Repeating Decimal to Fraction Calculator
Convert repeating decimals to fractions with algebraic step-by-step solutions
Example: For 0.333... enter "0" and "3" | For 0.1666... enter "0.1" and "6" | For 0.142857... enter "0" and "142857"
How to Convert Repeating Decimals to Fractions
Repeating decimals (also called recurring decimals) can be converted to fractions using an algebraic technique. This method works by setting up equations and eliminating the repeating part.
The Algebraic Method
- Let x equal the repeating decimal
- Multiply x by powers of 10 to shift the decimal point
- Subtract equations to eliminate the repeating part
- Solve for x to get the fraction
- Simplify the fraction
Example: Converting 0.333...
- Let x = 0.333...
- Multiply by 10: 10x = 3.333...
- Subtract: 10x - x = 3.333... - 0.333...
- Simplify: 9x = 3
- Solve: x = 3/9 = 1/3
Example: Converting 0.1666... (mixed repeating)
- Let x = 0.1666...
- Multiply by 10: 10x = 1.666...
- Multiply by 100: 100x = 16.666...
- Subtract: 100x - 10x = 16.666... - 1.666...
- Simplify: 90x = 15
- Solve: x = 15/90 = 1/6
Common Repeating Decimals
Frequently Asked Questions
How do you convert a repeating decimal to a fraction?
Use the algebraic method: set x equal to the decimal, multiply by powers of 10 to shift digits, subtract to eliminate the repeating part, then solve for x.
What is 0.333... as a fraction?
0.333... equals 1/3. Using the algebraic method: 10x - x = 3, so 9x = 3, therefore x = 1/3.
Can all repeating decimals be converted to fractions?
Yes! All repeating decimals can be expressed as exact fractions. This is because they represent rational numbers (ratios of integers).
What's the difference between 0.3 and 0.333...?
0.3 is a terminating decimal equal to 3/10. 0.333... is a repeating decimal equal to 1/3. They are different numbers with different fractional representations.
How do you handle decimals that don't repeat immediately?
For decimals like 0.1666... (where 1 doesn't repeat but 6 does), you multiply by different powers of 10 to account for both the non-repeating and repeating parts.
Why do some fractions produce repeating decimals?
When a fraction in simplest form has a denominator with prime factors other than 2 or 5, it produces a repeating decimal. This is due to the base-10 number system.