Edexcel 4MA1 • 1.3 DecimalsGrade 9 Killer
Recurring Decimals: The Loop Breaker
Master algebraic proof techniques to convert infinite decimals into exact fractions
The Loop Breaker (Interactive)
Original Decimal
0.333...
Notation: 0.3̇
After multiplying by 10
3.333...
💡 EXAMINER TIP
1 repeating digit → multiply by 10. This creates 9x = integer.
The Logic Track
The Goal
We hate infinity. We use algebra to trap the infinite part and delete it by subtraction.
The Strategy
- →Let x = the recurring decimal
- →Multiply by 10ⁿ where n = number of repeating digits
- →The "tails" align perfectly
- →Subtract to eliminate the infinite part
- →Solve for x and simplify the fraction
🚨 Examiner Trap Alert
Don't confuse 0.3̇ with 0.3! The dot means the digit repeats FOREVER. 0.3̇ = 1/3 exactly, but 0.3 = 3/10.
Phase Progression
Phase 1: The Anchor (Foundation)
Simple single-digit recurring: 0.3̇ = 1/3
Target: Grades 4-5 confidence builder
Phase 2: The Twist (Higher Tier)
Two-digit cycle: 0.1̇2̇ = 4/33
Target: Grades 6-7 pattern recognition
Phase 3: The Killer (Grade 8-9)
Mixed recurring: 0.2̇5̇4̇ requires double multiplication
Separator question - algebraic mastery
Dynamic Solver: Mark Scheme Working
Question Type: 0.3̇
Step 1: Let x = 0.333...
Step 2: Multiply by 10: 10x = 3.333...
Step 3: Subtract: 10x − x
9x = 3
Step 4: Solve for x
x = 3/9 = 1/3
Mark Scheme Notes
✓ Method Mark (M1)
Correctly identifying the multiplication factor (10ⁿ where n = cycle length)
✓ Accuracy Mark (A1)
Correctly performing the subtraction to eliminate the recurring part
✓ Final Answer (A1)
Simplifying the fraction to lowest terms
⚠ Common Error
Forgetting to subtract x after multiplication. The infinite tails must CANCEL OUT.
Final Conversion Result
0.333... = 1/3
This is an EXACT fraction - no approximation!
🎯 Exam Strategy Summary
Rule 1: Count the Cycle
1 digit repeating → ×10
2 digits repeating → ×100
3 digits repeating → ×1000
2 digits repeating → ×100
3 digits repeating → ×1000
Rule 2: The Subtraction
Always subtract the original x from the multiplied version. The tails MUST cancel perfectly.
Rule 3: Simplify
Find the HCF of numerator and denominator. Full marks require simplest form.
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