1/x Patterns
Magic Fractions: 1/7, 1/9, 1/11.
Core Concepts
Some fractions have beautiful repeating patterns. Learning these patterns makes you look like a human calculator!
The 9's Family
Denominator 9 repeats the numerator.
- 1/9 = 0.111...
- 2/9 = 0.222...
- 5/9 = 0.555...
The 11's Family
Denominator 11 repeats (9 × numerator).
- 1/11 = 0.0909... (9×1)
- 2/11 = 0.1818... (9×2)
- 3/11 = 0.2727... (9×3)
The Magic 7
1/7 has a special 6-digit repeating cycle: 142857.
For 2/7, 3/7 etc., the cycle just starts at a different digit!
Solved Examples
Example 1: 4/9
Question: Convert 4/9 to decimal.
• Denominator is 9.
• Rule: Repeat the numerator.
• Answer: 0.444...
Example 2: 4/11
Question: Convert 4/11 to decimal.
• Denominator is 11.
• Rule: Multiply numerator by 9. (4 × 9 = 36).
• This number repeats.
• Answer: 0.3636...
Example 3: 1/7
Question: Convert 1/7 to decimal.
• Memorize the magic number: 142857.
• Answer: 0.142857...
Example 4: 2/7
Question: Convert 2/7 to decimal.
• Cycle is 1-4-2-8-5-7.
• 2/7 starts with the second smallest digit in the cycle (2).
• Start at 2: 285714.
• Answer: 0.285714...
