Decimal to Fraction Calculator
The universal converter for translating terminating and repeating decimals into exact, simplified fractions and mixed numbers. Features complete algebraic derivations and Greatest Common Divisor (GCD) reductions.
Enter Standard Decimal
e.g. 0.75, 2.625, -3.4Repeating Decimal Input
e.g. 0.333... = 1/3Fraction Output
Mathematical Derivation Steps
How to Convert Decimals to Fractions
A decimal represents a fraction whose denominator is a power of 10. To convert a terminating decimal to a fraction, place the decimal digits over 10^n (where n is the number of digits after the decimal point) and simplify the fraction by dividing both numerator and denominator by their Greatest Common Divisor (GCD).
Place Value & Decimal Anatomy
The base-10 decimal system is an extension of whole-number place values into fractional units. Each position to the right of the decimal point represents a reciprocal power of 10:
Understanding this positional value allows any terminating decimal to be directly rewritten as an integer divided by 10ⁿ, where n is the count of decimal positions.
Converting Terminating Decimals (Powers of 10 & GCD)
A terminating decimal has a finite number of digits after the decimal separator. Converting it to its simplest fraction requires three systematic steps:
Count the number of digits (n) to the right of the decimal point. The unsimplified denominator is 10ⁿ.
Remove the decimal point to form the numerator, and place it over the power of 10 denominator.
Calculate the GCD of the numerator and denominator using prime factorization or Euclid's algorithm. Divide both terms by the GCD.
The Algebraic Method for Repeating Decimals
Repeating decimals (such as 0.333... or 0.1666...) cannot be placed over powers of 10 because their fractional tail is infinite. Instead, we use an algebraic cancellation technique:
1. Let x = 0.1666...
2. Shift decimal to end of non-repeating part (1 digit ⟹ multiply by 10):
10x = 1.666... (Equation 1)
3. Shift decimal past one repeating period (1 more digit ⟹ multiply by 100):
100x = 16.666... (Equation 2)
4. Subtract Equation 1 from Equation 2 to eliminate the repeating tail:
100x − 10x = 16.666... − 1.666...
90x = 15
5. Solve for x and reduce:
x = 15 / 90 = (15 ÷ 15) / (90 ÷ 15) = 1 / 6
Common Decimals to Fractions Reference Chart
Standard conversions encountered in engineering, woodworking, cooking, and finance:
| Decimal | Fraction | Type | Percentage |
|---|---|---|---|
| 0.1 | 1/10 | Terminating | 10% |
| 0.125 | 1/8 | Terminating (1/8th) | 12.5% |
| 0.166... | 1/6 | Repeating | 16.67% |
| 0.2 | 1/5 | Terminating | 20% |
| 0.25 | 1/4 | Terminating (Quarter) | 25% |
| 0.333... | 1/3 | Repeating (Third) | 33.33% |
| 0.375 | 3/8 | Terminating (3/8ths) | 37.5% |
| 0.5 | 1/2 | Terminating (Half) | 50% |
| 0.625 | 5/8 | Terminating (5/8ths) | 62.5% |
| 0.666... | 2/3 | Repeating (Two-Thirds) | 66.67% |
| 0.75 | 3/4 | Terminating (Three-Quarters) | 75% |
| 0.875 | 7/8 | Terminating (7/8ths) | 87.5% |
Real-World Applications & Use Cases
Converting decimals to fractions is an essential practical skill across tradecraft, cooking, engineering, and data science:
Carpentry & Drill Bit Sizing
Tape measures and fractional wrench sets use 16ths, 32nds, and 64ths of an inch. A digital caliper reading of 0.4375" converts to an exact 7/16" wrench.
Culinary & Baking Measurement
When scaling recipes, decimal quantities like 1.333 cups translate to 1 1/3 cups on standard volumetric measuring cups.
Financial Ratios & Stock Splits
Stock splits and debt-to-equity ratios often output decimal values like 1.5 which represent 3:2 (3/2) shares.
Odds & Probability Theory
A statistical probability of p = 0.05 is understood intuitively when expressed as 1 in 20 (1/20).
Step-by-Step Worked Examples
Convert 0.36 into a fraction in lowest terms.
1. Count decimal digits: 2 digits ⟹ Denominator = 10² = 100.
2. Unreduced fraction = 36 / 100.
3. Calculate GCD(36, 100): Both divide by 4 ⟹ GCD = 4.
4. Divide numerator and denominator: (36 ÷ 4) / (100 ÷ 4) = 9 / 25.
Convert 3.875 into an improper fraction and mixed number.
1. Separate whole part (3) and fractional part (0.875).
2. Convert 0.875: 3 decimal digits ⟹ 875 / 1000.
3. GCD(875, 1000) = 125 ⟹ (875 ÷ 125) / (1000 ÷ 125) = 7 / 8.
4. Mixed Number: 3 7/8.
5. Improper Fraction: (3 × 8 + 7) / 8 = 31 / 8.
Common Calculation Pitfalls
Forgetting to Reduce to Lowest Terms
Writing 75/100 or 50/100 without dividing by their GCD leaves the fraction unsimplified and mathematically incomplete.
Miscounting Decimal Place Values
Treating 0.05 as 5/10 instead of 5/100 = 1/20. Leading zeros after the decimal point must be counted as place values.
Treating Repeating Decimals as Terminating
Approximating 0.333 as 333/1000 instead of using the algebraic repeating method which yields the exact fraction 1/3.
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