How to Express 0.05 Repeating as a Fraction
What Does "0.05 Repeating" Actually Mean?
First, let's get one thing straight. When people say "0.05 repeating," they usually mean one of two things:
- 0.05555... — the digit 5 repeats infinitely after the decimal
- 0.050505... — the block "05" repeats infinitely
Both are valid interpretations. I'll show you how to handle each case because math doesn't care about ambiguity.
The Method: Converting Repeating Decimals to Fractions
Every repeating decimal follows the same logic:
- Set the decimal equal to x
- Multiply to shift the decimal point past one full repeating block
- Subtract to eliminate the repeating portion
- Solve for x
That's it. No shortcuts, no tricks. Just algebra.
Case 1: 0.05555... (5 Repeating)
Let x = 0.05555...
Step 1: Multiply by 10 to get one full repeat into view
10x = 0.5555...
Step 2: Multiply by 100 to get two full repeats
100x = 5.5555...
Step 3: Subtract
100x - 10x = 5.5555... - 0.5555...
90x = 5
Step 4: Solve
x = 5/90
Reduce: x = 1/18
Quick Check
1 ÷ 18 = 0.05555... ✓
Case 2: 0.050505... (05 Repeating)
Let x = 0.050505...
Step 1: The repeating block is "05" — that's 2 digits. Multiply by 100.
100x = 5.0505...
Step 2: Multiply by 10,000 to shift past two repeats
10000x = 505.0505...
Step 3: Subtract
10000x - 100x = 505.0505... - 5.0505...
9900x = 500
Step 4: Solve
x = 500/9900
Reduce: x = 5/99
Quick Check
5 ÷ 99 = 0.050505... ✓
Comparison Table
| Decimal Form | Fraction Form | Reduced To |
|---|---|---|
| 0.05555... | 5/90 | 1/18 |
| 0.050505... | 500/9900 | 5/99 |
Getting Started: Practice Problems
Try these on your own before checking answers:
- 0.07777... → 7/90
- 0.121212... → 12/99 = 4/33
- 0.03333... → 3/90 = 1/30
The process never changes. Count your repeating digits, multiply accordingly, subtract, and simplify.
Why This Works
You're exploiting a simple fact: when you subtract a repeating decimal from a shifted version of itself, the repeating parts cancel out. What's left is a clean integer equation.
No memorization needed. Just multiplication and subtraction.
If 0.05 repeating gives you 0.05555..., your answer is 1/18. If it gives you 0.050505..., your answer is 5/99.