IGCSE Maths (0580) Recurring Decimals to Fractions: Easy Step-by-Step Revision Notes
Hello Grade 9 superstars! Welcome back to your ultimate math portal. Today, we are cracking open one of the most common topics in your Cambridge IGCSE Mathematics (0580) Extended syllabus: converting recurring decimals into fractions using clear algebraic methods.
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What is a Recurring Decimal?
A recurring decimal (or repeating decimal) is a decimal number where a digit or a specific pattern of digits repeats infinitely after the decimal point. Instead of writing digits forever, Cambridge pattern notations use a dot above the digits to mark where the repetition starts and stops.
- 0.3̇ means the 3 repeats forever: 0.33333...
- 0.2̇7̇ means the block '27' repeats forever: 0.272727...
- 0.41̇2̇ means only the '12' block repeats: 0.4121212...
The 2 Main Types of Exam Questions
In your IGCSE 0580 exam papers, recurring decimal questions typically fall into two categories:
- Type 1: Pure Recurring Decimals – The repeating pattern starts immediately after the decimal point (e.g., convert 0.7̇ or 0.3̇6̇ into a fraction).
- Type 2: Mixed Recurring Decimals – There are one or more non-repeating digits right after the decimal point before the pattern begins (e.g., convert 0.23̇ or 0.14̇5̇ into a fraction).
Step-by-Step Algebraic Conversion Method
To score full marks on extended structured questions, you must show your clear algebraic working. Follow these systematic steps:
- Step 1: Set up your base equation. Let a variable x equal your given decimal, and expand the pattern out a few iterations.
- Step 2: Multiply by a power of 10. Multiply your base equation by 10, 100, or 1000 so that the repeating blocks line up perfectly under each other.
- Step 3: Subtract the equations. Subtract your original equation from your new equation. This cancels out the infinite recurring tail completely!
- Step 4: Solve for x. Simplify the remaining values to form a fraction, making sure to reduce it to its lowest terms.
Stepwise Solved Worked Examples
Example 1: Converting a Type 1 Pure Recurring Decimal
Question: Show that the recurring decimal 0.4̇7̇ can be written as the fraction 47/99.
Example 2: Converting a Type 2 Mixed Recurring Decimal
Question: Convert 0.21̇6̇ into a fraction in its simplest form.
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📄 Practice Worksheet & Revision Material
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