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Converting Decimals to Fractions: Repeating

Master converting repeating decimals to fractions with interactive lessons and practice problems! Designed for students 10 to 16 years old.

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Converting Decimals to Fractions: Repeating

Study Guide

Key Definition

A repeating decimal is a decimal number that after a certain place repeats the same digits in a pattern.

Important Notes

To convert a repeating decimal 0.a̅b into a fraction, let k = number of non-repeating digits and n = number of repeating digits.
Form the integer A = all digits (non-repeating + one block of repeating), and B = integer of the non-repeating part.
Numerator = A – B; Denominator = a number consisting of n 9's followed by k 0's (i.e. 10ᵏ⁺ⁿ – 10ᵏ).
Always simplify the final fraction.

Mathematical Notation

$N$ = the full repeating decimal (e.g., N = 0.a b̅)

$k$ = number of non-repeating digits after the decimal

$n$ = number of repeating digits

$\frac{10^{k+n}N - 10^k N}{10^{k+n} - 10^k}$ = conversion formula

Use overlines to show repeating digits, e.g., 0.12̅3.

Why It Works

When multiplying by 10ᵏ⁺ⁿ and 10ᵏ and subtracting, the repeating part cancels, leaving an integer over another integer.

Remember

Simplify the fraction at the end.

Quick Reference

Decimal to Fraction (mixed or pure):$\frac{10^{k+n}N - 10^k N}{10^{k+n} - 10^k}$

Understanding Converting Decimals to Fractions: Repeating

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A repeating decimal repeats one or more digits infinitely after the decimal point.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the fraction form of the repeating decimal $0.\overline{3}$?

Real-World Problem

Question Exercise

Intermediate

Pizza Scenario

Imagine you ate $0.\overline{6}$ of a pizza. What fraction of the pizza did you eat?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Convert the repeating decimal $0.\overline{9}$ into a fraction.

Challenge Quiz

Single Choice Quiz

Advanced

What is the fraction form of the repeating decimal $0.\overline{8}$?

Recap

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Review key concepts and takeaways