 \\(\\frac{2}{7}\\)
\nSo,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n5 \u00f7 \\(\\frac{2}{7}\\) = \\(\\frac{35}{2}\\)
\nThe quotient for the given division equation is: \\(\\frac{35}{2}\\)<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Scan for Multimedia<\/strong><\/p>\n

Leveled Practice<\/strong> In 23 and 24, complete each division sentence.<\/strong><\/p>\n

Question 23.
\n\"Envision
\nThe number line shows 6 wholes.
\nAnswer:
\nThe given number line is:
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe completed division sentence is:
\n\"\"<\/p>\n

Question 24.
\n\"Envision
\nAnswer:
\nThe given diagram is:
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe completed division sentence is:
\n\"\"<\/p>\n

In 25 and 26, find each quotient. Draw a diagram to help.<\/strong><\/p>\n

Question 25.
\n\\(\\frac{3}{5}\\) \u00f7 3
\nAnswer:
\nThe given division equation is:
\n\\(\\frac{3}{5}\\) \u00f7 3
\nNow,
\nThe representation of the given division equation in the form of an area model is:
\n\"\"
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{3}{5}\\) \u00f7 3 = \\(\\frac{1}{5}\\)
\nThe quotient for the given division equation is: \\(\\frac{1}{5}\\)<\/p>\n

Question 26.
\n2 \u00f7 \\(\\frac{2}{5}\\)
\nAnswer:
\nThe given division equation is:
\n2 \u00f7 \\(\\frac{2}{5}\\)
\nNow,
\nThe representation of the given division equation in the form of an area model is:
\n\"\"
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n2 \u00f7 \\(\\frac{2}{5}\\) = 5
\nThe quotient for the given division equation is: 5<\/p>\n

In 27-30, find each reciprocal.<\/strong><\/p>\n

Question 27.
\n\\(\\frac{3}{10}\\)
\nAnswer:
\nWe know that,
\nTwo numbers whose product is “1” are called “Reciprocals” of each other. If a nonzero number is named as a fraction \\(\\frac{a}{b}\\), then its reciprocal is \\(\\frac{b}{a}\\)
\nHence, from the above,
\nWe can conclude that
\nThe reciprocal of \\(\\frac{3}{10}\\) is: \\(\\frac{10}{3}\\)<\/p>\n

Question 28.
\n6
\nAnswer:
\nWe know that,
\nTwo numbers whose product is “1” are called “Reciprocals” of each other. If a nonzero number is named as a fraction \\(\\frac{a}{b}\\), then its reciprocal is \\(\\frac{b}{a}\\)
\nHence, from the above,
\nWe can conclude that
\nThe reciprocal of 6 is: \\(\\frac{1}{6}\\)<\/p>\n

Question 29.
\n\\(\\frac{1}{15}\\)
\nAnswer:
\nWe know that,
\nTwo numbers whose product is “1” are called “Reciprocals” of each other. If a nonzero number is named as a fraction \\(\\frac{a}{b}\\), then its reciprocal is \\(\\frac{b}{a}\\)
\nHence, from the above,
\nWe can conclude that
\nThe reciprocal of \\(\\frac{1}{15}\\) is: 15<\/p>\n

Question 30.
\n3
\nAnswer:
\nWe know that,
\nTwo numbers whose product is “1” are called “Reciprocals” of each other. If a nonzero number is named as a fraction \\(\\frac{a}{b}\\), then its reciprocal is \\(\\frac{b}{a}\\)
\nHence, from the above,
\nWe can conclude that
\nThe reciprocal of 3 is: \\(\\frac{1}{3}\\)<\/p>\n

In 31-38, find each quotient.<\/strong><\/p>\n

Question 31.
\n36 \u00f7 \\(\\frac{3}{4}\\)
\nAnswer:
\nThe given division equation is: 36 \u00f7 \\(\\frac{3}{4}\\)
\nSo,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n36 \u00f7 \\(\\frac{3}{4}\\) = 48
\nThe quotient for the given division equation is: 48<\/p>\n

Question 32.
\n2 \u00f7 \\(\\frac{3}{8}\\)
\nAnswer:
\nThe given division equation is: 2 \u00f7 \\(\\frac{3}{8}\\)
\nSo,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n2 \u00f7 \\(\\frac{3}{8}\\) = \\(\\frac{16}{3}\\)
\nThe quotient for the given division equation is: \\(\\frac{16}{3}\\)<\/p>\n

Question 33.
\n18 \u00f7 \\(\\frac{2}{3}\\)
\nAnswer:
\nThe given division equation is: 18 \u00f7 \\(\\frac{2}{3}\\)
\nSo,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n18 \u00f7 \\(\\frac{2}{3}\\) = 27
\nThe quotient for the given division equation is: 27<\/p>\n

Question 34.
\n9 \u00f7 \\(\\frac{4}{5}\\)
\nAnswer:
\nThe given division equation is: 9 \u00f7 \\(\\frac{4}{5}\\)
\nSo,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n9 \u00f7 \\(\\frac{4}{5}\\) = \\(\\frac{45}{4}\\)
\nThe quotient for the given division equation is: \\(\\frac{45}{4}\\)<\/p>\n

Question 35.
\n\\(\\frac{1}{6}\\) \u00f7 2
\nAnswer:
\nThe given division equation is: \\(\\frac{1}{6}\\) \u00f7 2
\nSo,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{1}{6}\\) \u00f7 2 = \\(\\frac{1}{12}\\)
\nThe quotient for the given division equation is: \\(\\frac{1}{12}\\)<\/p>\n

Question 36.
\n\\(\\frac{2}{3}\\) \u00f7 3
\nAnswer:
\nThe given division equation is: \\(\\frac{2}{3}\\) \u00f7 3
\nSo,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{2}{3}\\) \u00f7 3 = \\(\\frac{2}{9}\\)
\nThe quotient for the given division equation is: \\(\\frac{2}{9}\\)<\/p>\n

Question 37.
\n\\(\\frac{3}{5}\\) \u00f7 2
\nAnswer:
\nThe given division equation is: \\(\\frac{3}{5}\\) \u00f7 2
\nSo,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{3}{5}\\) \u00f7 2 = \\(\\frac{3}{10}\\)
\nThe quotient for the given division equation is: \\(\\frac{3}{10}\\)<\/p>\n

Question 38.
\n\\(\\frac{1}{4}\\) \u00f7 4
\nAnswer:
\nThe given division equation is: \\(\\frac{1}{4}\\) \u00f7 4
\nSo,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{1}{4}\\) \u00f7 4 = \\(\\frac{1}{16}\\)
\nThe quotient for the given division equation is: \\(\\frac{1}{16}\\)<\/p>\n

Question 39.
\nA worker is pouring 3 quarts of liquid into \\(\\frac{3}{8}\\)-quart containers. How many of the containers can she fill? Write and solve a division equation.
\n\"Envision
\nAnswer:
\nIt is given that
\nA worker is pouring 3 quarts of liquid into \\(\\frac{3}{8}\\)-quart containers
\nSo,
\nThe number of containers she can fill = (The number of quarts of liquid a worker poured) \u00f7 (The size of each container)
\n= 3 \u00f7 \\(\\frac{3}{8}\\)
\nNow,
\nBy using the Long Division,
\n\"\"<\/p>\n

In 40-43, use the given information.<\/strong><\/p>\n

\"Envision<\/p>\n

Question 40.
\nHigher-Order Thinking<\/strong> Without doing any calculations, how can you use the information given to tell which animal moves the fastest?
\nAnswer:
\nThe given information is:
\n\"Envision
\nNow,
\nThe distance covered by a snail in 1 hour = 120 \u00f7 \\(\\frac{3}{4}\\)
\nSo,
\nBy using the Long Division,
\n\"\"
\nSo,
\nThe distance covered by a snail in 1 hour is: 160 ft
\nNow,
\nThe distance covered by a tortoise in 1 hour = 600 \u00f7 \\(\\frac{2}{3}\\)
\nSo,
\nBy using the Long Division,
\n\"\"
\nSo,
\nThe distance covered by a tortoise in 1 hour is: 900 ft
\nNow,
\nThe distance covered by a Sloth in 1 hour = 250 \u00f7 \\(\\frac{5}{8}\\)
\nSo,
\nBy using the Long Division,
\n\"\"
\nSo,
\nThe distance covered by a Sloth in 1 hour is: 400 ft
\nHence, from the above,
\nWe can conclude that
\nThe animal which moves the fastest is: Tortoise<\/p>\n

Question 41.
\nReasoning<\/strong> The quotient 250 \u00f7 \\(\\frac{5}{8}\\) tells about how far a sloth may move in one hour. How far can a sloth go in 90 minutes? Justify your reasoning.
\nAnswer:
\nIt is given that
\nThe quotient 250 \u00f7 \\(\\frac{5}{8}\\) tells about how far a sloth may move in one hour
\nNow,
\nWe know that,
\n1 hour = 60 minutes
\n1\\(\\frac{1}{2}\\) hour = 90 minutes
\nSo,
\nThe representation of 1\\(\\frac{1}{2}\\) into a fraction is:
\n\"\"
\nSo,
\nThe distance covered by a Sloth in 90 minutes = 250 \u00f7 \\(\\frac{3}{2}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe distance covered by a Sloth in 90 minutes is: \\(\\frac{500}{3}\\) ft<\/p>\n

Question 42.
\nThe quotient 600 \u00f7 \\(\\frac{2}{3}\\) tells about how far a tortoise may move in one hour. Find that distance.
\nAnswer:
\nIt is given that
\nThe quotient 600 \u00f7 \\(\\frac{2}{3}\\) tells about how far a tortoise may move in one hour
\nNow,
\nAccording to the given information,
\nThe distance covered by a tortoise in 1 hour = 600 \u00f7 \\(\\frac{2}{3}\\)
\nSo,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe distance covered by a tortoise in 1 hour is: 900 ft<\/p>\n

Question 43.
\nWrite and solve an equation to find how far a snail can go in one hour.
\nAnswer:
\nThe given information is:
\n\"Envision
\nNow,
\nThe distance covered by a snail in 1 hour = 120 \u00f7 \\(\\frac{3}{4}\\)
\nSo,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe distance covered by a snail in 1 hour is: 160 ft<\/p>\n

Question 44.
\nA waitress pours \\(\\frac{3}{4}\\) gallon of orange juice equally into 5 pitchers. What fraction of a gallon of orange juice is in each pitcher? Use the rectangle to represent the problem. Then write an equation to show the solution.
\n\"Envision
\nAnswer:
\nIt is given that
\nA waitress pours \\(\\frac{3}{4}\\) gallon of orange juice equally into 5 pitchers.
\nNow,
\nThe representation of the given information in the form of an Area model is:
\n\"\"
\nNow,
\nLet the fraction of a gallon of orange juice that is present in each pitcher be x
\nSo,
\nAccording to the given information,
\n5 \u00f7 x = \\(\\frac{3}{4}\\)
\nSo,
\nx = 5 \u00f7 \\(\\frac{3}{4}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe fraction of a gallon of orange juice that is present in each pitcher is: \\(\\frac{20}{3}\\) gallons<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 45.
\nSelect all the math statements that have the same quotient.
\n\"Envision 12 \u00f7 \\(\\frac{2}{3}\\)
\n\"Envision \\(\\frac{2}{3}\\) \u00f7 \\(\\frac{1}{27}\\)
\n\"Envision 16 \u00f7 \\(\\frac{4}{5}\\)
\n\"Envision 12 \u00f7 \\(\\frac{3}{2}\\)
\n\"Envision 24 \u00f7 \\(\\frac{4}{3}\\)
\nAnswer:
\nThe given math statements and their quotients are:
\na. 12 \u00f7 \\(\\frac{2}{3}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nb. \\(\\frac{2}{3}\\) \u00f7 \\(\\frac{1}{27}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nc. 16 \u00f7 \\(\\frac{4}{5}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nd. 12 \u00f7 \\(\\frac{3}{2}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\ne. 24 \u00f7 \\(\\frac{4}{3}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nAll the math statements that have the same quotient are:
\n\"\"<\/p>\n

Question 46.
\nSelect all the math statements that are true.
\n\"Envision
\nAnswer:
\nAll the math statements that are true are:
\n\"\"<\/p>\n

Lesson 1.5 Divide Fractions by Fractions<\/h3>\n

ACTIVITY<\/strong><\/p>\n

Solve & Discuss It!<\/strong><\/p>\n

A granola bar was cut into 6 equal pieces. Someone ate part of the granola bar so that \\(\\frac{2}{3}\\) of the original bar remains. How many \\(\\frac{1}{6}\\) parts are left? Use the picture to draw a model to represent and find \\(\\frac{2}{3}\\) \u00f7 \\(\\frac{1}{6}\\)<\/p>\n

\"Envision
\nAnswer:
\nIt is given that
\nA granola bar was cut into 6 equal pieces. Someone ate part of the granola bar so that \\(\\frac{2}{3}\\) of the original bar remains
\nNow,
\nThe given picture is:
\n\"\"
\nNow,
\nThe representation of the given information in the form of a bar diagram is:
\n\"\"
\nNow,
\nThe number of \\(\\frac{1}{6}\\) parts that are left = (The part that is remained in the original granola bar) \u00f7 \\(\\frac{1}{6}\\)
\n= \\(\\frac{2}{3}\\) \u00f7 \\(\\frac{1}{6}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe number of \\(\\frac{1}{6}\\) parts that are left is: 4<\/p>\n

A model with Math<\/strong>
\nYou can model with math by dividing a whole into equal parts.
\nAnswer:
\nThe representation of the above problem in the form of a bar diagram is:
\n\"\"<\/p>\n

Focus on math practices<\/strong>
\nUse Structure<\/strong> How can you use multiplication to check your answer?
\nAnswer:
\nYou can use multiplication to check your division answer the following way:
\na. Do the division problem.
\nb. Multiply the quotient times the divisor.
\nc. If there is a remainder, add it to the multiplication product.
\nd. Compare this answer to the dividend. They should be the same number<\/p>\n

? Essential Question<\/strong> How can you divide a fraction by a fraction?
\nAnswer:
\nThe easiest way to divide fractions is to follow three simple steps:
\na. Flip the divisor into a reciprocal.
\nb. Change the division sign into a multiplication sign and multiply.
\nc. Simplify if possible.<\/p>\n

Try It!<\/strong>
\nUse the number line below to represent \\(\\frac{1}{6}\\) \u00d7 3 = \\(\\frac{1}{2}\\). Then write an equivalent division sentence.
\n\"Envision
\nAnswer:
\nThe given multiplication equation is:
\n\\(\\frac{1}{6}\\) \u00d7 3 = \\(\\frac{1}{2}\\)
\nSo,
\nThe representation of the given division equation in the given number line is:
\n\"\"
\nNow,
\nThe representation of the given multiplication equation into division equation is:
\n\\(\\frac{1}{2}\\) \u00f7 \\(\\frac{1}{6}\\) = 3
\nHence, from the above,
\nWe can conclude that
\nThe equivalent division equation for the given multiplication equation is:
\n\\(\\frac{1}{2}\\) \u00f7 \\(\\frac{1}{6}\\) = 3<\/p>\n

Convince Me!<\/strong> How are the dividend, divisor, and quotient represented on the number line?
\nAnswer:
\nIn a number line,
\nThe dividend represents the “Product”
\nThe divisor represents the “The division of the dividend into equal parts”
\nThe quotient represents the “The number of times the divisor divided”<\/p>\n

Try It!<\/strong>
\nFind \\(\\frac{1}{4}\\) \u00f7 \\(\\frac{3}{8}\\). Draw an area model.
\nAnswer:
\nThe given division expression is: \\(\\frac{1}{4}\\) \u00f7 \\(\\frac{3}{8}\\)
\nHence,
\nThe representation of the given division expression by using the area model is:
\n\"\"<\/p>\n

Try It!<\/strong>
\nHow wide is a rectangular strip of land with a length of \\(\\frac{3}{4}\\) mile and an area of \\(\\frac{1}{2}\\) square mile? Use the area formula: A = l \u00d7 w.
\n\"Envision
\nAnswer:
\nIt is given that
\nThe length of a rectangular strip is \\(\\frac{3}{4}\\) mile and an area of a rectangular strip is \\(\\frac{1}{2}\\) square mile
\nNow,
\nIt is given that
\nArea (A) = Length (L) \u00d7 Width (W)
\nSo,
\n\\(\\frac{1}{2}\\) = \\(\\frac{3}{4}\\) \u00d7 Width (W)
\nWidth (W) = \\(\\frac{1}{2}\\) \u00f7 \\(\\frac{3}{4}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\"\"<\/p>\n

KEY CONCEPT<\/strong><\/p>\n

To divide a fraction by a fraction, rewrite the division equation as a multiplication equation.
\n\"Envision<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n?Essential Question<\/strong> How can you divide a fraction by a fraction?
\nAnswer:
\nThe easiest way to divide fractions is to follow three simple steps:
\na. Flip the divisor into a reciprocal.
\nb. Change the division sign into a multiplication sign and multiply.
\nc. Simplify if possible.<\/p>\n

Question 2.
\nCritique Reasoning<\/strong> To find the quotient of \\(\\frac{2}{5}\\) \u00f7 \\(\\frac{8}{5}\\), Corey rewrites the problem as \\(\\frac{5}{2}\\) \u00d7 \\(\\frac{8}{5}\\). Explain Corey’s mistake and how to correct it.
\nAnswer:
\nIt is given that
\nTo find the quotient of \\(\\frac{2}{5}\\) \u00f7 \\(\\frac{8}{5}\\), Corey rewrites the problem as \\(\\frac{5}{2}\\) \u00d7 \\(\\frac{8}{5}\\)
\nNow,
\nThe given division expression is: \\(\\frac{2}{5}\\) \u00f7 \\(\\frac{8}{5}\\)
\nNow,
\nWe know that,
\nWe have to find the reciprocal of a fraction that is present to the right side of the division symbol
\nSo,
\n\\(\\frac{2}{5}\\) \u00f7 \\(\\frac{8}{5}\\)
\n= \\(\\frac{2}{5}\\) \u00d7 \\(\\frac{5}{8}\\)
\nHence, from the above,
\nWe can conclude that
\nCorey’s mistake is: He found the reciprocal of a fraction that is present to the left side of the division symbol instead of the right side of the division symbol
\nThe correct form of the given division expression is:
\n\\(\\frac{2}{5}\\) \u00d7 \\(\\frac{5}{8}\\)<\/p>\n

Question 3.
\nReasoning<\/strong> is the quotient of \\(\\frac{3}{5}\\) \u00f7 \\(\\frac{6}{7}\\) greater than or less than \\(\\frac{3}{5}\\)? Explain.
\nAnswer:
\nThe given division expression is: \\(\\frac{3}{5}\\) \u00f7 \\(\\frac{6}{7}\\)
\nNow,
\nThe representation of the given division expression in the decimal form is:
\n\\(\\frac{3}{5}\\) \u00f7 \\(\\frac{6}{7}\\)
\n= 0.60 \u00f7 0.85
\nNow,
\nBy using the Long Division,
\n\"\"
\nSo,
\nThe representation of the quotient value for the given divison expression in the decimal form is: 0.7
\nHence, from the above,
\nWe can conclude that
\nThe quotient of \\(\\frac{3}{5}\\) \u00f7 \\(\\frac{6}{7}\\) is greater than \\(\\frac{3}{5}\\)<\/p>\n

Question 4.
\nHow is dividing a whole number by a fraction different from dividing a fraction by a fraction?
\nAnswer:
\nThe steps to divide a fraction by a Whole Number are:
\n1. Turn the whole number into a fraction.
\n2. Multiply the fraction by the whole number fraction.
\n3. Multiply the tops together and the bottoms together
\nThe steps to divide a fraction by a fraction are:
\n1. Arrange the equation
\n2. Keep the first fraction
\n3. Change the sign
\n4. Flip the fraction
\n5. Multiply the tops together and the bottoms together<\/p>\n

Do You Know How?<\/strong><\/p>\n

In 5\u20137, write a division sentence to represent each model.<\/strong><\/p>\n

Question 5.
\n\"Envision
\nAnswer:
\nThe given model is:
\n\"Envision
\nNow,
\nFrom the given model,
\nWe can observe that
\nThe division sentence to represent the given model is:
\n\\(\\frac{4}{5}\\) \u00f7 4
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe division sentence to represent the given model is:
\n\\(\\frac{4}{5}\\) \u00f7 4<\/p>\n

Question 6.
\n\"Envision
\nAnswer:
\nThe given model is:
\n\"Envision
\nNow,
\nFrom the given model,
\nWe can observe that
\nThe division sentence to represent the given model is:
\n\\(\\frac{1}{2}\\) \u00f7 \\(\\frac{2}{3}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe division sentence to represent the given model is:
\n\\(\\frac{1}{2}\\) \u00f7 \\(\\frac{2}{3}\\)<\/p>\n

Question 7.
\n\"Envision
\nAnswer:
\nThe given model is:
\n\"Envision
\nNow,
\nFrom the given model,
\nWe can observe that
\nThe division sentence for the given model is:
\n\\(\\frac{3}{4}\\) \u00f7 \\(\\frac{1}{8}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe division sentence for the given model is:
\n\\(\\frac{3}{4}\\) \u00f7 \\(\\frac{1}{8}\\)<\/p>\n

In 8-11, find each quotient.<\/strong><\/p>\n

Question 8.
\n\\(\\frac{3}{4}\\) \u00f7 \\(\\frac{2}{3}\\)
\nAnswer:
\nThe given division expression is:
\n\\(\\frac{3}{4}\\) \u00f7 \\(\\frac{2}{3}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{3}{4}\\) \u00f7 \\(\\frac{2}{3}\\) = \\(\\frac{9}{8}\\)
\nThe quotient for the given division expression is: \\(\\frac{9}{8}\\)<\/p>\n

Question 9.
\n\\(\\frac{3}{12}\\) \u00f7 \\(\\frac{1}{8}\\)
\nAnswer:
\nThe given division expression is:
\n\\(\\frac{3}{12}\\) \u00f7 \\(\\frac{1}{8}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{3}{12}\\) \u00f7 \\(\\frac{1}{8}\\) = 2
\nThe quotient for the given division expression is: 2<\/p>\n

Question 10.
\n\\(\\frac{1}{2}\\) \u00f7 \\(\\frac{4}{5}\\)
\nAnswer:
\nThe given division expression is:
\n\\(\\frac{1}{2}\\) \u00f7 \\(\\frac{4}{5}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{1}{2}\\) \u00f7 \\(\\frac{4}{5}\\) = \\(\\frac{5}{8}\\)
\nThe quotient for the given division expression is: \\(\\frac{5}{8}\\)<\/p>\n

Question 11.
\n\\(\\frac{7}{10}\\) \u00f7 \\(\\frac{2}{5}\\)
\nAnswer:
\nThe given division expression is:
\n\\(\\frac{7}{10}\\) \u00f7 \\(\\frac{2}{5}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{7}{10}\\) \u00f7 \\(\\frac{2}{5}\\) = \\(\\frac{7}{4}\\)
\nThe quotient for the given division expression is: \\(\\frac{7}{4}\\)<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Scan for Multimedia<\/strong><\/p>\n

In 12 and 13, complete each division sentence using the models provided.<\/strong><\/p>\n

Question 12.
\n\"Envision
\nAnswer:
\nThe given division expression is:
\n\\(\\frac{1}{3}\\) \u00f7 \\(\\frac{1}{12}\\)
\nNow,
\nThe given model is:
\n\"\"
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{1}{3}\\) \u00f7 \\(\\frac{1}{12}\\) = 4<\/p>\n

Question 13.
\n\"Envision
\nAnswer:
\nThe given division expression is:
\n\\(\\frac{2}{5}\\) \u00f7 \\(\\frac{1}{10}\\)
\nNow,
\nThe given model is:
\n\"\"
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{2}{5}\\) \u00f7 \\(\\frac{1}{10}\\) = 4<\/p>\n

In 14-25, find each quotient.<\/strong><\/p>\n

Question 14.
\n\\(\\frac{2}{3}\\) \u00f7 \\(\\frac{1}{3}\\)
\nAnswer:
\nThe given division expression is:
\n\\(\\frac{2}{3}\\) \u00f7 \\(\\frac{1}{3}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{2}{3}\\) \u00f7 \\(\\frac{1}{3}\\) = 2
\nThe quotient for the given division expresion is: 2<\/p>\n

Question 15.
\n\\(\\frac{1}{2}\\) \u00f7 \\(\\frac{1}{16}\\)
\nAnswer:
\nThe given division expression is:
\n\\(\\frac{1}{2}\\) \u00f7 \\(\\frac{1}{16}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{1}{2}\\) \u00f7 \\(\\frac{1}{16}\\) = 8
\nThe quotient for the given division expresion is: 8<\/p>\n

Question 16.
\n\\(\\frac{1}{4}\\) \u00f7 \\(\\frac{1}{12}\\)
\nAnswer:
\nThe given division expression is:
\n\\(\\frac{1}{4}\\) \u00f7 \\(\\frac{1}{12}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{1}{4}\\) \u00f7 \\(\\frac{1}{12}\\) = 3
\nThe quotient for the given division expresion is: 3<\/p>\n

Question 17.
\n\\(\\frac{6}{7}\\) \u00f7 \\(\\frac{3}{7}\\)
\nAnswer:
\nThe given division expression is:
\n\\(\\frac{6}{7}\\) \u00f7 \\(\\frac{3}{7}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{6}{7}\\) \u00f7 \\(\\frac{3}{7}\\) = 2
\nThe quotient for the given division expresion is: 2<\/p>\n

Question 18.
\n\\(\\frac{5}{14}\\) \u00f7 \\(\\frac{4}{7}\\)
\nAnswer:
\nThe given division expression is:
\n\\(\\frac{5}{14}\\) \u00f7 \\(\\frac{4}{7}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{5}{14}\\) \u00f7 \\(\\frac{4}{7}\\) = \\(\\frac{5}{8}\\)
\nThe quotient for the given division expresion is: \\(\\frac{5}{8}\\)<\/p>\n

Question 19.
\n\\(\\frac{5}{8}\\) \u00f7 \\(\\frac{1}{2}\\)
\nAnswer:
\nThe given division expression is:
\n\\(\\frac{5}{8}\\) \u00f7 \\(\\frac{1}{2}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{5}{8}\\) \u00f7 \\(\\frac{1}{2}\\) = \\(\\frac{5}{4}\\)
\nThe quotient for the given division expresion is: \\(\\frac{5}{4}\\)<\/p>\n

Question 20.
\n\\(\\frac{7}{12}\\) \u00f7 \\(\\frac{3}{4}\\)
\nAnswer:
\nThe given division expression is:
\n\\(\\frac{7}{12}\\) \u00f7 \\(\\frac{3}{4}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{7}{12}\\) \u00f7 \\(\\frac{3}{4}\\) = \\(\\frac{7}{9}\\)
\nThe quotient for the given division expresion is: \\(\\frac{7}{9}\\)<\/p>\n

Question 21.
\n\\(\\frac{2}{7}\\) \u00f7 \\(\\frac{1}{2}\\)
\nAnswer:
\nThe given division expression is:
\n\\(\\frac{2}{7}\\) \u00f7 \\(\\frac{1}{2}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{2}{7}\\) \u00f7 \\(\\frac{1}{2}\\) = \\(\\frac{4}{7}\\)
\nThe quotient for the given division expresion is: \\(\\frac{4}{7}\\)<\/p>\n

Question 22.
\n\\(\\frac{3}{10}\\) \u00f7 \\(\\frac{3}{5}\\)
\nAnswer:
\nThe given division expression is:
\n\\(\\frac{3}{10}\\) \u00f7 \\(\\frac{3}{5}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{3}{10}\\) \u00f7 \\(\\frac{3}{5}\\) = \\(\\frac{1}{2}\\)
\nThe quotient for the given division expresion is: \\(\\frac{1}{2}\\)<\/p>\n

Question 23.
\n\\(\\frac{7}{12}\\) \u00f7 \\(\\frac{1}{8}\\)
\nAnswer:
\nThe given division expression is:
\n\\(\\frac{7}{12}\\) \u00f7 \\(\\frac{1}{8}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{7}{12}\\) \u00f7 \\(\\frac{1}{8}\\) = \\(\\frac{14}{3}\\)
\nThe quotient for the given division expresion is: \\(\\frac{14}{3}\\)<\/p>\n

Question 24.
\n\\(\\frac{3}{10}\\) \u00f7 \\(\\frac{3}{5}\\)
\nAnswer:
\nThe given division expression is:
\n\\(\\frac{3}{10}\\) \u00f7 \\(\\frac{3}{5}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{3}{10}\\) \u00f7 \\(\\frac{3}{5}\\) = \\(\\frac{1}{2}\\)
\nThe quotient for the given division expresion is: \\(\\frac{1}{2}\\)<\/p>\n

Question 25.
\n\\(\\frac{2}{5}\\) \u00f7 \\(\\frac{1}{8}\\)
\nAnswer:
\nThe given division expression is:
\n\\(\\frac{2}{5}\\) \u00f7 \\(\\frac{1}{8}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{2}{5}\\) \u00f7 \\(\\frac{1}{8}\\) = \\(\\frac{16}{5}\\)
\nThe quotient for the given division expresion is: \\(\\frac{16}{5}\\)<\/p>\n

Question 26.
\nBe Precise<\/strong> A large bag contains \\(\\frac{12}{15}\\) pound of granola. How many \\(\\frac{1}{3}\\) pound bags can be filled with this amount of granola? How much granola is left over?
\n\"Envision
\nAnswer:
\nIt is given that
\nA large bag contains \\(\\frac{12}{15}\\) pound of granola
\nNow,
\nLet the number of \\(\\frac{1}{3}\\) pound bags that can be filled with the given amount of granola be: x
\nNow,
\nAccording to the given information,
\n\\(\\frac{1}{3}\\) \u00d7 x = \\(\\frac{12}{15}\\)
\nx = \\(\\frac{12}{15}\\) \u00f7 \\(\\frac{1}{3}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nSo,
\nThe number of \\(\\frac{1}{3}\\) pound bags that can be filled with the given amount of granola is about 2 bags
\nNow,
\nThe amount of granola that is leftover = (The number of \\(\\frac{1}{3}\\) pound bags that can be filled with the given amount of granola) – (The total amount of granola)
\n= \\(\\frac{12}{5}\\) – \\(\\frac{12}{15}\\)
\nNow,
\nBy using the Long Subtraction,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe amount of granola that is leftover is: \\(\\frac{8}{5}\\) pounds<\/p>\n

Question 27.
\nHigher-Order Thinking<\/strong> Find \\(\\frac{3}{4}\\) \u00f7 \\(\\frac{2}{3}\\). Then draw a picture and write an explanation describing how to get the answer.
\nAnswer:
\nThe given division expression is:
\n\\(\\frac{3}{4}\\) \u00f7 \\(\\frac{2}{3}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence,
\nThe representation of the given division expression in the form of an area model is:
\n\"\"
\n\\(\\frac{3}{4}\\) \u00f7 \\(\\frac{2}{3}\\) = \\(\\frac{9}{8}\\)<\/p>\n

Question 28.
\nThe area of a rectangular painting is \\(\\frac{1}{6}\\) square yard. The width is \\(\\frac{2}{3}\\) yard. What is the length of the painting? Use the formula A = l \u00d7 w.
\nAnswer:
\nIt is given that
\nThe area of a rectangular painting is \\(\\frac{1}{6}\\) square yard. The width is \\(\\frac{2}{3}\\) yard.
\nNow,
\nLet the length of the painting be x yard
\nNow,
\nFrom the given information,
\nA = l \u00d7 w
\nSo,
\n\\(\\frac{1}{6}\\) = x \u00d7 \\(\\frac{2}{3}\\)
\nx = \\(\\frac{1}{6}\\) \u00f7 \\(\\frac{2}{3}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe length of the painting is: \\(\\frac{1}{4}\\) yard<\/p>\n

Question 29.
\nSolve for n in the equation \\(\\frac{13}{16}\\) \u00f7 \\(\\frac{1}{6}\\) = n.
\nAnswer:
\nThe given division expression is:
\n\\(\\frac{13}{16}\\) \u00f7 \\(\\frac{1}{6}\\) = n
\nWhere,
\n‘n’ is the quotient for the given division expression
\nNow,
\nBy using the Long Division,
\n\"\"
\nSo,
\n\\(\\frac{13}{16}\\) \u00f7 \\(\\frac{1}{6}\\) = \\(\\frac{39}{8}\\)
\nHence, from the above,
\nWe can conclude that
\nThe value of ‘n’ is: \\(\\frac{39}{8}\\)<\/p>\n

Question 30.
\nModel with Math<\/strong>
\nA cafeteria uses \\(\\frac{1}{6}\\) pound of coffee to fill a large coffee dispenser. The cafeteria has \\(\\frac{2}{3}\\) pound of coffee to use.
\na. Complete the model at the right to find how many coffee dispensers the cafeteria can fill.
\nb. Write a division sentence that describes the model and tells how many dispensers can be filled.
\n\"Envision
\nAnswer:
\nIt is given that
\nA cafeteria uses \\(\\frac{1}{6}\\) pound of coffee to fill a large coffee dispenser. The cafeteria has \\(\\frac{2}{3}\\) pound of coffee to use.
\nNow,
\nThe representation of the given information in the form of a model is:
\n\"\"
\nNow,
\nFrom the given model,
\nWe can observe that
\nThe number of coffee dispensers the cafeteria can fill = (The number of pounds of the coffee cafeteria has to use) \u00f7 (The number of pounds of the coffee a cafeteria has to use)
\n= \\(\\frac{2}{3}\\) \u00f7 \\(\\frac{1}{6}\\)
\nNow,
\nBy using the Long division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\na. The model to find how many coffee dispensers the cafeteria can fill is:
\n\"\"
\nb. The division sentence that describes the model and tells how many dispensers can be filled is:
\n\\(\\frac{2}{3}\\) \u00f7 \\(\\frac{1}{6}\\)<\/p>\n

Question 31.
\nModel with Math<\/strong> A full load for a small truck to haul is \\(\\frac{2}{3}\\) ton of gravel. The truck is hauling \\(\\frac{1}{2}\\) ton of gravel.
\n\"Envision
\na. Complete the model below to find how much of a full load the truck is hauling.
\nb. Write a division sentence that describes the model and tells how much of a full load the truck is hauling.
\n\"Envision
\nAnswer:
\nIt is given that
\nA full load for a small truck to haul is \\(\\frac{2}{3}\\) ton of gravel. The truck is hauling \\(\\frac{1}{2}\\) ton of gravel.
\nNow,
\nThe representation of the given information in the form of a model is:
\n\"\"
\nNow,
\nFrom the given model,
\nWe can observe that
\nThe amount of a full load the truck is hauling = (The amount of a full load for a small truck to haul) \u00f7 (The amount of gravel a truck is hauling)
\n= \\(\\frac{2}{3}\\) \u00f7 \\(\\frac{1}{2}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\na. The model below to find how much of a full load the truck is hauling is:
\n\"\"
\nb. The division sentence describes the model and tells how much of a full load the truck is hauling is:
\n\\(\\frac{2}{3}\\) \u00f7 \\(\\frac{1}{2}\\)<\/p>\n

Question 32.
\nUse Structure<\/strong> How many \\(\\frac{1}{4}\\)-inch pieces can be cut from a piece of metal \\(\\frac{5}{4}\\) inch long?
\nAnswer:
\nIt is given that
\n\\(\\frac{1}{4}\\)-inch pieces can be cut from a piece of metal \\(\\frac{5}{4}\\) inch long
\nSo,
\nThe number of \\(\\frac{1}{4}\\)-inch pieces that can be cut from \\(\\frac{5}{4}\\) inch long
\n= \\(\\frac{5}{4}\\) \u00f7 \\(\\frac{1}{4}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe number of \\(\\frac{1}{4}\\)-inch pieces that can be cut from \\(\\frac{5}{4}\\) inch long is: 5 pieces<\/p>\n

Question 33.
\nWrite a problem that could be solved by finding \\(\\frac{5}{8}\\) \u00f7 \\(\\frac{2}{5}\\).
\nAnswer:
\nThe given division expression is:
\n\\(\\frac{5}{8}\\) \u00f7 \\(\\frac{2}{5}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{5}{8}\\) \u00f7 \\(\\frac{2}{5}\\) = \\(\\frac{25}{16}\\)<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 34.
\nWhich division sentence is shown by the model at the right?
\n\"Envision
\nA. \\(\\frac{2}{3}\\) \u00f7 \\(\\frac{1}{9}\\) = 6
\nB. \\(\\frac{1}{9}\\) \u00f7 \\(\\frac{2}{3}\\) = \\(\\frac{1}{6}\\)
\nC. 6 \u00f7 \\(\\frac{1}{9}\\) = 54
\nD. 6 \u00f7 \\(\\frac{2}{3}\\) = 9
\nAnswer:
\nThe given model is:
\n\"Envision
\nNow,
\nFrom the given model,
\nWe can observe that
\n\\(\\frac{2}{3}\\) \u00f7 \\(\\frac{1}{9}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe division sentence that is shown by the model is:
\n\"\"<\/p>\n

Lesson 1.6 Divide Mixed Numbers<\/h3>\n

Solve & Discuss It!<\/strong><\/p>\n

A jeweler has a 5\\(\\frac{1}{2}\\)inch strip of silver wire that she is cutting into 1\\(\\frac{3}{8}\\)-inch pieces. How many pieces can she make?
\n\"Envision
\nAnswer:
\nIt is given that
\nA jeweler has a 5\\(\\frac{1}{2}\\)inch strip of silver wire that she is cutting into 1\\(\\frac{3}{8}\\)-inch pieces
\nSo,
\nThe number of pieces of silver wire she can make = (The total length of the silver wire) \u00f7 (The length of each cutting piece of the silver wire)
\n= 5\\(\\frac{1}{2}\\) \u00f7 1\\(\\frac{3}{8}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe number of pieces of silver wire she can make is: 4 pieces<\/p>\n

Generalize<\/strong> How can you use what you know about solving problems with fractions to find how many pieces the jeweler can make?
\nAnswer:
\nThe steps to solve problems with fractions with the help of what you know is:
\na. Read the problem carefully.
\nb. Think about what it is asking us to do.
\nc. Think about the information we need.
\nd. Solve it.
\ne Simplify, if necessary.
\nf. Think about whether our solution makes sense (in order to check it).<\/p>\n

Focus on math practices<\/strong>
\nGeneralize<\/strong> Explain how to use estimation to check whether your answer is reasonable.
\nAnswer:
\nTo use estimation, you need\u00a0to read and understand the problem. There will be clues in the problem to let you know if estimation is a good option for solving that specific problem. Estimating means finding an answer that is an approximate answer<\/p>\n

? Essential Question<\/strong> How can you divide with mixed numbers?
\nAnswer:
\nThe steps to divide the mixed numbers are:
\nStep 1:
\nMake sure everything is in fraction form\u00a0(mixed numbers need to be changed to improper fractions).
\nStep 2:
\nFlip the second fraction and change the operation to multiply.
\nStep 3:
\nMultiply across the top for the new numerator and across the bottom for the new denominator.
\nStep 4:
\nSimplify (if needed).<\/p>\n

Try It!<\/strong>
\nHow many medium bumper stickers can Damon fit side by side on his car bumper? Find 37\\(\\frac{1}{2}\\) \u00f7 10\\(\\frac{3}{4}\\).
\n\"Envision
\nAnswer:
\nThe given information is:
\n\"\"
\nNow,
\nThe number of medium bumper stickers can Damon fit side by side on his car bumper = (The total length of the car bumper) \u00f7 (The length of the medium bumper)
\n= 37\\(\\frac{1}{2}\\) \u00f7 10\\(\\frac{3}{4}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\"\"<\/p>\n

Convince Me!<\/strong> Why do you multiply \\(\\frac{75}{2}\\) by \\(\\frac{4}{43}\\) to divide 37\\(\\frac{1}{2}\\) by 10\\(\\frac{3}{4}\\)?
\nAnswer:
\nThe given division expression is:
\n37\\(\\frac{1}{2}\\) \u00f7 10\\(\\frac{3}{4}\\)
\nNow,
\nThe representation of the given mixed numbers into a fraction is:
\n\"\"
\n\"\"
\nSo,
\n37\\(\\frac{1}{2}\\) \u00f7 10\\(\\frac{3}{4}\\)
\n= \\(\\frac{75}{2}\\) \u00f7 \\(\\frac{43}{4}\\)
\nNow,
\nWe know that,
\nTwo numbers whose product is “1” are called “reciprocals” of each other. If a nonzero number is named as a fraction \\(\\frac{a}{b}\\), then its reciprocal is \\(\\frac{b}{a}\\)
\nSo,
\n\\(\\frac{75}{2}\\) \u00f7 \\(\\frac{43}{4}\\)
\n= \\(\\frac{75}{2}\\) \u00d7 \\(\\frac{4}{43}\\)
\nHence, from the above,
\nWe can conclude that
\nYou can multiply \\(\\frac{75}{2}\\) by \\(\\frac{4}{43}\\) to divide 37\\(\\frac{1}{2}\\) by 10\\(\\frac{3}{4}\\) due to the property of “Reciprocals”<\/p>\n

Try It!<\/strong>
\nDivide.<\/strong><\/p>\n

a. 20 \u00f7 2\\(\\frac{2}{3}\\)
\nAnswer:
\nThe given division expression is:
\n20 \u00f7 2\\(\\frac{2}{3}\\)
\nSo,
\nThe representtaion of 2\\(\\frac{2}{3}\\) into a fraction is:
\n\"\"
\nSo,
\n20 \u00f7 \\(\\frac{8}{3}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n20 \u00f7 2\\(\\frac{2}{3}\\) = \\(\\frac{15}{2}\\)<\/p>\n

b. 12\\(\\frac{1}{2}\\) \u00f7 6
\nAnswer:
\nThe given division expression is:
\n12\\(\\frac{1}{2}\\) \u00f7 6
\nSo,
\nThe representation of 12\\(\\frac{1}{2}\\) into a fraction is:
\n\"\"
\nSo,
\n\\(\\frac{25}{2}\\) \u00f7 6
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n12\\(\\frac{1}{2}\\) \u00f7 6 = \\(\\frac{25}{12}\\)<\/p>\n

KEY CONCEPT<\/strong>
\n\"Envision<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n? Essential Question<\/strong> How can you divide with mixed numbers?
\nAnswer:
\nThe steps to divide the mixed numbers are:
\nStep 1:
\nMake sure everything is in fraction form\u00a0(mixed numbers need to be changed to improper fractions).
\nStep 2:
\nFlip the second fraction and change the operation to multiply.
\nStep 3:
\nMultiply across the top for the new numerator and across the bottom for the new denominator.
\nStep 4:
\nSimplify (if needed).<\/p>\n

Question 2.
\nGeneralize<\/strong> When dividing mixed numbers, why is it important to estimate the quotient first?
\nAnswer:
\nYou can estimate to find the approximate quotient of fractions and mixed numbers. Estimating quotients of fractions is similar to estimating products of fractions. To estimate the quotient of two fractions, you\u00a0will first rewrite the problem as a multiplication problem<\/p>\n

Question 3.
\nReasoning<\/strong> In Example 1, how many long bumper stickers can Damon fit side by side on his car bumper? Will there be uncovered space? Explain.
\nAnswer:
\nIt is given that
\nThe total length of the bumper is: 37\\(\\frac{1}{2}\\) inches
\nNow,
\nThe given information is:
\n\"\"
\nNow,
\nFrom the given table,
\nWe can observe that
\nThe total length of the bumper = (The length of the short bumper) + (The length of the medium bumper) + (The length of the long bumper)
\n= 6\\(\\frac{1}{4}\\) + 10\\(\\frac{3}{4}\\) + 15
\nNow,
\nBy using the Long Addition,
\n\"\"
\nSo,
\nThe total length of the bumper = 17 + 15
\n= 32 inches
\nNow,
\nFrom the answer,
\nWe can observe that
\nThe given total length of the bumper > The calculated total length of the bumper
\nHence, from the above,
\nWe can conclude that there will be uncovered space in the bumper<\/p>\n

Question 4.
\nWhat is the difference between dividing fractions less than 1 and dividing mixed numbers?
\nAnswer:
\nDividing mixed numbers is very similar to multiplying mixed numbers. You just add one step i.e., after changing the divisor into an improper fraction, you then find its reciprocal and multiply.<\/p>\n

Do You Know How?<\/strong><\/p>\n

In 5\u201313, find each quotient.<\/strong><\/p>\n

Question 5.
\n\"Envision
\nAnswer:
\nThe given division expression is:
\n2\\(\\frac{5}{8}\\) \u00f7 2\\(\\frac{1}{4}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n2\\(\\frac{5}{8}\\) \u00f7 2\\(\\frac{1}{4}\\) = \\(\\frac{7}{8}\\)
\nThe quotient for the given division expression is: \\(\\frac{7}{6}\\)<\/p>\n

Question 6.
\n3 \u00f7 4\\(\\frac{1}{2}\\)
\nAnswer:
\nThe given division expression is:
\n3 \u00f7 4\\(\\frac{1}{2}\\)
\nNow,
\nThe representation of 4\\(\\frac{1}{2}\\) into a fraction is:
\n\"\"
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n3 \u00f7 4\\(\\frac{1}{2}\\) = \\(\\frac{2}{3}\\)
\nThe quotient for the given division expression is: \\(\\frac{2}{3}\\)<\/p>\n

Question 7.
\n18 \u00f7 3\\(\\frac{2}{3}\\)
\nAnswer:
\nThe given division expression is:
\n18 \u00f7 3\\(\\frac{2}{3}\\)
\nNow,
\nThe representation of 3\\(\\frac{2}{3}\\) into a fraction is:
\n\"\"
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n18 \u00f7 3\\(\\frac{2}{3}\\) = \\(\\frac{54}{11}\\)
\nThe quotient for the given division expression is: \\(\\frac{54}{11}\\)<\/p>\n

Question 8.
\n1\\(\\frac{2}{5}\\) \u00f7 7
\nAnswer:
\nThe given division expression is:
\n1\\(\\frac{2}{5}\\) \u00f7 7
\nNow,
\nThe representation of 1\\(\\frac{2}{5}\\) into a fraction is:
\n\"\"
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n1\\(\\frac{2}{5}\\) \u00f7 7 = \\(\\frac{1}{5}\\)
\nThe quotient for the given division expression is: \\(\\frac{1}{5}\\)<\/p>\n

Question 9.
\n5 \u00f7 6\\(\\frac{2}{5}\\)
\nAnswer:
\nThe given division expression is:
\n5 \u00f7 6\\(\\frac{2}{5}\\)
\nNow,
\nThe representation of 3\\(\\frac{2}{3}\\) into a fraction is:
\n\"\"
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n5 \u00f7 6\\(\\frac{2}{5}\\) = \\(\\frac{25}{32}\\)
\nThe quotient for the given division expression is: \\(\\frac{25}{32}\\)<\/p>\n

Question 10.
\n8\\(\\frac{1}{5}\\) \u00f7 3\\(\\frac{3}{4}\\)
\nAnswer:
\nThe given division expression is:
\n8\\(\\frac{1}{5}\\) \u00f7 3\\(\\frac{3}{4}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n8\\(\\frac{1}{5}\\) \u00f7 3\\(\\frac{3}{4}\\) = \\(\\frac{164}{75}\\)
\nThe quotient for the given division expression is: \\(\\frac{164}{75}\\)<\/p>\n

Question 11.
\n2\\(\\frac{1}{2}\\) \u00f7 4\\(\\frac{1}{10}\\)
\nAnswer:
\nThe given division expression is:
\n2\\(\\frac{1}{2}\\) \u00f7 4\\(\\frac{1}{10}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n2\\(\\frac{1}{2}\\) \u00f7 4\\(\\frac{1}{10}\\) = \\(\\frac{25}{41}\\)
\nThe quotient for the given division expression is: \\(\\frac{25}{41}\\)<\/p>\n

Question 12.
\n2\\(\\frac{2}{3}\\) \u00f7 6
\nAnswer:
\nThe given division expression is:
\n2\\(\\frac{2}{3}\\) \u00f7 6
\nNow,
\nThe representation of 2\\(\\frac{2}{3}\\) into a fraction is:
\n\"\"
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n2\\(\\frac{2}{3}\\) \u00f7 6 = \\(\\frac{4}{9}\\)
\nThe quotient for the given division expression is: \\(\\frac{4}{9}\\)<\/p>\n

Question 13.
\n6\\(\\frac{5}{9}\\) \u00f7 1\\(\\frac{7}{9}\\)
\nAnswer:
\nThe given division expression is:
\n6\\(\\frac{5}{9}\\) \u00f7 1\\(\\frac{7}{9}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n6\\(\\frac{5}{9}\\) \u00f7 1\\(\\frac{7}{9}\\) = \\(\\frac{59}{16}\\)
\nThe quotient for the given division expression is: \\(\\frac{59}{16}\\)<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Scan for Multimedia<\/strong><\/p>\n

Leveled Practice<\/strong> In 14-25, find each quotient.<\/strong><\/p>\n

Question 14.
\n\"Envision
\nAnswer:
\nThe given division expression is:
\n10 \u00f7 2\\(\\frac{1}{4}\\)
\nNow,
\nThe representation of 2\\(\\frac{1}{4}\\) into a fraction is:
\n\"\"
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n10 \u00f7 2\\(\\frac{1}{4}\\) = \\(\\frac{40}{9}\\)
\nThe quotient for the given division expression is: \\(\\frac{40}{9}\\)<\/p>\n

Question 15.
\n\"Envision
\nAnswer:
\nThe given division expression is:
\n9\\(\\frac{1}{3}\\) \u00f7 6
\nNow,
\nThe representation of 9\\(\\frac{1}{3}\\) into a fraction is:
\n\"\"
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n9\\(\\frac{1}{3}\\) \u00f7 6 = \\(\\frac{14}{9}\\)
\nThe quotient for the given division expression is: \\(\\frac{14}{9}\\)<\/p>\n

Question 16.
\n\"Envision
\nAnswer:
\nThe given division expression is:
\n1\\(\\frac{3}{8}\\) \u00f7 4\\(\\frac{1}{8}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n1\\(\\frac{3}{8}\\) \u00f7 4\\(\\frac{1}{8}\\) = \\(\\frac{1}{3}\\)
\nThe quotient for the given division expression is: \\(\\frac{1}{3}\\)<\/p>\n

Question 17.
\n\"Envision
\nAnswer:
\nThe given division expression is:
\n2\\(\\frac{2}{3}\\) \u00f7 8
\nNow,
\nThe representation of 2\\(\\frac{2}{3}\\) into a fraction is:
\n\"\"
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n2\\(\\frac{2}{3}\\) \u00f7 8 = \\(\\frac{1}{3}\\)
\nThe quotient for the given division expression is: \\(\\frac{1}{3}\\)<\/p>\n

Question 18.
\n\"Envision
\nAnswer:
\nThe given division expression is:
\n4\\(\\frac{1}{3}\\) \u00f7 3\\(\\frac{1}{4}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n4\\(\\frac{1}{3}\\) \u00f7 3\\(\\frac{1}{4}\\) = \\(\\frac{4}{3}\\)
\nThe quotient for the given division expression is: \\(\\frac{4}{3}\\)<\/p>\n

Question 19.
\n\"Envision
\nAnswer:
\nThe given division expression is:
\n1 \u00f7 8\\(\\frac{5}{9}\\)
\nNow,
\nThe representation of 8\\(\\frac{5}{9}\\) into a fraction is:
\n\"\"
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n1 \u00f7 8\\(\\frac{5}{9}\\) = \\(\\frac{9}{77}\\)
\nThe quotient for the given division expression is: \\(\\frac{9}{77}\\)<\/p>\n

Question 20.
\n3\\(\\frac{5}{6}\\) \u00f7 9\\(\\frac{5}{6}\\)
\nAnswer:
\nThe given division expression is:
\n3\\(\\frac{5}{6}\\) \u00f7 9\\(\\frac{5}{6}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n3\\(\\frac{5}{6}\\) \u00f7 9\\(\\frac{5}{6}\\) = \\(\\frac{23}{59}\\)
\nThe quotient for the given division expression is: \\(\\frac{23}{59}\\)<\/p>\n

Question 21.
\n16 \u00f7 2\\(\\frac{2}{3}\\)
\nAnswer:
\nThe given division expression is:
\n16 \u00f7 2\\(\\frac{2}{3}\\)
\nNow,
\nThe representation of 2\\(\\frac{2}{3}\\) into a fraction is:
\n\"\"
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n16 \u00f7 2\\(\\frac{2}{3}\\) = 6
\nThe quotient for the given division expression is: 6<\/p>\n

Question 22.
\n2\\(\\frac{5}{8}\\) \u00f7 13
\nAnswer:
\nThe given division expression is:
\n2\\(\\frac{5}{8}\\) \u00f7 13
\nNow,
\nThe representation of 2\\(\\frac{5}{8}\\) into a fraction is:
\n\"\"
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n2\\(\\frac{5}{8}\\) \u00f7 13 = \\(\\frac{21}{104}\\)
\nThe quotient for the given division expression is: \\(\\frac{21}{104}\\)<\/p>\n

Question 23.
\n3\\(\\frac{6}{7}\\) \u00f7 6\\(\\frac{3}{4}\\)
\nAnswer:
\nThe given division expression is:
\n3\\(\\frac{6}{7}\\) \u00f7 6\\(\\frac{3}{4}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n3\\(\\frac{6}{7}\\) \u00f7 6\\(\\frac{3}{4}\\) = \\(\\frac{4}{7}\\)
\nThe quotient for the given division expression is: \\(\\frac{4}{7}\\)<\/p>\n

Question 24.
\n2\\(\\frac{1}{3}\\) \u00f7 1\\(\\frac{1}{3}\\)
\nAnswer:
\nThe given division expression is:
\n2\\(\\frac{1}{3}\\) \u00f7 1\\(\\frac{1}{3}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n2\\(\\frac{1}{3}\\) \u00f7 1\\(\\frac{1}{3}\\) = \\(\\frac{7}{4}\\)
\nThe quotient for the given division expression is: \\(\\frac{7}{4}\\)<\/p>\n

Question 25.
\n3\\(\\frac{3}{4}\\) \u00f7 1\\(\\frac{1}{2}\\)
\nAnswer:
\nThe given division expression is:
\n3\\(\\frac{3}{4}\\) \u00f7 1\\(\\frac{1}{2}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n3\\(\\frac{3}{4}\\) \u00f7 1\\(\\frac{1}{2}\\) = \\(\\frac{5}{2}\\)
\nThe quotient for the given division expression is: \\(\\frac{5}{2}\\)<\/p>\n

Question 26.
\nBeth is making a rope ladder. Each step of the ladder is 2\\(\\frac{1}{3}\\), feet wide. Beth has a rope that is 21 feet long. How many steps can she make from the rope?
\n\"Envision
\nAnswer:
\nIt is given that
\nBeth is making a rope ladder. Each step of the ladder is 2\\(\\frac{1}{3}\\), feet wide. Beth has a rope that is 21 feet long
\nSo,
\nThe number of steps Berth can make from the rope = (The total length of the rope) \u00f7 (The length of each step of the ladder)
\n= 21 \u00f7 2\\(\\frac{1}{3}\\)
\nNow,
\nThe representation of 2\\(\\frac{1}{3}\\) into a fraction form is:
\n\"\"
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe number of steps Berth can make from the rope is: 9 steps<\/p>\n

Question 27.
\nThe area of this rectangle is 257\\(\\frac{1}{4}\\) in.2<\/sup>. Find side length w.
\n\"Envision
\nAnswer:
\nIt is given that
\nThe area of this rectangle is 257\\(\\frac{1}{4}\\) in.2<\/sup>
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nWe know that,
\nThe area of a rectangle = Length \u00d7 Width
\nSo,
\n257\\(\\frac{1}{4}\\) = 10\\(\\frac{1}{2}\\) \u00d7 w
\nw = 257\\(\\frac{1}{4}\\) \u00f7 10\\(\\frac{1}{2}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe side length (w) of the given rectangle is: \\(\\frac{49}{2}\\) in.<\/p>\n

In 28 and 29, use the picture.<\/strong><\/p>\n

Question 28.
\nThe larger room is twice as long as the smaller room. How long is the larger room?
\n\"Envision
\nAnswer:
\nIt is given that
\nThe larger room is twice as long as the smaller room.
\nNow,
\nThe given figure is:
\n\"Envision
\nSo,
\nThe length of the larger room = 2 \u00d7 (The length of the smaller room)
\n= 2 \u00d7 20\\(\\frac{4}{5}\\)
\nNow,
\nThe representation of 20\\(\\frac{4}{5}\\) into a fraction is:
\n\"\"
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe length of the larger room is: \\(\\frac{208}{5}\\) feet<\/p>\n

Question 29.
\nIf the length of the smaller room is divided into 4 equal parts, how long is each part?
\nAnswer:
\nIt is given that
\nThe length of the smaller room is divided into 4 equal parts
\nSo,
\nAccording to the given information,
\nThe length of each part in the smaller room = 20\\(\\frac{4}{5}\\) \u00f7 4
\nNow,
\nThe representation of 20\\(\\frac{4}{5}\\) into a fraction is:
\n\"\"
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe length of each part in the smaller room is: [latx]\\frac{26}{5}[\/latex] feet<\/p>\n

Question 30.
\nMake Sense and Persevere<\/strong> Luis has 3 pounds of ground turkey to make turkey burgers. He uses \\(\\frac{3}{8}\\) pound per burger to make 6 burgers. How many \\(\\frac{1}{4}\\)-pound burgers can Luis make with the remaining turkey?
\nAnswer:
\nIt is given that
\nLuis has 3 pounds of ground turkey to make turkey burgers. He uses \\(\\frac{3}{8}\\) pound per burger to make 6 burgers
\nSo,
\nThe remaining amount of ground Turkey Luis has = 3 – \\(\\frac{3}{8}\\)
\nNow,
\nBy using the Long Subtraction,
\n\"\"
\nSo,
\nThe remaining amount of ground turkey Luis has is: \\(\\frac{21}{8}\\) pounds
\nNow,
\nThe number of \\(\\frac{1}{4}\\)-pound burgers can Luis make with the remaining turkey =
\n\\(\\frac{21}{8}\\) \u00f7 \\(\\frac{1}{4}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe number of \\(\\frac{1}{4}\\)-pound burgers can Luis make with the remaining turkey is about 11 burgers<\/p>\n

Question 31.
\nHigher Order Thinking<\/strong> If 9 \u00d7 \\(\\frac{n}{5}\\) = 9 \u00f7 \\(\\frac{n}{5}\\), then what does n equal? Explain.
\nAnswer:
\nIt is given that
\n9 \u00d7 \\(\\frac{n}{5}\\) = 9 \u00f7 \\(\\frac{n}{5}\\)
\nSo,
\nAccording to the Property of Reciprocals,
\n9 \u00d7 \\(\\frac{n}{5}\\) = 9 \u00f7 \\(\\frac{n}{5}\\)
\n9 \u00d7 \\(\\frac{n}{5}\\) = 9 \u00d7 \\(\\frac{5}{n}\\)
\n\\(\\frac{n}{5}\\) = \\(\\frac{5}{n}\\)
\nn \u00d7 n = 5 \u00d7 5
\nn\u00b2 = 25
\nn = \\(\\sqrt{25}\\)
\nn = 5
\nHence, from the above,
\nWe can conclude that
\nThe value of “n” is: 5<\/p>\n

Question 32.
\nMargaret uses 1\\(\\frac{3}{4}\\) teaspoons of key lime zest to make 12 key lime cupcakes. She wants to make 30 cupcakes. How much key lime zest will Margaret use?
\nAnswer:
\nIt is given that
\nMargaret uses 1\\(\\frac{3}{4}\\) teaspoons of key lime zest to make 1\\(\\frac{1}{2}\\) key lime cupcakes
\nNow,
\nLet the number of teaspoons of key lime zest to make 30 cupcakes be x
\nSo,
\n1\\(\\frac{3}{4}\\) \u00d7 12 = x \u00d7 30
\nNow,
\nThe representation of 1\\(\\frac{3}{4}\\) into a fraction is:
\n\"\"
\nSo,
\nx = \\(\\frac{7}{4}\\) \u00d7 \\(\\frac{12}{30}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe number of teaspoons of key lime zest to make 30 cupcakes is: \\(\\frac{7}{10}\\) teaspoons of key lime zest<\/p>\n

Question 33.
\nUse Structure<\/strong> A gem store in Fort Lauderdale received a shipment of 1\\(\\frac{1}{2}\\) pounds of moonstone crystals. If these moonstone crystals were separated into 6 equal bags, how much would each bag weigh?
\nAnswer:
\nIt is given that
\nA gem store in Fort Lauderdale received a shipment of 1\\(\\frac{1}{2}\\) pounds of moonstone crystals.
\nSo,
\nThe weight of each bag of moonstone crystals = (The total weight of moonstone crystals) \u00f7 (The number of bags that moonstone crystals equally divided)
\n= 1\\(\\frac{1}{2}\\) \u00f7 6
\nNow,
\nThe representation of 1\\(\\frac{1}{2}\\) into a fraction is:
\n\"\"
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe weight of each bag of moonstone crystals is: \\(\\frac{1}{4}\\) pounds<\/p>\n

Question 34.
\nThe owner of an aquatic store used 17\\(\\frac{1}{2}\\) gallons of water to fill aquariums. He put 5\\(\\frac{5}{6}\\) gallons of water in each aquarium. How many aquariums did he fill?
\nAnswer:
\nIt is given that
\nThe owner of an aquatic store used 17\\(\\frac{1}{2}\\) gallons of water to fill aquariums. He put 5\\(\\frac{5}{6}\\) gallons of water in each aquarium.
\nSo,
\nThe number of aquariums did he fill = (The total amount of water used to fill aquariums) \u00f7 (The amount of water in each aquarium)
\n= 17\\(\\frac{1}{2}\\) \u00f7 5\\(\\frac{5}{6}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe number of aquariums did he fill is: 3 aquariums<\/p>\n

Question 35.
\nWrite an explanation to a friend about how you would estimate 17\\(\\frac{1}{5}\\) \u00f7 3\\(\\frac{4}{5}\\).
\nAnswer:
\nThe given division expression is:
\n17\\(\\frac{1}{5}\\) \u00f7 3\\(\\frac{4}{5}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n17\\(\\frac{1}{5}\\) \u00f7 3\\(\\frac{4}{5}\\) = \\(\\frac{86}{19}\\)<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 36.
\nA restaurant has 15\\(\\frac{1}{5}\\) pounds of alligator meat to make tasty alligator dishes.
\nPART A<\/strong>
\nEach pot of alligator stew requires 2\\(\\frac{3}{8}\\) pounds of alligator meat. Which solution shows how many pots of alligator stew can be made?
\nA. 36 pots; 15\\(\\frac{1}{5}\\) \u00d7 2\\(\\frac{3}{8}\\)
\nB. \\(\\frac{5}{32}\\) pot; 2\\(\\frac{3}{8}\\) \u00f7 15\\(\\frac{1}{5}\\)
\nC. 7 pots; 15\\(\\frac{1}{5}\\) \u00f7 2\\(\\frac{3}{8}\\)
\nD. 6 pots; 15\\(\\frac{1}{5}\\) \u00f7 2\\(\\frac{3}{8}\\)
\nAnswer:
\nIt is given that
\nA restaurant has 15\\(\\frac{1}{5}\\) pounds of alligator meat to make tasty alligator dishes. Each pot of alligator stew requires 2\\(\\frac{3}{8}\\) pounds of alligator meat
\nSo,
\nThe number of pots of alligator stew can be made = 15\\(\\frac{1}{5}\\) \u00f7 2\\(\\frac{3}{8}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nSo,
\nThe number of pots of alligator stew that can be made is about 6 pots
\nHence, from the above,
\nWe can conclude that
\nThe solution that shows how many pots of alligator stew can be made is:
\n\"\"<\/p>\n

PART B<\/strong>
\nThe restaurant could make a smaller pot of alligator stew that uses 1\\(\\frac{3}{5}\\) pounds of alligator meat. How many smaller pots of alligator stew can be made than the larger pots?
\nAnswer:
\nIt is given that
\nThe restaurant could make a smaller pot of alligator stew that uses 1\\(\\frac{3}{5}\\) pounds of alligator meat
\nSo,
\nThe number of smaller pots of stew that can be made = 15\\(\\frac{1}{5}\\) \u00f7 1\\(\\frac{3}{5}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nSo,
\nThe number of smaller pots of stew that can be made is about 10 pots
\nNow,
\nFrom Part (A),
\nWe can observe that
\nThe number of larger pots of stew that can be made is about 6 pots
\nSo,
\nThe number of smaller pots more than the larger pots = 10 – 6
\n= 4 pots
\nHence, from the above,
\nWe can conclude that
\nThe number of smaller pots more than the larger pots are: 4 pots<\/p>\n

Lesson 1.7 Solve Problems with Rational Numbers<\/h3>\n

ACTIVITY<\/strong><\/p>\n

Explain It!<\/strong>
\nJenna feeds her cat twice a day. She gives her cat \\(\\frac{3}{4}\\) can of cat food each time. Jenna is having a friend take care of her cat for 5 days. To prepare, she bought 8 cans of cat food. Did Jenna buy enough cat food?
\n\"Envision<\/p>\n

A. What do you need to know before you can answer the question?
\nAnswer:
\nIt is given that
\nJenna feeds her cat twice a day. She gives her cat \\(\\frac{3}{4}\\) can of cat food each time. Jenna is having a friend take care of her cat for 5 days. To prepare, she bought 8 cans of cat food.
\nHence,
\nThe hidden question for the given information is:
\nHow much food will be needed for Jenna’s cat for 5 days?<\/p>\n

B. How can you determine which operations to use to solve the problem?
\nAnswer:
\nIt is given that
\nJenna feeds her cat twice a day. She gives her cat \\(\\frac{3}{4}\\) can of cat food each time. Jenna is having a friend take care of her cat for 5 days. To prepare, she bought 8 cans of cat food.
\nSo,
\nThe operations we will use to solve the given problem is:
\na.
\nTo find the total amount of food Jenna gives her cat ——> Multiplication
\nb.
\nTo find the amount of food Jenna’s cat will need for 5 days ——->Multiplication
\nc.
\nTo find the amount of food Jenna’s cat will need for 8 days ——-> Multiplication
\nd.
\nTo know whether Jenna bought enough food or not ——> Comparison<\/p>\n

Focus on math practices<\/strong>
\nReasoning<\/strong> To find out whether she has enough cat food, Jenna multiplies, divides, and compares. Explain how Jenna may have solved the problem.
\nAnswer:
\nThe steps Jenna took to solve the given problem are:
\na.
\nTo find the total amount of food Jenna gives her cat ——> Multiplication
\nb.
\nTo find the amount of food Jenna’s cat will need for 5 days ——->Multiplication
\nc.
\nTo find the amount of food Jenna’s cat will need for 8 days ——-> Multiplication
\nd.
\nTo know whether Jenna bought enough food or not ——> Comparison<\/p>\n

?Essential Question<\/strong> How can you solve problems with rational numbers?
\nAnswer:
\nWe know that,
\nA “Rational number” is a number that can be written as a fraction.
\nFractions, integers, and decimals are all rational numbers.
\nExample:
\nFraction to Decimal:
\n\\(\\frac{1}{2}\\) = 0.5
\nDecimal to Fraction:
\na. 0.15= \\(\\frac{3}{20}\\)
\nb. 15\u00f75=3
\nc. 100\u00f75=20<\/p>\n

Try It!<\/strong>
\nThe farmer decided that he ordered enough planks for an arena that measured 115\\(\\frac{1}{4}\\) ft by 63\\(\\frac{1}{2}\\) ft. Is he correct? Explain.
\nAnswer:
\nIt is given that
\nThe farmer decided that he ordered enough planks for an arena that measured 115\\(\\frac{1}{4}\\) ft by 63\\(\\frac{1}{2}\\) ft.
\nSo,
\nThe perimeter of the arena = 2 \u00d7 (115\\(\\frac{1}{4}\\) + 63\\(\\frac{1}{2}\\))
\nNow,
\nBy using the Long Addition,
\n\"\"
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nSo,
\nThe perimeter of the arena is: \\(\\frac{715}{2}\\) ft
\nNow,
\nThe number of 8\\(\\frac{1}{2}\\)-foot planks needed = (The perimeter of the arena) \u00f7 8\\(\\frac{1}{2}\\)
\n= \\(\\frac{715}{2}\\) \u00f7 8\\(\\frac{1}{2}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nSo,
\nThe number of 8\\(\\frac{1}{2}\\)-foot planks needed is about 47 planks
\nBut,
\nIt is given that
\nA farmer needed 130 8\\(\\frac{1}{2}\\)-foot planks
\nSo,
\n47 < 130
\nHence, from the above,
\nWe can conclude that
\nThe farmer is not correct<\/p>\n

Convince Me!<\/strong> What questions do you need to answer to solve the Try It!?
\nAnswer:
\nThe questions you need to answer to solve the Try It is:
\na. The perimeter of the arena with the given length and width of the arena
\nb. The number of 8\\(\\frac{1}{2}\\)-foot planks for the perimeter we find in part (a)<\/p>\n

Try It!<\/strong>
\nThe number of runners who finish the marathon is 320. Runners donate $2.50 for each mile they run. How much money is donated? Explain.
\n\"Envision
\nAnswer:
\nIt is given that
\nThe number of runners who finish the marathon is 320. Runners donate $2.50 for each mile they run
\nNow,
\nThe total number of miles run by 320 runners = (The total length of the marathon) \u00d7 320
\n= 26.2 \u00d7 320
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nSo,
\nThe total number of miles run by 320 runners is: 8,384 miles
\nNow,
\nThe amount of money donated = (The total number of miles run by 320 runners) \u00d7 (The amount of money donated by each runner for each mile)
\n= 8,384 \u00d7 $2.50
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nSo,
\nThe amount of money donated is: $20,960
\nHence, from the above,
\nWe can conclude that
\n\"\"<\/p>\n

KEY CONCEPT<\/strong>
\nWhen solving multistep problems with fractions or decimals:
\n\u2022 decide the steps to use to solve the problem.
\n\u2022 choose the correct operations.
\n\u2022 identify the information you need from the problem.
\n\u2022 correctly use the information.
\n\u2022 calculate accurately.
\n\u2022 interpret solutions and check that the answer is reasonable.<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n?Essential Question<\/strong> How can you solve problems with rational numbers?
\nAnswer:
\nWe know that,
\nA “Rational number” is a number that can be written as a fraction.
\nFractions, integers, and decimals are all rational numbers.
\nExample:
\nFraction to Decimal:
\n\\(\\frac{1}{2}\\) = 0.5
\nDecimal to Fraction:
\na. 0.15= \\(\\frac{3}{20}\\)
\nb. 15\u00f75=3
\nc. 100\u00f75=20<\/p>\n

Question 2.
\nBe Precise<\/strong> Meghan has 5\\(\\frac{1}{4}\\) yards of fabric. She plans to use \\(\\frac{2}{3}\\) of the fabric to make 4 identical backpacks. To find how much fabric she will use to make the backpacks, Meghan multiplies 5\\(\\frac{1}{4}\\) by \\(\\frac{2}{3}\\). What else does Meghan need to do to find how much fabric she needs for each backpack?
\nAnswer:
\nIt is given that
\nMeghan has 5\\(\\frac{1}{4}\\) yards of fabric. She plans to use \\(\\frac{2}{3}\\) of the fabric to make 4 identical backpacks. To find how much fabric she will use to make the backpacks, Meghan multiplies 5\\(\\frac{1}{4}\\) by \\(\\frac{2}{3}\\)
\nNow,
\nThe amount of fabric needed to make 4 identical backpacks = (The amount of fabric she planned to use to make the backpacks) \u00d7 (The total length of the fabric) \u00d7 (The number of identical backpacks)
\n= \\(\\frac{2}{3}\\) \u00d7 5\\(\\frac{1}{4}\\) \u00d7 4
\nSo,
\nAside From the given information,
\nMeghan needed to multiply the number of identical backpacks to the result of \\(\\frac{2}{3}\\) \u00d7 5\\(\\frac{1}{4}\\)
\nHence, from the above,
\nWe can conclude that
\nTo find how much fabric Meghan needs for each backpack,
\nMeghan needed to multiply the number of identical backpacks to the result of \\(\\frac{2}{3}\\) \u00d7 5\\(\\frac{1}{4}\\)<\/p>\n

Question 3.
\nCritique Reasoning<\/strong> Each side of a square patio is 10.5 feet. The patio is made up of 1.5-foot by 1.5-foot square stones. What is the number of stones on the patio? Look at the solution below. Does it include all the steps needed to solve the problem? Explain.
\n\"Envision
\nAnswer:
\nIt is given that
\nEach side of a square patio is 10.5 feet. The patio is made up of 1.5-foot by 1.5-foot square stones
\nNow,
\nWe know that,
\nThe area of a square = Side \u00d7 Side
\nSo,
\nThe area of a square patio = 10.5 \u00d7 10.5
\nSo,
\nThe area of the square stones = 1.5 \u00d7 1.5
\nSo,
\nThe number of stones in the patio = (The area of a square patio) \u00f7 (The area of the square stones)
\n= (10.5 \u00d7 10.5) \u00f7 (1.5 \u00d7 1.5)
\n= 110.25 \u00f7 2.25
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe given solution did not include all the steps needed to solve the given problem
\nThe number of stones in the patio is: 49 stones<\/p>\n

Do You Know How?<\/strong><\/p>\n

Question 4.
\nDevon records 4 hours of reality shows on her DVR. She records comedy shows for \\(\\frac{3}{8}\\) of that amount of time. Devon watches all the reality and comedy shows in half-hour sittings.
\na. Find the number of hours of comedy shows that Devon records.
\nAnswer:
\nIt is given that
\nDevon records 4 hours of reality shows on her DVR. She records comedy shows for \\(\\frac{3}{8}\\) of that amount of time. Devon watches all the reality and comedy shows in half-hour sittings.
\nSo,
\nThe number of hours of comedy shows that Devon records = \\(\\frac{3}{8}\\) \u00d7 (The amount of time Devon records the reality shows on her DVR)
\n= \\(\\frac{3}{8}\\) \u00d7 4
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe number of hours of comedy shows that Devon records are: \\(\\frac{3}{2}\\) hours<\/p>\n

b. Find the total number of hours of reality and comedy shows that Devon records.
\nAnswer:
\nIt is given that
\nDevon records 4 hours of reality shows on her DVR. She records comedy shows for \\(\\frac{3}{8}\\) of that amount of time. Devon watches all the reality and comedy shows in half-hour sittings.
\nNow,
\nFrom part (a),
\nWe can observe that
\nThe number of hours of comedy shows that Devon records are: \\(\\frac{3}{2}\\) hours
\nSo,
\nThe total number of hours of reality and comedy shows that Devon records = (The number of hours of reality shows that Devon records) + (The number of hours of comedy shows that Devon records)
\n= \\(\\frac{3}{2}\\) + 4
\nNow,
\nBy using the Long Addition,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe total number of hours of reality and comedy shows that Devon records are: 5\\(\\frac{1}{2}\\) hours<\/p>\n

c. Find the number of half-hour sittings needed to watch all the shows.
\nAnswer:
\nIt is given that
\nDevon records 4 hours of reality shows on her DVR. She records comedy shows for \\(\\frac{3}{8}\\) of that amount of time. Devon watches all the reality and comedy shows in half-hour sittings.
\nNow,
\nFrom part (b),
\nWe can observe that
\nThe total number of hours of reality and comedy shows that Devon records are: 5\\(\\frac{1}{2}\\) hours
\nSo,
\nThe number of half-hour sittings needed to watch all the shows = (The total number of hours of reality and comedy shows that Devon records) \u00f7 \\(\\frac{1}{2}\\)
\n= 5\\(\\frac{1}{2}\\) \u00f7 \\(\\frac{1}{2}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe number of half-hour sittings needed to watch all the shows is: 11 half-hour sittings<\/p>\n

Question 5.
\nAn auto mechanic earns $498.75 in 35 hours during the week. His pay is $2.50 more per hour on weekends. If he works 6 hours on the weekend in addition to 35 hours during the week, how much does he earn?
\na. What questions do you need to answer to solve the problem?
\nAnswer:
\nIt is given that
\nAn auto mechanic earns $498.75 in 35 hours during the week. His pay is $2.50 more per hour on weekends. If he works 6 hours on the weekend in addition to 35 hours during the week
\nHence,
\nThe questions you need to answer to solve the given problem are:
\na. How much does an auto mechanic earn per hour?
\nb. How much does an auto mechanic earn per hour on weekends?<\/p>\n

b. How much does the auto mechanic earn? Explain.
\nAnswer:
\nIt is given that
\nAn auto mechanic earns $498.75 in 35 hours during the week. His pay is $2.50 more per hour on weekends. If he works 6 hours on the weekend in addition to 35 hours during the week
\nSo,
\nThe amount of money does an auto mechanic earn per hour = (The amount an auto mechanic earns during the week) \u00f7 (Number of hours)
\n= $498.75 \u00f7 35
\nNow,
\nBy using the Long Division,
\n\"\"
\nSo,
\nThe amount of money an auto mechanic earns per hour is: $14.25
\nNow,
\nThe amount of money an auto mechanic earns on weekends per hour = $2.50 + $14.25
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nSo,
\nThe amount of money an auto mechanic earns during weekends is: $16.75
\nNow,
\nThe total amount of money an auto mechanic earns = (The total amount of money an auto mechanic earns during weekdays) + (The total amount of money an auto mechanic earns on weekends)
\n= $498.75 + (6 \u00d7 $16.75)
\n= $498.75 + $100.5
\n= $599.25
\nHence, from the above,
\nWe can conclude that
\nThe total amount of money an auto mechanic earns is: $599.25<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Scan for Multimedia<\/strong><\/p>\n

In 6-8, use the picture at the right.<\/strong><\/p>\n

Question 6.
\nYou buy 3.17 pounds of apples, 1.25 pounds of pears, and 2.56 pounds of oranges. What is your total bill rounded to the nearest cent?
\n\"Envision
\nAnswer:
\nIt is given that
\nYou buy 3.17 pounds of apples, 1.25 pounds of pears, and 2.56 pounds of oranges
\nSo,
\nYour total bill rounded to the nearest cent = [(The total weight of apples) \u00d7 (The cost of each apple)] + [(The total weight of pears) \u00d7 (The cost of each pear)] + [(The total weight of oranges) \u00d7 (The cost of each orange)]
\n= (3.17 \u00d7 $0.99) + (1.25 \u00d7 $1.19) + (2.56 \u00d7 $1.09)
\n= $3.13 + $1.48 + 2.79
\n= $4.61 + $2.79
\n= $7.40
\nHence, from the above,
\nWe can conclude that
\nYour total bill rounded to the nearest cent is: $7.40<\/p>\n

Question 7.
\nA student pays for 8.9 pounds of apples with a $10 bill. How much change does the student receive?
\na. What do you do first to solve the problem?
\nAnswer:
\nIt is given that
\nA student pays for 8.9 pounds of apples with a $10 bill
\nHence,
\nThe hidden question you need to answer for the given problem is:
\nWhat is the total bill you have to pay for 8.9 pounds of apples for $0.99 each?<\/p>\n

b. What do you do next?
\nAnswer:
\nAfter completing the solution in part (a),
\nThe thing you have to do next is:
\nSubtract the amount of money the student received by subtracting the value of the total bill you have to pay for 8.9 pounds of apples for $0.99 each from $10<\/p>\n

Question 8.
\nA customer pays $3.27 for oranges and $4.76 for pears. How many pounds of fruit does the customer buy?
\na. What do you do first to solve the problem?
\nAnswer:
\nIt is given that
\nA customer pays $3.27 for oranges and $4.76 for pears
\nHence,
\nThe hidden question you have to answer for the given problem is:
\nHow many pounds of oranges and pears does the customer can buy?<\/p>\n

b. What do you do next?
\nAnswer:
\nAfter find the solution to the question present in part (a),
\nThe thing you have to do next is:
\nAdd the number of pounds of oranges and pears to find the total number of pounds of fruit doe the customer bought<\/p>\n

Question 9.
\nCritique Reasoning<\/strong> Students put 2\\(\\frac{1}{4}\\) pounds of trail mix into bags that each weigh \\(\\frac{3}{8}\\) pound. They bring \\(\\frac{2}{3}\\) of the bags of trail mix on a hiking trip. Can you determine how many bags of trail mix are left by completing just one step? Explain.
\nAnswer:
\nIt is given that
\nStudents put 2\\(\\frac{1}{4}\\) pounds of trail mix into bags that each weigh \\(\\frac{3}{8}\\) pound. They bring \\(\\frac{2}{3}\\) of the bags of trail mix on a hiking trip.
\nSo,
\nThe number of bags of trail mix = (The total number of pounds of trail mix) \u00f7 (The weight of each bag of trail mix)
\n= 2\\(\\frac{1}{4}\\) \u00f7 \\(\\frac{3}{8}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nSo,
\nThe number of bags of trail mix is: 6 bags
\nNow,
\nThe \\(\\frac{2}{3}\\) of the bags of trail mix = 6 \u00d7 \\(\\frac{2}{3}\\)
\n= \\(\\frac{6 \u00d7 2}{3}\\)
\n= 4 bags
\nSo,
\nThe number of bags of trail mix left = (The total number of bags of trail mix) – (The \\(\\frac{2}{3}\\) of the bags of trail mix)
\n= 6 – 4
\n= 2 bags
\nHence, from the above,
\nWe can conclude that
\nThe number of bags of trail mix that are left is: 2 bags
\nWe can’t determine the number of bags of trail mix by completing just one step<\/p>\n

Question 10.
\nThree-fifths of the T-shirts in a T-shirt shop are blue. Five-eighths of those T-shirts are on sale. One-third of the blue T-shirts that are on sale are size medium. What fraction of the shop’s T-shirts are blue T-shirts that are on sale and are size medium? Explain.
\nAnswer:
\nIt is given that
\nThree-fifths of the T-shirts in a T-shirt shop are blue. Five-eighths of those T-shirts are on sale. One-third of the blue T-shirts that are on sale are size medium.
\nSo,
\nThe number of blue T-shirts are: \\(\\frac{3}{5}\\) shirts
\nThe number of blue T-shirts that are on sale = \\(\\frac{3}{5}\\) \u00d7 \\(\\frac{5}{8}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nSo,
\nThe number of blue T-shirts that are on sale is: \\(\\frac{3}{8}\\) shirts
\nNow,
\nThe number of blue T-shirts that are on sale and are medium size = \\(\\frac{3}{8}\\) \u00d7 \\(\\frac{1}{3}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nSo,
\nThe number of blue shirts that are on sale and are medium size is: \\(\\frac{1}{8}\\) shirts
\nHence, from the above,
\nWe can conclude that
\nThe number of blue T-shirts that are on sale is: \\(\\frac{3}{8}\\) shirts
\nThe number of blue shirts that are on sale and are medium size is: \\(\\frac{1}{8}\\) shirts<\/p>\n

In 11 and 12, use the diagram.<\/strong>
\nA community garden is made up of three gardens: a vegetable garden, an herb garden, and a flower garden.<\/p>\n

Question 11.
\nThe area of the vegetable garden is 0.4 of the area of the community garden. What is the area of the vegetable garden?
\n\"Envision
\nAnswer:
\nIt is given that
\nA community garden is made up of three gardens: a vegetable garden, an herb garden, and a flower garden.
\nThe area of the vegetable garden is 0.4 of the area of the community garden.
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nThe area of the community garden = (Length of the community garden) \u00d7 (Width of the community garden)
\n= 6.2 \u00d7 4.5
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nSo,
\nThe area of the community garden is: 27.9 m\u00b2
\nNow,
\nThe area of the vegetable garden = 0.4 \u00d7 (The area of the community garden)
\n= 0.4 \u00d7 27.9
\n= 11.16 m\u00b2
\nHence, from the above,
\nWe can conclude that
\nThe area of the vegetable garden is: 11.16 m\u00b2<\/p>\n

Question 12.
\nThe area of the flower garden is 9.7 square meters greater than the herb garden. What is the area of the flower garden?
\nAnswer:
\nIt is given that
\nThe area of the flower garden is 9.7 square meters greater than the herb garden.
\nNow,
\nThe given figure is:
\n\"Envision
\nNow,
\nFrom the given figure,
\nWe can observe that
\nThe area of the Herb garden = 2.2 \u00d7 1.6
\n= 3.52 m\u00b2
\nSo,
\nThe area of the flower garden = 9.7 + (The area of the flower garden)
\n= 9.7 + 3.52
\n= 13.22 m\u00b2
\nHence, from the above,
\nWe can conclude that
\nThe area of the flower garden is: 13.22 m\u00b2<\/p>\n

Question 13.
\nReasoning<\/strong> At the end of a party, \\(\\frac{3}{4}\\) cup of smoked fish dip is left. Jim divides \\(\\frac{4}{5}\\) of the leftover smoked fish dip equally between 2 friends. How much dip does each friend get?
\nAnswer:
\nIt is given that
\nAt the end of a party, \\(\\frac{3}{4}\\) cup of smoked fish dip is left. Jim divides \\(\\frac{4}{5}\\) of the leftover smoked fish dip equally between 2 friends.
\nNow,
\nThe \\(\\frac{4}{5}\\) of the leftover smoked fish = \\(\\frac{4}{5}\\) \u00d7 \\(\\frac{3}{4}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nSo,
\nThe \\(\\frac{4}{5}\\) of the leftover smoked fish is: \\(\\frac{3}{5}\\)
\nNow,
\nThe amount of dip each friend get = \\(\\frac{3}{5}\\) \u00f7 2
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe amount of dip each friend get is: \\(\\frac{3}{10}\\) of the leftover smoked fish dip<\/p>\n

Question 14.
\nStudents are planning a 3-day hiking trip in the Everglades. The hike covers a distance of 18.5 kilometers. The students hike 0.28 of the total distance on the first day. If they split the remaining distance equally between the second and third days, how far will they hike on day 3?
\nAnswer:
\nIt is given that
\nStudents are planning a 3-day hiking trip in the Everglades. The hike covers a distance of 18.5 kilometers. The students hike 0.28 of the total distance on the first day and they split the remaining distance equally between the second and third days
\nSo,
\nThe remaining distance of the hike after the first day = (The total distance of hike) – (The distance of hike on the first day)
\n= 18.5 – (0.28 \u00d7 18.5)
\n= 18.5 – 5.18
\n= 13.32 kilometers
\nNow,
\nThe distance of the hike on the second and the third days which are split equally = (The remaining distance of the hike after the first day) \u00f7 2
\n= 13.32 \u00f72
\n= 6.66 kilometers
\nSo,
\nThe amount of distance the students hiked on the third day is: 6.66 kilometers
\nHence, from the above,
\nWe can conclude that
\nThe amount of distance the students hiked on the third day is: 6.66 kilometers<\/p>\n

Question 15.
\nHigher-Order Thinking<\/strong> Kelly buys three containers of potato salad at the deli. She brings \\(\\frac{4}{5}\\) of the potato salad to a picnic. How many pounds of potato salad does Kelly bring to the picnic? Describe two different ways to solve the problem.
\n\"Envision
\nAnswer:
\nIt is given that
\nKelly buys three containers of potato salad at the deli. She brings \\(\\frac{4}{5}\\) of the potato salad to a picnic
\nNow,
\nThe given figure is:
\n\"Envision
\nSo,
\nThe total weight of the containers of potato salad = 1.03 + 1.12 + 1.6
\n= 3.75 pounds
\nSo,
\nThe total weight of potato salad Kelly brought = 3 \u00d7 (The total weight of the containers of potato salad)
\n= 3 \u00d7 3.75
\n= 11.25 pounds
\nNow,
\nThe amount of potato salad Kelly brought to picnic = \\(\\frac{4}{5}\\) \u00d7 (The total weight of potato salad Kelly brought)
\n= \\(\\frac{4}{5}\\) \u00d7 11.25
\n= 9 pounds
\nHence, from the above,
\nWe can conclude that
\nThe amount of potato salad Kelly brought to the picnic is: 9 pounds<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 16.
\nStudents make 84\\(\\frac{1}{2}\\) ounces of liquid soap for a craft fair. They put the soap in 6\\(\\frac{1}{2}\\)-ounce bottles and sell each bottle for $5.50. Which expression shows how much students earn if they sell all the bottles of liquid soap?
\n\"Envision
\nAnswer:
\nIt s given that
\nStudents make 84\\(\\frac{1}{2}\\) ounces of liquid soap for a craft fair. They put the soap in 6\\(\\frac{1}{2}\\)-ounce bottles and sell each bottle for $5.50.
\nSo,
\nThe number of bottles of liquid soap = 84\\(\\frac{1}{2}\\) \u00f7 6\\(\\frac{1}{2}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nSo,
\nThe number of bottles of liquid soap is: 13 bottles
\nNow,
\nThe cost of the 13 bottles of liquid soap = 13 \u00d7 (The cost of each liquid soap)
\n= 13 \u00d7 $5.50
\n= $71.5
\nHence, from the above,
\nWe can conclude that
\nThe expression shows how much students earn if they sell all the bottles of liquid soap is:
\n\"\"<\/p>\n

Question 17.
\nClaire mowed 5 lawns last week. She mowed each lawn in \\(\\frac{7}{12}\\) hour. She mowed the same lawns this week in \\(\\frac{5}{12}\\) hour each using her new lawnmower. How many times longer was Claire’s time to mow all the lawns last week than this week?
\nAnswer:
\nIt is given that
\nClaire mowed 5 lawns last week. She mowed each lawn in \\(\\frac{7}{12}\\) hour. She mowed the same lawns this week in \\(\\frac{5}{12}\\) hour each using her new lawnmower
\nSo,
\nThe number of hours Claire mowed last week = (The number of lawns) \u00d7 (The amount of time each lawn was mowed last week)
\n= 5 \u00d7 \\(\\frac{7}{12}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nSo,
\nThe number of hours Claire mowed last week is: \\(\\frac{35}{12}\\) hours
\nNow,
\nThe number of hours Claire mowed this week = (The number of lawns) \u00d7 (The amount of time each lawn was mowed this week)
\n= 5 \u00d7 \\(\\frac{5}{12}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nSo,
\nThe number of hours Claire mowed this week is: \\(\\frac{25}{12}\\) hours
\nNow,
\nThe difference between the number of hours Claire mowed the last week and this week = \\(\\frac{35}{12}\\) – \\(\\frac{25}{12}\\)
\n= \\(\\frac{35 – 25}{12}\\)
\n= \\(\\frac{10}{12}\\)
\n= \\(\\frac{5}{6}\\) hours
\nHence, from the above,
\nWe can conclude that
\nClaire’s time to mow all the lawns last week is \\(\\frac{5}{6}\\) times longer than this week<\/p>\n

Topic 1 REVIEW<\/h3>\n

? Topic Essential Question<\/strong>
\nHow can you fluently add, subtract, multiply, and divide decimals? How can you multiply and divide fractions?
\nAnswer:
\nThe steps to add, subtract, multiply, and divide decimals are:
\na. Write the numbers vertically with the decimals lined up (if the number is a whole number then the decimal goes on the right end of the number)
\nb. Add zeros to make sure there is the same number of digits in each number.
\nc. Add or subtract as normal.
\nd. Multiply just as you would if the numbers were all whole numbers and add at the end.
\ne. After that count how many decimals places the two factors have (start at the far right of each number) and then give the answer that the total number of decimal places.
\nf. To divide decimals, always use long division.
\nThe process to multiply and divide fractions is:
\na. Dividing two fractions is the same as multiplying the first fraction by the reciprocal of the second fraction.
\nb. The first step to dividing fractions is to find the reciprocal (reverse the numerator and denominator) of the second fraction.
\nc. Next, multiply the two numerators. Then, multiply the two denominators.<\/p>\n

Vocabulary Review<\/strong><\/p>\n

Complete each definition and then provide an example of each vocabulary word.<\/strong><\/p>\n

Vocabulary<\/strong><\/p>\n

reciprocal
\ndividend
\nfraction
\nproduct<\/p>\n

\"Envision
\nAnswer:
\n\"\"<\/p>\n

Use Vocabulary in Writing<\/strong><\/p>\n

Explain how to use multiplication to find the value of \\(\\frac{1}{3}\\) \u00f7 \\(\\frac{9}{5}\\) Use the words multiplication, divisor, quotient, and reciprocal in your explanation.
\nAnswer:
\nThe given division expression is:
\n\\(\\frac{1}{3}\\) \u00f7 \\(\\frac{9}{5}\\)
\nWhere,
\n\\(\\frac{1}{3}\\) is the dividend
\n\\(\\frac{9}{5}\\) is the divisor
\nNow,
\nThe “Reciprocal” of \\(\\frac{9}{5}\\) is: \\(\\frac{5}{9}\\)
\nSo,
\n\\(\\frac{1}{3}\\) \u00f7 \\(\\frac{9}{5}\\)
\n= \\(\\frac{1}{3}\\) \u00d7 \\(\\frac{5}{9}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{1}{3}\\) \u00f7 \\(\\frac{9}{5}\\) = \\(\\frac{5}{27}\\)<\/p>\n

Concepts and Skills Review<\/strong><\/p>\n

LESSON 1.1 Fluently Add, Subtract and Multiply Decimals<\/strong><\/p>\n

Quick Review<\/strong>
\nTo add or subtract decimals, line up the decimal points so that place-value positions correspond. Add or subtract as you would with whole numbers, and place the decimal point in the answer. To multiply decimals, multiply as you would with whole numbers, then place the decimal point in the product by starting at the right and counting the number of places equal to the sum of the number of decimal places in each factor.<\/p>\n

Example<\/strong>
\nAdd, subtract, or multiply.
\n\"Envision<\/p>\n

Practice<\/strong>
\nAdd, subtract, or multiply.<\/strong><\/p>\n

Question 1.
\n91.2 + 89.9
\nAnswer:
\nThe given Addition expression is: 91.2 + 89.9
\nNow,
\nBy using the Long Addition,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n91.2 + 89.9 = 181.1<\/p>\n

Question 2.
\n902.3 – 8.8
\nAnswer:
\nThe given Subtraction expression is: 902.3 – 8.8
\nNow,
\nBy using the Long Subtraction,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n902.3 – 8.8 = 893.5<\/p>\n

Question 3.
\n5 \u00d7 98.2
\nAnswer:
\nThe given Multiplication Expression is: 5 \u00d7 98.2
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n5 \u00d7 98.2 = 491<\/p>\n

Question 4.
\n4 \u00d7 0.21
\nAnswer:
\nThe given Multiplication Expression is: 4 \u00d7 0.21
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n4 \u00d7 0.21 = 0.84<\/p>\n

Question 5.
\n62.99 \u2013 10.83
\nAnswer:
\nThe given Subtraction Expression is: 62.99 – 10.83
\nNow,
\nBy using the Long Subtraction,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n62.99 – 10.83 = 52.16<\/p>\n

Question 6.
\n423.22 + 98.30
\nAnswer:
\nThe given Addition Expression is: 423.22 + 98.30
\nNow,
\nBy using the Long Addition,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n423.22 + 98.30 = 521.52<\/p>\n

Question 7.
\n4.4 \u00d7 6
\nAnswer:
\nThe given Multiplication Expression is: 4.4 \u00d7 6
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n4.4 \u00d7 6 = 26.4<\/p>\n

Question 8.
\n7 \u00d7 21.6
\nAnswer:
\nThe given Multiplication Expression is: 7 \u00d7 21.6
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n7 \u00d7 21.6 = 151.2<\/p>\n

Question 9.
\n24.52 – 9.6
\nAnswer:
\nThe given Subtraction Expression is: 24.52 – 9.6
\nNow,
\nBy using the Long Subtraction,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n24.52 – 9.6 = 14.92<\/p>\n

Question 10.
\n369.45 + 32.42
\nAnswer:
\nThe given Addition Expression is: 369.45 + 32.42
\nNow,
\nBy using the Long Addition,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n369.45 + 32.42 = 401.87<\/p>\n

Question 11.
\n12.5 \u00d7 163.2
\nAnswer:
\nThe given Multiplication Expression is: 12.5 \u00d7 163.2
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n12.5 \u00d7 163.2 = 2,040<\/p>\n

Question 12.
\n16 \u00d7 52.3
\nAnswer:
\nThe given Multiplication Expression is: 16 \u00d7 52.3
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n16 \u00d7 52.3 = 836.8<\/p>\n

Question 13.
\n121.3 + 435.7
\nAnswer:
\nThe given Addition Expression is: 121.3 + 435.7
\nNow,
\nBy using the Long Addition,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n121.3 + 435.7 = 557<\/p>\n

Question 14.
\n201.7 – 104.6
\nAnswer:
\nThe given Subtraction Expression is: 201.7 – 104.6
\nNow,
\nBy using the Long Subtraction,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n201.7 – 104.6 = 97.1<\/p>\n

LESSON 1.2 Fluently Divide Whole Numbers and Decimals<\/strong><\/p>\n

Quick Review<\/strong>
\nTo divide decimals, multiply the divisor and the dividend by the same power of 10 so that the divisor is a whole number. Then use an algorithm for whole-number division.<\/p>\n

Example<\/strong>
\nFind 2.75 \u00f7 0.05.
\n\"Envision<\/p>\n

Practice <\/strong>
\nDivide.<\/strong><\/p>\n

Question 1.
\n9.6 \u00f7 1.6
\nAnswer:
\nThe given Division Expression is: 9.6 \u00f7 1.6
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n9.6 \u00f7 1.6 = 6<\/p>\n

Question 2.
\n48.4 \u00f7 0.4
\nAnswer:
\nThe given Division Expression is: 48.4 \u00f7 0.4
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n48.4 \u00f7 0.4 = 121<\/p>\n

Question 3.
\n13.2 \u00f7 0.006
\nAnswer:
\nThe given Division Expression is: 13.2 \u00f7 0.006
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n13.2 \u00f7 0.006 = 2,200<\/p>\n

Question 4.
\n10.8 \u00f7 0.09
\nAnswer:
\nThe given Division Expression is: 10.8 \u00f7 0.09
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n10.8 \u00f7 0.09 = 120<\/p>\n

Question 5.
\n45 \u00f7 4.5
\nAnswer:
\nThe given Division expression is: 45 \u00f7 4.5
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n45 \u00f7 4.5 = 10<\/p>\n

Question 6.
\n1,008 \u00f7 1.8
\nAnswer:
\nThe given Division Expression is: 1,008 \u00f7 1.8
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n1,008 \u00f7 1.8 = 560<\/p>\n

Question 7.
\n1.26 \u00f7 0.2
\nAnswer:
\nThe given Division Expression is: 1.26 \u00f7 0.2
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n1.26 \u00f7 0.2 = 6.3<\/p>\n

Question 8.
\n2.24 \u00f7 3.2
\nAnswer:
\nThe given Division Expression is: 2.24 \u00f7 3.2
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n2.24 \u00f7 3.2 = 0.7<\/p>\n

Question 9.
\n35.75 \u00f7 55
\nAnswer:
\nThe given Division Expression is: 35.75 \u00f7 55
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n35.75 \u00f7 55 = 0.65<\/p>\n

Question 10.
\n120.4 \u00f7 602
\nAnswer:
\nThe given Division Expression is: 120.4 \u00f7 602
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n120.4 \u00f7 602 = 0.2<\/p>\n

Question 11.
\n330 \u00f7 5.5
\nAnswer:
\nThe given Division Expression is: 330 \u00f7 5.5
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n330 \u00f7 5.5 = 60<\/p>\n

Question 12.
\n1.08 \u00f7 0.027
\nAnswer:
\nThe given Division Expression is: 1.08 \u00f7 0.027
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n1.08 \u00f7 0.027 = 40<\/p>\n

LESSON 1.3 Multiply Fractions<\/strong><\/p>\n

Quick Review<\/strong>
\nMultiply the numerators to find the numerator of the product. Multiply the denominators to find the denominator of the product.<\/p>\n

Example<\/strong>
\n\"Envision
\n10 of the 18 rectangles are in the overlap area. So, \\(\\frac{2}{3}\\) \u00d7 \\(\\frac{5}{6}\\) = \\(\\frac{10}{8}\\) or \\(\\frac{5}{9}\\).<\/p>\n

Practice<\/strong>
\nFind each product.<\/strong><\/p>\n

Question 1.
\n\\(\\frac{2}{3}\\) \u00d7 \\(\\frac{3}{8}\\)
\nAnswer:
\nThe given Multiplication Expression is:
\n\\(\\frac{2}{3}\\) \u00d7 \\(\\frac{3}{8}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{2}{3}\\) \u00d7 \\(\\frac{3}{8}\\) = \\(\\frac{1}{4}\\)<\/p>\n

Question 2.
\n\\(\\frac{1}{4}\\) \u00d7 \\(\\frac{3}{5}\\)
\nAnswer:
\nThe given Multiplication Expression is:
\n\\(\\frac{1}{4}\\) \u00d7 \\(\\frac{3}{5}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{1}{4}\\) \u00d7 \\(\\frac{3}{5}\\) = \\(\\frac{3}{20}\\)<\/p>\n

Question 3.
\n\\(\\frac{1}{6}\\) \u00d7 \\(\\frac{1}{8}\\)
\nAnswer:
\nThe given Multiplication Expression is:
\n\\(\\frac{1}{6}\\) \u00d7 \\(\\frac{1}{8}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{1}{6}\\) \u00d7 \\(\\frac{1}{8}\\) = \\(\\frac{1}{48}\\)<\/p>\n

Question 4.
\n\\(\\frac{4}{7}\\) \u00d7 \\(\\frac{4}{7}\\)
\nAnswer:
\nThe given Multiplication Expression is:
\n\\(\\frac{4}{7}\\) \u00d7 \\(\\frac{4}{7}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{4}{7}\\) \u00d7 \\(\\frac{4}{7}\\) = \\(\\frac{16}{49}\\)<\/p>\n

Question 5.
\n\\(\\frac{6}{7}\\) \u00d7 \\(\\frac{1}{2}\\)
\nAnswer:
\nThe given Multiplication Expression is:
\n\\(\\frac{6}{7}\\) \u00d7 \\(\\frac{1}{2}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{6}{7}\\) \u00d7 \\(\\frac{1}{2}\\) = \\(\\frac{3}{7}\\)<\/p>\n

Question 6.
\n\\(\\frac{3}{8}\\) \u00d7 \\(\\frac{8}{3}\\)
\nAnswer:
\nThe given Multiplication Expression is:
\n\\(\\frac{3}{8}\\) \u00d7 \\(\\frac{8}{3}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{3}{8}\\) \u00d7 \\(\\frac{8}{3}\\) = 1<\/p>\n

Question 7.
\n\\(\\frac{2}{3}\\) \u00d7 \\(\\frac{1}{3}\\)
\nAnswer:
\nThe given Multiplication Expression is:
\n\\(\\frac{2}{3}\\) \u00d7 \\(\\frac{1}{3}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{2}{3}\\) \u00d7 \\(\\frac{1}{3}\\) = \\(\\frac{2}{9}\\)<\/p>\n

Question 8.
\n\\(\\frac{7}{8}\\) \u00d7 \\(\\frac{3}{2}\\)
\nAnswer:
\nThe given Multiplication Expression is:
\n\\(\\frac{7}{8}\\) \u00d7 \\(\\frac{3}{2}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{7}{8}\\) \u00d7 \\(\\frac{3}{2}\\) = \\(\\frac{21}{16}\\)<\/p>\n

Question 9.
\n2\\(\\frac{1}{3}\\) \u00d7 4\\(\\frac{1}{5}\\)
\nAnswer:
\nThe given Multiplication Expression is:
\n2\\(\\frac{1}{3}\\) \u00d7 4\\(\\frac{1}{5}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n2\\(\\frac{1}{3}\\) \u00d7 4\\(\\frac{1}{5}\\) = \\(\\frac{49}{5}\\)<\/p>\n

Question 10.
\n4\\(\\frac{1}{2}\\) \u00d7 6\\(\\frac{2}{3}\\)
\nAnswer:
\nThe given Multiplication Expression is:
\n4\\(\\frac{1}{2}\\) \u00d7 6\\(\\frac{2}{3}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n4\\(\\frac{1}{2}\\) \u00d7 6\\(\\frac{2}{3}\\) = 30<\/p>\n

Question 11.
\n3\\(\\frac{3}{5}\\) \u00d7 2\\(\\frac{5}{7}\\)
\nAnswer:
\nThe given Multiplication Expression is:
\n3\\(\\frac{3}{5}\\) \u00d7 2\\(\\frac{5}{7}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n3\\(\\frac{3}{5}\\) \u00d7 2\\(\\frac{5}{7}\\) = \\(\\frac{342}{35}\\)<\/p>\n

Question 12.
\n14\\(\\frac{2}{7}\\) \u00d7 4\\(\\frac{3}{10}\\)
\nAnswer:
\nThe given Multiplication Expression is:
\n14\\(\\frac{2}{7}\\) \u00d7 4\\(\\frac{3}{10}\\)
\nNow,
\nBy using the Long Multiplication,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n14\\(\\frac{2}{7}\\) \u00d7 4\\(\\frac{3}{10}\\) = \\(\\frac{430}{7}\\)<\/p>\n

LESSONS 1.4 AND 1.5 Understand and Divide with Fractions<\/strong><\/p>\n

Quick Review<\/strong>
\nTo divide by a fraction, use the reciprocal of the divisor to rewrite the problem as a multiplication problem.
\nPractice Find each quotient.<\/p>\n

Example<\/strong>
\nFind 4 \u00f7 \\(\\frac{4}{5}\\).
\n\"Envision<\/p>\n

Find \\(\\frac{3}{4}\\) \u00f7 \\(\\frac{4}{5}\\).
\n\"Envision<\/p>\n

Practice<\/strong><\/p>\n

Find each Quotient<\/strong><\/p>\n

Question 1.
\n7 \u00f7 \\(\\frac{1}{2}\\)
\nAnswer:
\nThe given Division Expression is:
\n7 \u00f7 \\(\\frac{1}{2}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n7 \u00f7 \\(\\frac{1}{2}\\) = 14
\nThe quotient for the given division expression is: 14<\/p>\n

Question 2.
\n6 \u00f7 \\(\\frac{2}{5}\\)
\nAnswer:
\nThe given Division Expression is:
\n6 \u00f7 \\(\\frac{2}{5}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n6 \u00f7 \\(\\frac{2}{5}\\) = 15
\nThe quotient for the given division expression is: 15<\/p>\n

Question 3.
\n2 \u00f7 \\(\\frac{1}{8}\\)
\nAnswer:
\nThe given Division Expression is:
\n2 \u00f7 \\(\\frac{1}{8}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n2 \u00f7 \\(\\frac{1}{8}\\) = 16
\nThe quotient for the given division expression is: 16<\/p>\n

Question 4.
\n8 \u00f7 \\(\\frac{4}{9}\\)
\nAnswer:
\nThe given Division Expression is:
\n8 \u00f7 \\(\\frac{4}{9}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n8 \u00f7 \\(\\frac{4}{9}\\) = 18
\nThe quotient for the given division expression is: 18<\/p>\n

Question 5.
\n\\(\\frac{1}{2}\\) \u00f7 \\(\\frac{1}{4}\\)
\nAnswer:
\nThe given Division Expression is:
\n\\(\\frac{1}{2}\\) \u00f7 \\(\\frac{1}{4}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{1}{2}\\) \u00f7 \\(\\frac{1}{4}\\) = 2
\nThe quotient for the given division expression is: 2<\/p>\n

Question 6.
\n\\(\\frac{8}{10}\\) \u00f7 \\(\\frac{1}{5}\\)
\nAnswer:
\nThe given Division Expression is:
\n\\(\\frac{8}{10}\\) \u00f7 \\(\\frac{1}{5}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{8}{10}\\) \u00f7 \\(\\frac{1}{5}\\) = 4
\nThe quotient for the given division expression is: 4<\/p>\n

Question 7.
\n\\(\\frac{5}{6}\\) \u00f7 \\(\\frac{3}{8}\\)
\nAnswer:
\nThe given Division Expression is:
\n\\(\\frac{5}{6}\\) \u00f7 \\(\\frac{3}{8}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{5}{6}\\) \u00f7 \\(\\frac{3}{8}\\) = \\(\\frac{20}{9}\\)
\nThe quotient for the given division expression is: \\(\\frac{20}{9}\\)<\/p>\n

Question 8.
\n\\(\\frac{1}{3}\\) \u00f7 \\(\\frac{1}{2}\\)
\nAnswer:
\nThe given Division Expression is:
\n\\(\\frac{1}{3}\\) \u00f7 \\(\\frac{1}{2}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nwe can conclude that
\n\\(\\frac{1}{3}\\) \u00f7 \\(\\frac{1}{2}\\) = \\(\\frac{2}{3}\\)
\nThe quotient for the given division expression is: \\(\\frac{2}{3}\\)<\/p>\n

Question 9.
\n5 \u00f7 \\(\\frac{5}{16}\\)
\nAnswer:
\nThe given Division Expression is:
\n5 \u00f7 \\(\\frac{5}{16}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n5 \u00f7 \\(\\frac{5}{16}\\) = 16
\nThe quotient for the given division expression is: 16<\/p>\n

Question 10.
\n\\(\\frac{7}{12}\\) \u00f7 \\(\\frac{3}{4}\\)
\nAnswer:
\nThe given Divisio Expression is:
\n\\(\\frac{7}{12}\\) \u00f7 \\(\\frac{3}{4}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{7}{12}\\) \u00f7 \\(\\frac{3}{4}\\) = \\(\\frac{7}{9}\\)
\nThe quotient for the given division expression is: \\(\\frac{7}{9}\\)<\/p>\n

Question 11.
\n20 \u00f7 \\(\\frac{5}{6}\\)
\nAnswer;
\nThe given Division Expression is:
\n20 \u00f7 \\(\\frac{5}{6}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n20 \u00f7 \\(\\frac{5}{6}\\) = 24
\nThe quotient for the given division expression is: 24<\/p>\n

Question 12.
\n16 \u00f7 \\(\\frac{1}{4}\\)
\nAnswer:
\nThe given Division Expression is:
\n16 \u00f7 \\(\\frac{1}{4}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n16 \u00f7 \\(\\frac{1}{4}\\) = 64
\nThe quotient for the given division expression is: 64<\/p>\n

Question 13.
\n\\(\\frac{4}{5}\\) \u00f7 \\(\\frac{1}{8}\\)
\nAnswer:
\nThe given Division Expression is:
\n\\(\\frac{4}{5}\\) \u00f7 \\(\\frac{1}{8}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{4}{5}\\) \u00f7 \\(\\frac{1}{8}\\) = \\(\\frac{32}{5}\\)
\nThe quotient for the given division expression is: \\(\\frac{32}{5}\\)<\/p>\n

Question 14.
\n5 \u00f7 \\(\\frac{1}{10}\\)
\nAnswer:
\nThe given Division Expression is:
\n5 \u00f7 \\(\\frac{1}{10}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n5 \u00f7 \\(\\frac{1}{10}\\) = 50
\nThe quotient for the given division expression is: 50<\/p>\n

Question 15.
\n\\(\\frac{7}{11}\\) \u00f7 \\(\\frac{1}{11}\\)
\nAnswer:
\nThe given Division Expression is:
\n\\(\\frac{7}{11}\\) \u00f7 \\(\\frac{1}{11}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n\\(\\frac{7}{11}\\) \u00f7 \\(\\frac{1}{11}\\) = 7
\nThe quotient for the given division expression is: 7<\/p>\n

Question 16.
\n4 \u00f7 \\(\\frac{2}{8}\\)
\nAnswer:
\nThe given Division Expression is:
\n4 \u00f7 \\(\\frac{2}{8}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe ca conclude that
\n4 \u00f7 \\(\\frac{2}{8}\\) = 16
\nThe quotient for the given division expression is: 16<\/p>\n

LESSON 1.6 Divide Mixed Numbers<\/strong><\/p>\n

Quick Review<\/strong>
\nTo divide by a mixed number, rename each mixed number as a fraction. Then use the reciprocal of the divisor to rewrite the problem as a multiplication problem.<\/p>\n

Example<\/strong>
\n\"Envision
\nPractice
\n<\/strong>Find each quotient
\n<\/strong><\/p>\n

Question 1.
\n6\\(\\frac{3}{8}\\) \u00f7 4\\(\\frac{1}{4}\\)
\nAnswer:
\nThe given Division Expression is:
\n6\\(\\frac{3}{8}\\) \u00f7 4\\(\\frac{1}{4}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n6\\(\\frac{3}{8}\\) \u00f7 4\\(\\frac{1}{4}\\) = \\(\\frac{3}{2}\\)
\nThe quotient for the given division expression is: \\(\\frac{3}{2}\\)<\/p>\n

Question 2.
\n9 \u00f7 2\\(\\frac{2}{7}\\)
\nAnswer:
\nThe given Division Expression is:
\n9 \u00f7 2\\(\\frac{2}{7}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n9 \u00f7 2\\(\\frac{2}{7}\\) = \\(\\frac{63}{16}\\)
\nThe quotient for the given division expression is: \\(\\frac{63}{16}\\)<\/p>\n

Question 3.
\n3\\(\\frac{3}{5}\\) \u00f7 1\\(\\frac{1}{5}\\)
\nAnswer:
\nThe given Division Expression is:
\n3\\(\\frac{3}{5}\\) \u00f7 1\\(\\frac{1}{5}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n3\\(\\frac{3}{5}\\) \u00f7 1\\(\\frac{1}{5}\\) = 3
\nThe quotient for the given division expression is: 3<\/p>\n

Question 4.
\n5\\(\\frac{1}{2}\\) \u00f7 3\\(\\frac{3}{8}\\)
\nAnswer:
\nThe given Division Expression is:
\n5\\(\\frac{1}{2}\\) \u00f7 3\\(\\frac{3}{8}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n5\\(\\frac{1}{2}\\) \u00f7 3\\(\\frac{3}{8}\\) = \\(\\frac{44}{27}\\)
\nThe quotient for the given division expression is: \\(\\frac{4}{27}\\)<\/p>\n

Question 5.
\n3\\(\\frac{2}{5}\\) \u00f71\\(\\frac{1}{5}\\)
\nAnswer:
\nThe given Division Expression is:
\n3\\(\\frac{2}{5}\\) \u00f71\\(\\frac{1}{5}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n3\\(\\frac{2}{5}\\) \u00f71\\(\\frac{1}{5}\\) = \\(\\frac{17}{6}\\)
\nThe quotient for the given division expression is: \\(\\frac{17}{6}\\)<\/p>\n

Question 6.
\n12\\(\\frac{1}{6}[\/ltex] \u00f7 3
\nAnswer:
\nThe given Division Expression is:
\n12[latex]\\frac{1}{6}[\/ltex] \u00f7 3
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe ca conclude that
\n12[latex]\\frac{1}{6}[\/ltex] \u00f7 3 = [latex]\\frac{73}{18}\\)
\nThe quotient for the given division expression is: \\(\\frac{73}{18}\\)<\/p>\n

Question 7.
\n12 \u00f7 1\\(\\frac{1}{2}\\)
\nAnswer:
\nThe given Division Expression is:
\n12 \u00f7 1\\(\\frac{1}{2}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n12 \u00f7 1\\(\\frac{1}{2}\\) = 8
\nThe quotient for the given division expression is: 8<\/p>\n

Question 8.
\n3\\(\\frac{1}{2}\\) \u00f7 2\\(\\frac{1}{4}\\)
\nAnswer:
\nThe given Division Expression is:
\n3\\(\\frac{1}{2}\\) \u00f7 2\\(\\frac{1}{4}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n3\\(\\frac{1}{2}\\) \u00f7 2\\(\\frac{1}{4}\\) = \\(\\frac{14}{9}\\)
\nThe quotient for the given division expression is: \\(\\frac{14}{9}\\)<\/p>\n

Question 9.
\n8 \u00f7 1\\(\\frac{1}{4}\\)
\nAnswer:
\nThe given Division Expression is:
\n8 \u00f7 1\\(\\frac{1}{4}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n8 \u00f7 1\\(\\frac{1}{4}\\) = \\(\\frac{32}{5}\\)
\nThe quotient for the given division expression is: \\(\\frac{32}{5}\\)<\/p>\n

Question 10.
\n10\\(\\frac{1}{2}\\) \u00f7 1\\(\\frac{3}{4}\\)
\nAnswer:
\nThe given Division Expression is:
\n10\\(\\frac{1}{2}\\) \u00f7 1\\(\\frac{3}{4}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n10\\(\\frac{1}{2}\\) \u00f7 1\\(\\frac{3}{4}\\) = 6
\nThe quotient for the given division expression is: 6<\/p>\n

Question 11.
\n3\\(\\frac{3}{4}\\) \u00f7 2\\(\\frac{1}{2}\\)
\nAnswer:
\nThe given Division Expression is:
\n3\\(\\frac{3}{4}\\) \u00f7 2\\(\\frac{1}{2}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n3\\(\\frac{3}{4}\\) \u00f7 2\\(\\frac{1}{2}\\) = \\(\\frac{3}{2}\\)
\nThe quotient for the given division expression is: \\(\\frac{3}{2}\\)<\/p>\n

Question 12.
\n60 \u00f7 3\\(\\frac{1}{3}\\)
\nAnswer:
\nThe given Division Expression is:
\n60 \u00f7 3\\(\\frac{1}{3}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\n60 \u00f7 3\\(\\frac{1}{3}\\) = 18
\nThe quotient for the given division expression is: 18<\/p>\n

LESSON 1.7 Solve Problems with Rational Numbers<\/strong><\/p>\n

Quick Review<\/strong>
\nWhen solving multistep problems:
\n\u2022 decide the steps to solve the problem.
\n\u2022 choose the correct operations.
\n\u2022 identify the information you need from the problem.
\n\u2022 correctly use the information.
\n\u2022 calculate accurately.
\n\u2022 check if the answer is reasonable.<\/p>\n

Example<\/strong>
\nJane’s garden is 3.4 meters by 6.5 meters. If fencing costs $2.25 per meter, how much will it cost to enclose Jane’s garden?
\nStep 1: Find how much fence is needed.
\n3.4 + 3.4 + 6.5 + 6.5 = 19.8 meters
\nStep 2: Multiply to find the cost.
\n19.8 \u00d7 2.25 = $44.55
\nStep 3: Estimate to check.
\n3 + 3 + 7 + 7 = 20 meters
\n20 \u00d7 2.00 = $40.00
\n$40 is close to $44.55, so the answer is reasonable.<\/p>\n

Practice<\/strong>
\nDaisy has one cucumber that is 3 inches long and another cucumber that is 5 inches long. She cuts the cucumbers into \\(\\frac{3}{8}\\)-inch-thick slices and adds them to a salad. How many \\(\\frac{3}{8}\\)-inch-thick slices does Daisy have?<\/p>\n

Question 1.
\nWrite division expressions to represent the first steps in the problem.
\nAnswer:
\nIt is given that
\nDaisy has one cucumber that is 3 inches long and another cucumber that is 5 inches long. She cuts the cucumbers into \\(\\frac{3}{8}\\)-inch-thick slices and adds them to a salad.
\nSo,
\nThe total length of 2 cucumbers = (The length of the first cucumber) + (The length of the second cucumber)
\n= 3 + 5
\n= 8 inches
\nNow,
\nThe number of \\(\\frac{3}{8}\\)-inch-thick slices does Daisy have = (The total length of 2 cucumbers) \u00f7 \\(\\frac{3}{8}\\)
\n= 8 \u00f7 \\(\\frac{3}{8}\\)
\nHence, from the above,
\nWe can conclude that
\nThe division expression to represent the given information is:
\nThe number of \\(\\frac{3}{8}\\)-inch-thick slices do Daisy have = 8 \u00f7 \\(\\frac{3}{8}\\)<\/p>\n

Question 2.
\nSolve. Then explain your answer.
\nAnswer:
\nFrom Question 1,
\nWe can observe that
\nThe number of \\(\\frac{3}{8}\\)-inch-thick slices does Daisy have = (The total length of 2 cucumbers) \u00f7 \\(\\frac{3}{8}\\)
\n= 8 \u00f7 \\(\\frac{3}{8}\\)
\nNow,
\nBy using the Long Division,
\n\"\"
\nHence, from the above,
\nWe can conclude that
\nThe number of \\(\\frac{3}{8}\\)-inch-thick slices does Daisy have is about 21 pieces<\/p>\n

Topic 1 Fluency Practice<\/h3>\n

Pathfinder<\/strong>
\nShade a path from START to FINISH. Follow the solutions in which the digit in the hundredths place is greater than the digit in the tenths place. You can only move up, down, right, or left.
\n\"Envision<\/p>\n","protected":false},"excerpt":{"rendered":"

Go through the\u00a0enVision Math Common Core Grade 6 Answer Key\u00a0Topic 1 Use Positive Rational Numbers\u00a0regularly and improve your accuracy in solving questions. enVision Math Common Core 6th Grade Answers Key Topic 1 Use Positive, Rational Numbers Topic Essential Question How can you fluently add, subtract, multiply, and divide decimals? How can you multiply and divide …<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/posts\/15930"}],"collection":[{"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/comments?post=15930"}],"version-history":[{"count":2,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/posts\/15930\/revisions"}],"predecessor-version":[{"id":27333,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/posts\/15930\/revisions\/27333"}],"wp:attachment":[{"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/media?parent=15930"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/categories?post=15930"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/tags?post=15930"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}