Converting Repeating Decimals to Fractions

Examples, solutions, and videos to help Grade 8 students know the intuitive reason why every repeating decimal is equal to a fraction and how to convert a decimal expansion that eventually repeats into a fraction.

New York State Common Core Math Grade 8, Module 7, Lesson 10

Worksheets for Grade 8

The following diagram shows how to convert a repeating decimal to a fraction. Scroll down the page for more examples and solutions of how to convert a repeating decimal to a fraction.


Lesson 10 Student Outcomes

• Students know the intuitive reason why every repeating decimal is equal to a fraction. Students convert a decimal expansion that eventually repeats into a fraction.
• Students know that the decimal expansions of rational numbers repeat eventually.
• Students understand that irrational numbers are numbers that are not rational. Irrational numbers cannot be be represented as a fraction and have infinite decimals that never repeat.

Lesson 10 Summary

Numbers with decimal expansions that repeat are rational numbers and can be converted to fractions using a linear equation.
Irrational numbers are numbers that are not rational. They have infinite decimals that do not repeat and cannot be represented as a fraction.

Lesson 10 Classwork

Example 1
Find the fraction that is equal to the infinite decimal \\(0.\\overline {81} \\)

Exercises 1–2
1. a. Let x = \\(0.\\overline {123} \\) Explain why multiplying both sides of this equation by 10³ will help us determine the fractional representation of x.
b. After multiplying both sides of the equation by 10³, rewrite the resulting equation by making a substitution that will help determine the fractional value of x. Explain how you were able to make the substitution.
c. Solve the equation to determine the value of x.
d. Is your answer reasonable? Check your answer using a calculator.
2. Find the fraction equal to x = \\(0.\\overline 4 \\). Check that you are correct using a calculator.

Example 2
Find the fraction that is equal to the infinite decimal \\(2.13\\overline 8 \\).

Exercises 3–4
3. Find the fraction equal to \\(1.6\\overline {23} \\). Check that you are correct using a calculator.
4. Find the fraction equal to \\(2.9\\overline {60} \\). Check that you are correct using a calculator.

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