{"id":82532,"date":"2021-04-01T19:09:26","date_gmt":"2021-04-01T13:39:26","guid":{"rendered":"https:\/\/ccssmathanswers.com\/?p=82532"},"modified":"2022-04-01T10:26:06","modified_gmt":"2022-04-01T04:56:06","slug":"eureka-math-grade-8-module-7-lesson-8","status":"publish","type":"post","link":"https:\/\/bigideasmathanswers.com\/eureka-math-grade-8-module-7-lesson-8\/","title":{"rendered":"Eureka Math Grade 8 Module 7 Lesson 8 Answer Key"},"content":{"rendered":"

Engage NY Eureka Math 8th Grade Module 7 Lesson 8 Answer Key<\/h2>\n

Eureka Math Grade 8 Module 7 Lesson 8 Example Answer Key<\/h3>\n

Example 1.
\nShow that the decimal expansion of \\(\\frac{26}{4}\\) is 6.5.
\nAnswer:
\nUse the example with students so they have a model to complete Exercises 1\u20135.
\n\u2192 Show that the decimal expansion of \\(\\frac{26}{4}\\) is 6.5.
\nStudents might use the long division algorithm, or they might simply observe \\(\\frac{26}{4}\\) = \\(\\frac{13}{2}\\) = 6.5.
\n\u2192 Here is another way to see this: What is the greatest number of groups of 4 that are in 26?
\nThere are 6 groups of 4 in 26.
\n\u2192 Is there a remainder?
\nYes, there are 2 left over.
\n\u2192 This means we can write 26 as
\n26 = 6 \u00d7 4 + 2.
\nThis means we could also compute \\(\\frac{26}{4}\\) as follows:
\n\\(\\frac{26}{4}\\) = \\(\\frac{6 \\times 4 + 2}{4}\\)
\n\\(\\frac{26}{4}\\) = \\(\\frac{6 \\times 4}{4}\\) + \\(\\frac{2}{4}\\)
\n\\(\\frac{26}{4}\\) = 6 + \\(\\frac{2}{4}\\)
\n\\(\\frac{26}{4}\\) = 6\\(\\frac{2}{4}\\) = 6 \\(\\frac{1}{2}\\).
\n(Some students might note we are simply rewriting the fraction as a mixed number.)
\nThe fraction \\(\\frac{26}{4}\\) is equal to the finite decimal 6.5. When the fraction is not equal to a finite decimal, then we need to use the long division algorithm to determine the decimal expansion of the number.<\/p>\n

Eureka Math Grade 8 Module 7 Lesson 8 Exploratory Challenge\/Exercise Answer Key<\/h3>\n

Exploratory Challenge\/Exercises 1\u20135<\/p>\n

Exercise 1.
\na. Use long division to determine the decimal expansion of \\(\\frac{142}{2}\\).
\nAnswer:
\n\"Engage<\/p>\n

b. Fill in the blanks to show another way to determine the decimal expansion of \\(\\frac{142}{2}\\).
\n\"Engage
\nAnswer:
\n\"Engage<\/p>\n

c. Does the number \\(\\frac{142}{2}\\) have a finite or an infinite decimal expansion?
\nAnswer:
\nThe decimal expansion of \\(\\frac{142}{2}\\) is 71.0 and is finite.<\/p>\n

Exercise 2.
\na. Use long division to determine the decimal expansion of \\(\\frac{142}{4}\\).
\nAnswer:
\n\"Engage<\/p>\n

b. Fill in the blanks to show another way to determine the decimal expansion of \\(\\frac{142}{4}\\).
\n\"Engage
\nAnswer:
\n\"Engage<\/p>\n

c. Does the number \\(\\frac{142}{4}\\) have a finite or an infinite decimal expansion?
\nAnswer:
\nThe decimal expansion of \\(\\frac{142}{4}\\) is 35.5 and is finite.<\/p>\n

Exercise 3.
\na. Use long division to determine the decimal expansion of \\(\\frac{142}{6}\\).
\nAnswer:
\n\"Engage<\/p>\n

b. Fill in the blanks to show another way to determine the decimal expansion of \\(\\frac{142}{6}\\).
\n\"Engage
\nAnswer:
\n\"Engage<\/p>\n

c. Does the number \\(\\frac{142}{6}\\) have a finite or an infinite decimal expansion?
\nAnswer:
\nThe decimal expansion of \\(\\frac{142}{6}\\) is 23.666\u2026 and is infinite.<\/p>\n

Exercise 4.
\na. Use long division to determine the decimal expansion of \\(\\frac{142}{11}\\).
\nAnswer:
\n\"Engage<\/p>\n

b. Fill in the blanks to show another way to determine the decimal expansion of \\(\\frac{142}{11}\\).
\n\"Engage
\nAnswer:
\n\"Engage<\/p>\n

c. Does the number \\(\\frac{142}{11}\\) have a finite or an infinite decimal expansion?
\nAnswer:
\nThe decimal expansion of \\(\\frac{142}{11}\\) is 12.90909\u2026 and is infinite.<\/p>\n

Exercise 5.
\nIn general, which fractions produce infinite decimal expansions?
\nAnswer:
\nWe discovered in Lesson 6 that fractions equivalent to ones with denominators that are a power of 10 are precisely the fractions with finite decimal expansions. These fractions, when written in simplified form, have denominators with factors composed of 2\u2018s and 5\u2018s. Thus any fraction, in simplified form, whose denominator contains a factor different from 2 or 5 must yield an infinite decimal expansion.<\/p>\n

Exercises 6\u201310<\/p>\n

Exercise 6.
\nDoes the number \\(\\frac{65}{13}\\) have a finite or an infinite decimal expansion? Does its decimal expansion have a repeating pattern?
\nAnswer:
\nThe number \\(\\frac{65}{13}\\) is rational and so has a decimal expansion with a repeating pattern. Actually, \\(\\frac{65}{13}\\) = \\(\\frac{5 \\times 13}{13}\\) = 5, so it is a finite decimal. Viewed as an infinite decimal, \\(\\frac{65}{13}\\) is 5.0000\u2026 with a repeat block of 0.<\/p>\n

Exercise 7.
\nDoes the number \\(\\frac{17}{11}\\) have a finite or an infinite decimal expansion? Does its decimal expansion have a repeating pattern?
\nAnswer:
\nThe rational \\(\\frac{17}{11}\\) is in simplest form, and we see that it is not equivalent to a fraction with a denominator that is a power of 10. Thus, the rational has an infinite decimal expansion with a repeating pattern.<\/p>\n

Exercise 8.
\nIs the number 0.212112111211112111112\u2026 rational? Explain. (Assume the pattern you see in the decimal expansion continues.)
\nAnswer:
\nAlthough the decimal expansion of this number has a pattern, it is not a repeating pattern. The number cannot be rational. It is irrational.<\/p>\n

Exercise 9.
\nDoes the number \\(\\frac{860}{999}\\) have a finite or an infinite decimal expansion? Does its decimal expansion have a
\nrepeating pattern?
\nAnswer:
\nThe number is rational and so has a decimal expansion with a repeating pattern. Since the fraction is not equivalent to one with a denominator that is a power of 10, it is an infinite decimal expansion.<\/p>\n

Exercise 10.
\nIs the number 0.1234567891011121314151617181920212223\u2026 rational? Explain. (Assume the pattern you see in the decimal expansion continues.)
\nAnswer:
\nAlthough the decimal expansion of this number has a pattern, it is not a repeating pattern. The number cannot be rational. It is irrational.<\/p>\n

Eureka Math Grade 8 Module 7 Lesson 8 Problem Set Answer Key<\/h3>\n

Question 1.
\nWrite the decimal expansion of \\(\\frac{7000}{9}\\) as an infinitely long repeating decimal.
\nAnswer:
\n\\(\\frac{7000}{9}\\) = \\(\\frac{777 \\times 9}{9}\\) + \\(\\frac{7}{9}\\)
\n= \\(\\frac{7777}{9}\\)
\n\"Eureka
\nThe decimal expansion of \\(\\frac{7000}{9}\\) is \\(777 . \\overline{7}\\)<\/p>\n

Question 2.
\nWrite the decimal expansion of \\(\\frac{6555555}{3}\\) as an infinitely long repeating decimal.
\nAnswer:
\n\\(\\frac{6555555}{3}\\) = \\(\\frac{2185185 \\times 3}{3}\\) + \\(\\frac{0}{3}\\)
\n= 2 185 185
\nThe decimal expansion of \\(\\frac{6555555}{3}\\) is \\(2,185,185 . \\overline{0}\\).<\/p>\n

Question 3.
\nWrite the decimal expansion of \\(\\frac{350000}{11}\\) as an infinitely long repeating decimal.
\nAnswer:
\n\\(\\frac{350000}{11}\\) = \\(\\frac{31818 \\times 11}{11}\\) + \\(\\frac{2}{11}\\)
\n= 31818\\(\\frac{2}{11}\\)
\n\"Eureka
\nThe decimal expansion of \\(\\frac{350000}{11}\\) is \\(31,818 . \\overline{18}\\).<\/p>\n

Question 4.
\nWrite the decimal expansion of \\(\\frac{12000000}{37}\\) as an infinitely long repeating decimal.
\nAnswer:
\n\\(\\frac{12000000}{37}\\) = \\(\\frac{324324 \\times 37}{37}\\) + \\(\\frac{12}{37}\\)
\n= 324324\\(\\frac{12}{37}\\)
\n\"Eureka
\nThe decimal expansion of \\(\\frac{12000000}{37}\\) is \\(324,324 . \\overline{324}\\).<\/p>\n

Question 5.
\nSomeone notices that the long division of 2,222,222 by 6 has a quotient of 370,370 and a remainder of 2 and wonders why there is a repeating block of digits in the quotient, namely 370. Explain to the person why this happens.
\nAnswer:
\n\\(\\frac{2222222}{6}\\) = \\(\\frac{370370 \\times 6}{6}\\) + \\(\\frac{2}{6}\\)
\n= 370370 \\(\\frac{2}{6}\\)
\n\"Eureka
\nThe block of digits 370 keeps repeating because the long division algorithm leads us to perform the same division over and over again. In the algorithm shown above, we see that there are three groups of 6 in 22, leaving a remainder of 4. When we bring down the next 2, we see that there are exactly seven groups of 6 in 42. When we bring down the next 2, we see that there are zero groups of 6 in 2, leaving a remainder of 2. It is then that the process starts over because the next step is to bring down another 2, giving us 22, which is what we started with. Since the division repeats, then the digits in the quotient will repeat.<\/p>\n

Question 6.
\nIs the answer to the division problem 10\u00f73.2 a rational number? Explain.
\nAnswer:
\nYes. This is equivalent to the division problem 100\u00f732, which can be written as \\(\\frac{100}{32}\\), and so it is a rational number.<\/p>\n

Question 7.
\nIs \\(\\frac{3 \\pi}{77 \\pi}\\) a rational number? Explain.
\nAnswer:
\nYes. \\(\\frac{3 \\pi}{77 \\pi}\\) is equal to \\(\\frac{3}{77}\\) and so it is a rational number.<\/p>\n

Question 8.
\nThe decimal expansion of a real number x has every digit 0 except the first digit, the tenth digit, the hundredth digit, the thousandth digit, and so on, are each 1. Is x a rational number? Explain.
\nAnswer:
\nNo. Although there is a pattern to this decimal expansion, it is not a repeating pattern. Thus, x cannot be rational.<\/p>\n

Eureka Math Grade 8 Module 7 Lesson 8 Exit Ticket Answer Key<\/h3>\n

Question 1.
\nWill the decimal expansion of \\(\\frac{125}{8}\\) be finite or infinite? Explain. If we were to write the decimal expansion of this rational number as an infinitely long decimal, which block of numbers repeat?
\nAnswer:
\nThe decimal expansion of \\(\\frac{125}{8}\\) will be finite because \\(\\frac{125}{8}\\) is equivalent to a fraction with a denominator that is a power of 10. (Multiply the numerator and denominator each by 5 \u00d7 5 \u00d7 5.) If we were to write the decimal as an infinitely long decimal, then we\u2019d have a repeating block consisting of 0.<\/p>\n

Question 2.
\nWrite the decimal expansion of \\(\\frac{13}{7}\\) as an infinitely long repeating decimal.
\nAnswer:
\n\\(\\frac{13}{7}\\) = \\(\\frac{1 \\times 7}{7}\\) + \\(\\frac{6}{7}\\)
\n= 1\\(\\frac{6}{7}\\)
\n\"Eureka
\nThe decimal expansion of \\(\\frac{13}{7}\\) is \\(1 . \\overline{857142}\\).<\/p>\n","protected":false},"excerpt":{"rendered":"

Engage NY Eureka Math 8th Grade Module 7 Lesson 8 Answer Key Eureka Math Grade 8 Module 7 Lesson 8 Example Answer Key Example 1. Show that the decimal expansion of is 6.5. Answer: Use the example with students so they have a model to complete Exercises 1\u20135. \u2192 Show that the decimal expansion of … Read more<\/a><\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[6],"tags":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/82532"}],"collection":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/comments?post=82532"}],"version-history":[{"count":1,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/82532\/revisions"}],"predecessor-version":[{"id":91537,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/82532\/revisions\/91537"}],"wp:attachment":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/media?parent=82532"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/categories?post=82532"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/tags?post=82532"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}