## Engage NY Eureka Math 8th Grade Module 7 Lesson 6 Answer Key

### Eureka Math Grade 8 Module 7 Lesson 6 Example Answer Key

Example 1.

Consider the fraction \(\frac{5}{8}\). Write an equivalent form of this fraction with a denominator that is a power of 10, and write the decimal expansion of this fraction.

Answer:

Consider the fraction \(\frac{5}{8}\). Is it equivalent to one with a denominator that is a power of 10? How do you know?

Yes. The fraction 5/8 has denominator 8 and so has factors that are products of 2â€™s only.

Write \(\frac{5}{8}\)as an equivalent fraction with a denominator that is a power of 10.

We have \(\frac{5}{8}\) = \(\frac{5}{2 \times 2 \times 2} = \frac{5 \times 5 \times 5 \times 5}{2 \times 2 \times 2 \times 5 \times 5 \times 5} = \frac{625}{10 \times 10 \times 10} = \frac{625}{10^{3}}\)

What is \(\frac{5}{8}\) as a finite decimal?

\(\frac{5}{8}\) = \(\frac{625}{1000}\) = 0.625

Example 2.

Consider the fraction \(\frac{17}{125}\). Is it equal to a finite or an infinite decimal? How do you know?

Answer:

â†’ Letâ€™s consider the fraction \(\frac{17}{125}\). We want the decimal value of this number. Will it be a finite or an infinite decimal? How do you know?

â†’ We know that the fraction \(\frac{17}{125}\) is equal to a finite decimal because the denominator 125 is a product of 5â€™s, specifically, 5^{3}, and so we can write the fraction as one with a denominator that is a power of 10.

â†’ What will we need to multiply 5^{3} by to obtain a power of 10?

We will need to multiply by 2^{3}. Then, 5^{3} Ã— 2^{3} = (5 Ã— 2)^{3} = ã€–10ã€—^{3}.

â†’ Write \(\frac{17}{125}\) or its equivalent 17/5^{3} as a finite decimal.

\(\frac{17}{125}\) = \(\frac{17}{5^{3}} = \frac{17 \times 2^{3}}{5^{3} \times 2^{3}} = \frac{17 \times 8}{(5 \times 2)^{3}} = \frac{136}{10^{3}}\) = 0.136

(If the above two points are too challenging for some students, have them write out:

\(\frac{17}{125}\) = \(\frac{17}{5 \times 5 \times 5} = \frac{17 \times 2 \times 2 \times 2}{5 \times 5 \times 5 \times 2 \times 2 \times 2}\) = \(\frac{136}{1000}\) = 0.136.)

Example 3.

Will the decimal expansion of \(\frac{7}{80}\) be finite or infinite? If it is finite, find it.

Answer:

â†’ Will \(\frac{7}{80}\) have a finite or infinite decimal expansion?

We know that the fraction \(\frac{7}{80}\) is equal to a finite decimal because the denominator 80 is a product of 2â€™s and 5â€™s. Specifically, 2^{4} Ã— 5. This means the fraction is equivalent to one with a denominator that is a power of 10.

â†’ What will we need to multiply 2^{4} Ã— 5 by so that it is equal to (2 Ã— 5)^{n} = 10^{n} for some n?

We will need to multiply by 5^{3} so that 2^{4} Ã— 5^{4} = (2 Ã— 5)^{4} = 10^{4}.

â†’ Begin with\(\frac{7}{80}\) or \(\frac{7}{2^{4} \times 5}\). Use what you know about equivalent fractions to rewrite \(\frac{7}{80}\) in the form \(\frac{k}{10^{n}}\) and then write the decimal form of the fraction.

\(\frac{7}{80}\) = \(\frac{7}{2^{4} \times 5} = \frac{7 \times 5^{3}}{2^{4} \times 5 \times 5^{3}} = \frac{7 \times 125}{(2 \times 5)^{4}} = \frac{875}{10^{4}}\) = 0.0875

Example 4.

Will the decimal expansion of \(\frac{3}{160}\) be finite or infinite? If it is finite, find it.

Answer:

â†’ Will \(\frac{3}{160}\) have a finite or infinite decimal expansion?

We know that the fraction \(\frac{3}{160}\) is equal to a finite decimal because the denominator 160 is a product of 2â€™s and 5â€™s. Specifically, 2^{5} Ã— 5. This means the fraction is equivalent to one with a denominator that is a power of 10.

â†’ What will we need to multiply 2^{5} Ã— 5 by so that it is equal to (2 Ã— 5)^{n} = 10^{n} for some n?

We will need to multiply by 5^{4} so that 2^{5} Ã— 5^{5} = (2 Ã— 5)^{5} = 10^{5}.

Begin with \(\frac{3}{160}\) or \(\frac{3}{2^{5} \times 5}\). Use what you know about equivalent fractions to rewrite \(\frac{3}{160}\) in the form \(\frac{k}{10^{n}}\) and then write the decimal form of the fraction.

\(\frac{3}{160}\) = \(\frac{3}{2^{5} \times 5}=\frac{3 \times 5^{4}}{2^{5} \times 5 \times 5^{4}}=\frac{3 \times 625}{(2 \times 5)^{5}}=\frac{1875}{10^{5}}\)Â = 0.01875

### Eureka Math Grade 8 Module 7 Lesson 6 Exercise Answer Key

Opening Exercise

a. Use long division to determine the decimal expansion of \(\frac{54}{20}\).

Answer:

\(\frac{54}{20}\) = 2.7

b. Use long division to determine the decimal expansion of \(\frac{7}{8}\).

Answer:

\(\frac{7}{8}\) = 0.875

c. Use long division to determine the decimal expansion of \(\frac{8}{9}\).

Answer:

\(\frac{8}{9}\) = 0.8888 “â€¦”

d. Use long division to determine the decimal expansion of \(\frac{22}{7}\).

Answer:

\(\frac{22}{7}\) = 3.142857 “â€¦”

e. What do you notice about the decimal expansions of parts (a) and (b) compared to the decimal expansions of parts (c) and (d)?

Answer:

The decimal expansions of parts (a) and (b) ended. That is, when I did the long division, I was able to stop after a few steps. That was different from the work I had to do in parts (c) and (d). In part (c), I noticed that the same number kept coming up in the steps of the division, but it kept going on. In part (d), when I did the long division, it did not end. I stopped dividing after I found a few decimal digits of the decimal expansion.

Exercises 1â€“5

You may use a calculator, but show your steps for each problem.

Exercise 1.

Consider the fraction \(\frac{3}{8}\) .

a. Write the denominator as a product of 2â€™s and/or 5â€™s. Explain why this way of rewriting the denominator helps to find the decimal representation of \(\frac{3}{8}\) .

Answer:

The denominator is equal to 2^{3}. It is helpful to know that 8 = 2^{3} because it shows how many factors of 5 will be needed to multiply the numerator and denominator by so that an equivalent fraction is produced with a denominator that is a multiple of 10. When the denominator is a multiple of 10, the fraction can easily be written as a decimal using what I know about place value.

b. Find the decimal representation of \(\frac{3}{8}\). Explain why your answer is reasonable.

Answer:

\(\frac{3}{8}\) = \(\frac{3}{2^{3}}=\frac{3 \times 5^{3}}{2^{3} \times 5^{3}}=\frac{375}{10^{3}}\) = 0.375

The answer is reasonable because the decimal value, 0.375, is less than \(\frac{1}{2}\), just like the fraction \(\frac{3}{8}\).

Exercise 2.

Find the first four places of the decimal expansion of the fraction \(\frac{43}{64}\).

Answer:

The denominator is equal to 2^{6}.

\(\frac{43}{64}\) = \(\frac{43}{2^{6}}=\frac{43 \times 5^{6}}{2^{6} \times 5^{6}}=\frac{671875}{10^{6}}\) = 0.671875

The decimal expansion to the first four decimal places is 0.6718.

Exercise 3.

Find the first four places of the decimal expansion of the fraction \(\frac{29}{125}\).

Answer:

The denominator is equal to 5^{3}.

\(\frac{29}{125}\) = \(\frac{29}{5^{3}} = \frac{29 \times 2^{3}}{5^{3} \times 2^{3}} = \frac{232}{10^{3}}\) = 0.232

The decimal expansion to the first four decimal places is 0.2320.

Exercise 4.

Find the first four decimal places of the decimal expansion of the fraction \(\frac{19}{34}\).

Answer:

Using long division, the decimal expansion to the first four places is 0.5588â€¦.

Exercise 5.

Identify the type of decimal expansion for each of the numbers in Exercises 1â€“4 as finite or infinite. Explain why their decimal expansion is such.

We know that the number \(\frac{7}{8}\) had a finite decimal expansion because the denominator 8 is a product of 2â€™s and so is equivalent to a fraction with a denominator that is a power of 10. We know that the number \(\frac{43}{64}\) had a finite decimal expansion because the denominator 64 is a product of 2â€™s and so is equivalent to a fraction with a denominator that is a power of 10. We know that the number \(\frac{29}{125}\) had a finite decimal expansion because the denominator 125 is a product of 5â€™s and so is equivalent to a fraction with a denominator that is a power of 10. We know that the number \(\frac{19}{34}\) had an infinite decimal expansion because the denominator was not a product of 2â€™s or 5â€™s; it had a factor of 17 and so is not equivalent to a fraction with a denominator that is a power of 10.

Exercises 6â€“8

You may use a calculator, but show your steps for each problem.

Exercise 6.

Convert the fraction \(\frac{37}{40}\) to a decimal.

a. Write the denominator as a product of 2â€™s and/or 5â€™s. Explain why this way of rewriting the denominator helps to find the decimal representation of \(\frac{37}{40}\).

Answer:

The denominator is equal to 2^{3} Ã— 5. It is helpful to know that 40 is equal to 2^{3} Ã— 5 because it shows by how many factors of 5 the numerator and denominator will need to be multiplied to produce an equivalent fraction with a denominator that is a power of 10. When the denominator is a power of 10, the fraction can easily be written as a decimal using what I know about place value.

b. Find the decimal representation of \(\frac{37}{40}\). Explain why your answer is reasonable.

Answer:

\(\frac{37}{40}\) = \(\frac{37}{2^{3} \times 5} = \frac{37 \times 5^{2}}{2^{3} \times 5 \times 5^{2}} = \frac{925}{10^{3}}\) = 0.925

The answer is reasonable because the decimal value, 0.925, is less than 1, just like the fraction 37/40. Also, it is reasonable and correct because the fraction \(\frac{925}{1000}\) = \(\frac{37}{40}\); therefore, it has the decimal expansion 0.925.

Exercise 7.

Convert the fraction \(\frac{3}{250}\) to a decimal.

Answer:

The denominator is equal to 2 Ã— 5^{3}.

\(\frac{3}{250}\) = \(\frac{3}{2 \times 5^{3}} = \frac{3 \times 2^{2}}{2 \times 2^{2} \times 5^{3}} = \frac{12}{10^{3}}\) = 0.012

Exercise 8.

Convert the fraction \(\frac{7}{1250}\) to a decimal.

Answer:

The denominator is equal to 2 Ã— 5^{4}.

\(\frac{7}{1250}\) = \(\frac{7}{2 \times 5^{4}} = \frac{7 \times 2^{3}}{2 \times 2^{3} \times 5^{4}} = \frac{56}{10^{4}}\) = 0.0056

### Eureka Math Grade 8 Module 7 Lesson 6 Problem Set Answer Key

Convert each fraction given to a finite decimal, if possible. If the fraction cannot be written as a finite decimal, then state how you know. You may use a calculator, but show your steps for each problem.

Question 1.

\(\frac{2}{32}\)

The fraction \(\frac{2}{32}\) simplifies to \(\frac{1}{16}\).

The denominator is equal to 2^{4}.

\(\frac{1}{16}\) = \(\frac{1}{2^{4}} = \frac{1 \times 5^{4}}{2^{4} \times 5^{4}} = \frac{625}{10^{4}}\) = 0.0625

Question 2.

\(\frac{99}{125}\)

Answer:

The denominator is equal to 5^{3}.

\(\frac{99}{125}\) = \(\frac{99}{125}\) = 0.792

Question 3.

\(\frac{15}{128}\)

The denominator is equal to 2^{7}.

\(\frac{15}{128}\) = \(\frac{15}{128}\) = 0.1171875

Question 4.

\(\frac{8}{15}\)

Answer:

The fraction \(\frac{8}{15}\) is not a finite decimal because the denominator is equal to 3 Ã— 5. Since the denominator cannot be expressed as a product of 2â€™s and 5â€™s, then \(\frac{8}{15}\) is not a finite decimal.

Question 5.

\(\frac{3}{28}\)

Answer:

The fraction \(\frac{3}{28}\) is not a finite decimal because the denominator is equal to 2^{2} Ã— 7. Since the denominator cannot be expressed as a product of 2â€™s and 5â€™s, then \(\frac{3}{28}\) is not a finite decimal.

Question 6.

\(\frac{13}{400}\)

Answer:

The denominator is equal to 2^{4} Ã— 5^{2}.

\(\frac{13}{400}\) = \(\frac{13}{2^{4} \times 5^{2}} = \frac{13 \times 5^{2}}{2^{4} \times 5^{2} \times 5^{2}} = \frac{325}{10^{4}}\) = 0.0325

Question 7.

\(\frac{5}{64}\)

Answer:

The denominator is equal to 2^{6}.

\(\frac{5}{64}\) = \(\frac{5}{2^{6}} = \frac{5 \times 5^{6}}{2^{6} \times 5^{6}} = \frac{78125}{10^{6}}\) = 0.078125

Question 8.

\(\frac{15}{35}\)

Answer:

The fraction \(\frac{15}{35}\) reduces to \(\frac{3}{7}\). The denominator 7 cannot be expressed as a product of 2â€™s and 5â€™s. Therefore, \(\frac{3}{7}\) is not a finite decimal.

Question 9.

\(\frac{199}{250}\)

Answer:

The denominator is equal to 2 Ã— 5^{3}.

\(\frac{199}{250}\) = \(\frac{199}{2 \times 5^{3}} = \frac{199 \times 2^{2}}{2 \times 2^{2} \times 5^{3}} = \frac{796}{10^{3}}\) = 0.796

Question 10.

\(\frac{219}{625}\)

Answer:

The denominator is equal to 5^{4}.

\(\frac{219}{625}\) = \(\frac{219}{5^{4}} = \frac{219 \times 2^{4}}{2^{4} \times 5^{4}} = \frac{3504}{10^{4}}\) = 0.3504

### Eureka Math Grade 8 Module 7 Lesson 6 Exit Ticket Answer Key

Convert each fraction to a finite decimal if possible. If the fraction cannot be written as a finite decimal, then state how you know. You may use a calculator, but show your steps for each problem.

Question 1.

\(\frac{9}{16}\)

Answer:

The denominator is equal to 2^{4}.

\(\frac{9}{16}\) = \(\frac{9}{2^{4}} = \frac{9 \times 5^{4}}{2^{4} \times 5^{4}} = \frac{9 \times 625}{10^{4}} = \frac{5625}{10^{4}}\) = 0.5625

Question 2.

\(\frac{8}{125}\)

Answer:

The denominator is equal to 5^{3}.

\(\frac{8}{125}\) = \(\frac{8}{5^{3}} = \frac{8 \times 2^{3}}{5^{3} \times 2^{3}} = \frac{8 \times 8}{10^{3}} = \frac{64}{10^{3}}\) = 0.064

Question 3.

\(\frac{4}{15}\)

Answer:

The fraction \(\frac{4}{15}\) is not a finite decimal because the denominator is equal to 5 Ã— 3. Since the denominator cannot be expressed as a product of 2â€™s and 5â€™s, then \(\frac{4}{15}\) is not a finite decimal.

Question 4.

\(\frac{1}{200}\)

Answer:

The denominator is equal to 2^{3} Ã— 5^{2}.

\(\frac{1}{200}\) = \(\frac{1}{2^{3} \times 5^{2}} = \frac{1 \times 5}{2^{3} \times 5^{2} \times 5} = \frac{5}{2^{3} \times 5^{3}} = \frac{5}{10^{3}}\) = 0.005