Recurring Decimals Quizzes
Visual overview of Recurring Decimals.
Introduction
Recurring decimals (also called repeating decimals) are decimals in which one or more digits repeat infinitely. They often appear when a fraction cannot be written as a finite decimal. Understanding recurring decimals is essential for GCSE Maths when converting between fractions and decimals, simplifying results, and solving algebraic problems.
Example: \( \tfrac{1}{3} = 0.\overline{3} \). The digit 3 repeats forever.
Core Concepts
Definition
A recurring decimal repeats a digit or group of digits forever. The repeating digits are shown with a bar above them.
- \(0.\overline{3}=0.333\ldots\)
- \(0.\overline{72}=0.727272\ldots\)
Identifying Recurring Decimals
Fractions with denominators that are not made from 2s or 5s usually give recurring decimals.
- \(\tfrac{1}{6}=0.1\overline{6}\)
- \(\tfrac{2}{11}=0.\overline{18}\)
Converting Fractions to Recurring Decimals
Divide the numerator by the denominator. If a remainder repeats, the decimal repeats too.
Example
\(\tfrac{1}{7}=0.\overline{142857}\) (period = 6)
Converting Recurring Decimals to Fractions
Use algebra to remove the repeating part:
- Let \(x = 0.\overline{3}\)
- Multiply by 10 → \(10x = 3.\overline{3}\)
- Subtract → \(9x = 3\)
- \(x = \tfrac{1}{3}\)
Example (Two Repeating Digits)
- Let \(x = 0.\overline{72}\)
- Multiply by 100 → \(100x = 72.\overline{72}\)
- Subtract → \(99x = 72\)
- \(x=\tfrac{72}{99}=\tfrac{8}{11}\)
Mixed Recurring Decimals
Some decimals have a non-repeating part followed by repeating digits.
Example
- Let \(x = 0.16\overline{6}\)
- \(10x = 1.6\overline{6}\)
- \(100x = 16.\overline{6}\)
- Subtract → \(90x = 15\)
- \(x = \tfrac{1}{6}\)
Terminating vs Recurring Decimals
- Terminating decimals: stop after a finite number of digits, e.g. \(0.25=\tfrac{1}{4}\)
- Recurring decimals: digits repeat forever, e.g. \(0.\overline{3}=\tfrac{1}{3}\)
- Rule: Fractions in lowest terms with denominators of the form \(2^m5^n\) terminate; all others recur.
Worked Examples
Example 1 (Foundation): Convert \(0.\overline{7}\) to a Fraction
- Let \(x = 0.\overline{7}\)
- \(10x = 7.\overline{7}\)
- Subtract → \(9x=7\)
- \(x=\tfrac{7}{9}\)
Example 2 (Higher): Convert \(0.1\overline{23}\) to a Fraction
- Let \(x = 0.1\overline{23}\)
- \(10x = 1.\overline{23}\), \(1000x = 123.\overline{23}\)
- Subtract → \(990x = 122\)
- \(x = \tfrac{122}{990}=\tfrac{61}{495}\)
Example 3: From Fraction to Decimal
- \(\tfrac{7}{11}=0.\overline{63}\)
Common Mistakes
- Not spotting the correct repeating digits.
- Using the wrong power of 10 in the algebraic method.
- Forgetting to simplify the fraction.
- Mixing terminating and recurring decimals incorrectly.
Applications
- Money, measurement, and probability problems.
- Converting between fractions, ratios, and percentages.
- Algebraic manipulation with repeating decimals.
Strategies & Tips
- Underline or note the repeating part clearly before working.
- Use powers of 10 matching the number of repeating digits.
- Always simplify your final fraction.
- Check by reconverting to decimal.
Summary / Call-to-Action
Recurring decimals bridge the gap between fractions and decimals. Mastering their identification and conversion gives you a strong foundation for ratio, proportion, and algebra topics.
- Practise converting both ways: fraction ↔ decimal.
- Start with one-digit repeats, then move to mixed forms.
- Check your work by reversing the process.