Recurring Decimals Quizzes
Introduction
Recurring decimals, also known as repeating decimals, are decimals in which one or more digits repeat infinitely. They appear frequently in fractions that cannot be expressed exactly as a finite decimal. Understanding recurring decimals is essential in GCSE Maths for converting between fractions and decimals, performing calculations, and solving algebraic problems.
For example, the fraction $$\frac{1}{3}$$ is equal to $$0.333\ldots$$, where the digit 3 repeats indefinitely. Mastering recurring decimals allows students to work confidently with fractions, ratios, percentages, and algebraic expressions involving repeating patterns.
Core Concepts
Definition of Recurring Decimals
A recurring decimal is a decimal number in which a digit or group of digits repeats indefinitely after the decimal point. These are usually indicated by a bar over the repeating digits.
- Example: $$0.\overline{3} = 0.333\ldots$$
- Example: $$0.\overline{72} = 0.727272\ldots$$
Identifying Recurring Decimals
Recurring decimals often arise when fractions have denominators that are not factors of 10.
- Example: $$\frac{1}{6} = 0.1\overline{6}$$
- Example: $$\frac{2}{11} = 0.\overline{18}$$
Converting Fractions to Recurring Decimals
Divide the numerator by the denominator using long division. If the remainder starts repeating, the decimal is recurring.
- Example: $$\frac{1}{7}$$
- 1 ÷ 7 = 0.142857142857…
- Recurring pattern: 142857 → $$0.\overline{142857}$$
Converting Recurring Decimals to Fractions
To convert a recurring decimal into a fraction, use algebra:
Step 1: Let $$x$$ equal the recurring decimal.
Step 2: Multiply $$x$$ by a power of 10 to shift the repeating digits.
Step 3: Subtract the original $$x$$ to eliminate the repeating part.
Step 4: Solve for $$x$$ as a fraction.
Example:
Convert $$0.\overline{3}$$ to a fraction:
- Step 1: $$x = 0.\overline{3}$$
- Step 2: Multiply by 10: $$10x = 3.\overline{3}$$
- Step 3: Subtract original: $$10x - x = 3.\overline{3} - 0.\overline{3} = 3$$
- Step 4: Solve: $$9x = 3 \Rightarrow x = \frac{1}{3}$$
Example with multiple repeating digits:
Convert $$0.\overline{72}$$ to a fraction:
- Step 1: $$x = 0.\overline{72}$$
- Step 2: Multiply by 100: $$100x = 72.\overline{72}$$
- Step 3: Subtract original: $$100x - x = 72.\overline{72} - 0.\overline{72} = 72$$
- Step 4: Solve: $$99x = 72 \Rightarrow x = \frac{72}{99} = \frac{8}{11}$$
Terminating vs Recurring Decimals
- Terminating decimals: Decimals that have a finite number of digits. Example: 0.25 = 1/4
- Recurring decimals: Decimals that repeat indefinitely. Example: 0.333… = 1/3
- Rule: Fractions in lowest terms with denominators of the form $$2^m 5^n$$ produce terminating decimals; other denominators produce recurring decimals.
Recurring Patterns and Period
The repeating part of a decimal is called the period.
- Example: $$0.\overline{142857}$$ has period 6 (142857 repeats)
- Example: $$0.1\overline{6}$$ has period 1 (6 repeats)
Mixed Recurring Decimals
Some decimals have non-repeating and repeating parts:
- Example: $$0.16\overline{6}$$
- Convert to fraction using algebra:
- Step 1: $$x = 0.16\overline{6}$$
- Step 2: Multiply by 10 (for one non-repeating decimal): $$10x = 1.6\overline{6}$$
- Step 3: Multiply by 10 (for repeating part): $$100x = 16.\overline{6}$$
- Step 4: Subtract: 100x - 10x = 16.\overline{6} - 1.6\overline{6} = 15
- Step 5: Solve: 90x = 15 → $$x = \frac{15}{90} = \frac{1}{6}$$
Worked Examples
Example 1 (Foundation): Simple recurring decimal
Convert $$0.\overline{7}$$ to a fraction:
- Let $$x = 0.\overline{7}$$
- Multiply by 10: $$10x = 7.\overline{7}$$
- Subtract: $$10x - x = 7.\overline{7} - 0.\overline{7} = 7$$
- $$9x = 7 \Rightarrow x = \frac{7}{9}$$
Example 2 (Higher): Two-digit repeating decimal
Convert $$0.\overline{36}$$ to a fraction:
- Let $$x = 0.\overline{36}$$
- Multiply by 100: $$100x = 36.\overline{36}$$
- Subtract: $$100x - x = 36.\overline{36} - 0.\overline{36} = 36$$
- $$99x = 36 \Rightarrow x = \frac{36}{99} = \frac{4}{11}$$
Example 3 (Higher): Mixed recurring decimal
Convert $$0.1\overline{23}$$ to a fraction:
- Let $$x = 0.1\overline{23}$$
- Multiply by 10: $$10x = 1.\overline{23}$$
- Multiply by 100 (length of repeating part): $$1000x = 123.\overline{23}$$
- Subtract: 1000x - 10x = 123.\overline{23} - 1.\overline{23} = 122
- 990x = 122 → $$x = \frac{122}{990} = \frac{61}{495}$$
Example 4 (Foundation): Recurring decimal from fraction
Convert $$\frac{1}{3}$$ to decimal:
- 1 ÷ 3 = 0.333… → $$0.\overline{3}$$
Example 5 (Higher): Recurring decimal from fraction
Convert $$\frac{7}{11}$$ to decimal:
- 7 ÷ 11 = 0.636363… → $$0.\overline{63}$$
Example 6 (Higher): Verify fraction from recurring decimal
Check that $$0.\overline{72} = \frac{8}{11}$$:
- x = 0.\overline{72}
- 100x = 72.\overline{72}
- Subtract: 100x - x = 72.\overline{72} - 0.\overline{72} = 72
- 99x = 72 → x = 72/99 = 8/11
Common Mistakes
- Not identifying the repeating part correctly
- Incorrectly applying powers of 10 when converting mixed recurring decimals
- Failing to subtract the correct equation in algebraic method
- Not simplifying the fraction to lowest terms
- Confusing terminating decimals with recurring decimals
Tips to avoid errors:
- Clearly mark the repeating digits
- Use algebraic method step by step
- Always simplify fractions after conversion
- Practice both fraction-to-decimal and decimal-to-fraction conversions
- Check patterns with long division to confirm the recurring sequence
Applications
- Converting fractions to decimals for calculations in money, probability, and ratios
- Solving algebra problems involving repeating decimals
- Checking the reasonableness of results in exams
- Real-life examples such as repeating patterns in measurements or percentages
Strategies & Tips
- Identify repeating part carefully before applying algebra
- Use powers of 10 aligned with the length of repeating digits
- Always subtract equations properly to eliminate the repeating part
- Check your answer by converting back to decimal to confirm the pattern
- Practice a variety of fractions with different lengths of repeating sequences
Summary / Call-to-Action
Recurring decimals are an important part of GCSE Maths, bridging fractions and decimals. By mastering identification, conversion, and simplification, students can solve problems confidently and accurately. Consistent practice with both simple and mixed recurring decimals strengthens understanding and application in exams and real-life situations.
Next Steps:
- Attempt quizzes on recurring decimals to reinforce learning
- Practice converting fractions to recurring decimals and vice versa
- Apply knowledge in problem-solving and algebraic contexts
- Challenge yourself with mixed recurring decimals for higher-level practice
Mastery of recurring decimals ensures confidence in fractions, percentages, ratios, and advanced calculations in GCSE Maths.