Recurring Decimals to Fractions for GCSE Mathematics
Introduction
A recurring decimal is a decimal that has a digit or group of digits that repeat indefinitely. These decimals can be converted into fractions using a simple method.
Key Concepts and Definitions
- Recurring Decimal: A decimal with a repeating pattern, e.g., 0.666...
- Fraction: A quotient of two integers, e.g., 2/3
- Period: The repeating sequence of digits in a recurring decimal, e.g., "6" in 0.666...
Step-by-Step Conversion
1. Identify the Period: Circle the repeating digits.
2. Set up the Equation: Let x = the recurring decimal. Write an equation where x is equal to itself, but with the period shifted one place to the left.
3. Subtract the Equations: Subtract the second equation from the first. The repeating period will cancel out.
4. Solve for x: Solve the resulting equation for x. This gives you the fraction equivalent to the recurring decimal.
Example
Convert the recurring decimal 0.666... to a fraction.
1. Period: 6
2. x = 0.666...
3. 10x = 6.666...
4. 10x - x = 6.666... - 0.666...
5. 9x = 6
6. x = 2/3
Common Mistakes to Avoid
- Not identifying the period correctly
- Forgetting to multiply and subtract by the appropriate power of 10
- Assuming that the denominator of the fraction is always the difference of the two numbers you subtract
Practice Problems
1. Convert 0.444... to a fraction.
2. Find the fraction equivalent of 0.333...
3. Write 0.1212... as a fraction.
Conclusion
Converting recurring decimals to fractions is a fundamental skill in GCSE Mathematics. By understanding the steps and avoiding common mistakes, students can confidently tackle this topic and improve their exam performance.
Practice Resources
- [Fraction Conversion Practice Problems](www.example.com/fractionpractice)
- [Converting Recurring Decimals to Fractions Game](www.example.com/fractiongame)
FAQ
- Can all recurring decimals be converted to fractions? Yes, any recurring decimal can be expressed as a fraction.
- What if the period is longer than one digit? The process remains the same, but you may need to use more steps.
- How do I find the fraction for a nonterminating decimal that doesn't repeat? Nonterminating decimals cannot be converted to fractions.