Finite and Infinite Decimals

Examples, solutions, and videos to help Grade 8 students learn when a fraction has a denominator that is the product of 2’s and/or 5’s then it is a finite decimal equivalent otherwise it has an infinite decimal equivalent.

New York State Common Core Math Grade 8, Module 7, Lesson 6

Worksheets for Grade 8

Lesson 6 Student Outcomes

• Students know that every number has a decimal expansion (i.e., is equal to a finite or infinite decimal).
  
• Students know that when a fraction has a denominator that is the product of 2’s and/or 5’s, it has a finite decimal expansion because the fraction can then be written in an equivalent form with a denominator that is a power of 10.

Lesson 6 Summary

Fractions with denominators that can be expressed as products of 2’s and/or 5’s have decimal expansions that are finite.

Lesson 6 Classwork

Opening Exercises 1–5

1. Use long division to determine the decimal expansion of 54/20.
2. Use long division to determine the decimal expansion of 7/8.
3. Use long division to determine the decimal expansion of 8/9.
4. Use long division to determine the decimal expansion of 22/7.
5. What do you notice about the decimal expansions of Exercises 1 and 2 compared to the decimal expansions of Exercises 3 and 4?

Example 1
Consider the fraction 5/8. Is it equal to a finite decimal? How do you know?

Example 2
Consider the fraction 17/125. Is it equal to a finite or infinite decimal? How do you know?

Exercises 6–10
Show your steps, but use a calculator for the multiplications.
6. Convert the fraction 7/8 to a decimal.
a. Write the denominator as a product of 2’s or 5’s. Explain why this way of rewriting the denominator helps to find the decimal representation of 7/8.
b. Find the decimal representation of 7/8. Explain why your answer is reasonable.
7. Convert the fraction 43/64 to a decimal.
8. Convert the fraction 29/125 to a decimal.
9. Convert the fraction 19/34 to a decimal.
10. Identify the type of decimal expansion for each of the numbers in Exercises 6–9 as finite or infinite. Explain why their decimal expansion is such.

Example 3
Write 7/80 as a decimal. Will it be finite or infinite? Explain.

Example 4
Write 3/160 as a decimal. Will it be finite or infinite? Explain.

Exercises 11–13
Show your steps, but use a calculator for the multiplications.
11. Convert the fraction 37/40 to a decimal.
a. Write the denominator as a product of 2’s and/or 5’s. Explain why this way of rewriting the denominator helps to find the decimal representation of 37/40.
b. Find the decimal representation of 37/40. Explain why your answer is reasonable.
12. Convert the fraction 3/250 to a decimal.
13. Convert the fraction 7/1,250 to a decimal.

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