Topics
1. Real Numbers
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United States of America
Grade 8

1.02 Convert decimal expansions

Lesson
Practice

Introduction

We've already learned how to convert between fractions and decimals. Now let's look at how to change decimal expansions into fractions.

Convert decimal expansions to fractions

Every fraction has a decimal expansion. If a real number has an infinitely long decimal expansion with a repeating pattern, it must be a rational number. By observing the effect of multiplying decimals by powers of ten, we can convert a repeating decimal into a fraction.

Examples

Example 1

Convert 0.777\ldots into a fraction.

Worked Solution
Create a strategy

Begin by assigning the decimal a variable name and multiply by some powers of 10 so that the repeating digits will be in front of the decimal.

Apply the idea
\displaystyle x\displaystyle =\displaystyle 0.777\ldotsLet x be the decimal
\displaystyle 10x\displaystyle =\displaystyle 7.777\ldotsMultiply both sides by 10
\displaystyle 10x-x\displaystyle =\displaystyle 7.777\ldots-0.777\ldotsSubtract the original equation from the new equation
\displaystyle 9x\displaystyle =\displaystyle 7
\displaystyle x\displaystyle =\displaystyle \dfrac{7}{9}Evaluate

Example 2

Convert 4.\overline{35} to a decimal.

Worked Solution
Apply the idea
\displaystyle x\displaystyle =\displaystyle 4.\overline{35}Let x be the number
\displaystyle 100x\displaystyle =\displaystyle 435.\overline{35}Multiply both sides by 100
\displaystyle 100x-x\displaystyle =\displaystyle 435.35\ldots-4.35\ldotsSubtract the original equation from the new equation
\displaystyle 99x\displaystyle =\displaystyle 431
\displaystyle x\displaystyle =\displaystyle \dfrac{431}{99}Evaluate
Idea summary

Every decimal with a repeating pattern is a rational number, and we can convert a given repeating decimal expansion into a fraction.

Outcomes

8.NS.A.1

Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

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