Converting between Recurring Decimals and Fractions for GCSE Mathematics
Introduction
Key Concepts
- Recurring Decimal: A decimal that has a sequence of digits that repeat indefinitely.
- NonRecurring Decimal: A decimal that does not have any digits that repeat indefinitely.
- Terminating Decimal: A decimal that has a finite number of digits.
Converting Recurring Decimals to Fractions
To convert a recurring decimal to a fraction, follow these steps:
- Identify the repeating digits: Mark the repeating digits with a line above them.
- Multiply the repeating digits by 9: This gives you the numerator of the fraction.
- Subtract the original decimal: This gives you the denominator of the fraction.
- Example:
Convert 0.555... to a fraction.
- Repeating digits: 5
- 5 9 = 45
- 0.555 0.5 = 0.055
- Fraction: 45 / 100
Converting Fractions to Recurring Decimals
To convert a fraction to a recurring decimal, follow these steps:
- Divide the numerator by the denominator: This will give you the decimal representation.
- Check if the decimal terminates or repeats: If it terminates, you are done. If it repeats, identify the repeating digits.
- Move the decimal to the end of the repeating digits: This will give you the recurring decimal representation.
- Example:
Convert the fraction 1/3 to a recurring decimal.
- 1 / 3 = 0.33333...
- Repeating digits: 3
- Recurring decimal: 0.3
Common Mistakes
- Not identifying the repeating digits correctly.
- Forgetting to multiply the repeating digits by 9.
- Not subtracting the original decimal when converting to a fraction.
- Not checking for termination when converting to a decimal.
Practice Problems
- Question 1: Convert 0.7272... to a fraction.
- Question 2: Convert the fraction 1/6 to a recurring decimal.
Conclusion
Converting between recurring decimals and fractions is an important skill in GCSE Mathematics. By understanding the key concepts and steps involved, you can master this topic and prepare for success in your exams.
Remember to practice regularly and refer to the formulas and steps provided in this article. Good luck with your studies!
FAQ
- Q: How do I know if a decimal terminates or repeats?
A: A decimal terminates if the quotient has a finite number of digits. Otherwise, it repeats.
- Q: Why do we multiply the repeating digits by 9?
A: Multiplying the repeating digits by 9 cancels out the decimal places in the original number.
- Q: What is the difference between a recurring decimal and a terminating decimal?
A: A recurring decimal has digits that repeat indefinitely, while a terminating decimal has a finite number of digits.