Decimal to Fraction Calculator
Convert decimals to fractions instantly with step-by-step solutions, including repeating decimals
🔢 Convert Decimal to Fraction
For repeating decimals, use notation like 0.333... or just 0.33 (will detect pattern)
💡 Enter any decimal number. The calculator handles terminating and repeating decimals automatically.
📐 Conversion Formulas & Methods
Terminating Decimals
Where n = number of decimal places:
- • 1 decimal place: multiply by 10/10
- • 2 decimal places: multiply by 100/100
- • 3 decimal places: multiply by 1000/1000
Repeating Decimals (Algebraic Method)
For 0.333...
Let x = 0.333...
Multiply by 10: 10x = 3.333...
Subtract: 10x - x = 3.333... - 0.333...
Simplify: 9x = 3
Result: x = 3/9 = 1/3
General Rule:
• 1 repeating digit: multiply by 10, subtract, divide by 9
• 2 repeating digits: multiply by 100, subtract, divide by 99
• 3 repeating digits: multiply by 1000, subtract, divide by 999
Example: Convert 0.75
Step-by-Step Process:
Step 1: Count decimal places: 0.75 has 2 decimal places
Step 2: Write as fraction: 0.75/1
Step 3: Multiply by 100/100: (0.75 × 100)/(1 × 100) = 75/100
Step 4: Find GCD: GCD(75, 100) = 25
Step 5: Simplify: (75÷25)/(100÷25) = 3/4
Final Answer: 0.75 = 3/4
What is Decimal to Fraction Conversion?
Decimal to fraction conversion is the mathematical process of expressing a decimal number as a ratio of two integers (a fraction)—while decimals represent parts of a whole using a base-10 positional system with a decimal point, fractions express the same values as numerator divided by denominator, providing an exact representation that's often more useful in mathematics, cooking, construction, and situations requiring precise proportions without approximation.
The conversion process depends on the type of decimal: terminating decimals (like 0.5, 0.75, 2.125) have a finite number of digits and convert by placing the decimal over a power of 10 then simplifying, while repeating decimals (like 0.333..., 0.142857142857...) require algebraic manipulation because their infinite pattern makes simple place-value methods inadequate—understanding both methods enables accurate conversion of any decimal number to its fractional equivalent.
This conversion reveals important mathematical relationships: 0.5 = 1/2 shows half precisely, 0.333... = 1/3 demonstrates that some fractions produce infinite repeating decimals, and 0.125 = 1/8 illustrates how powers of 2 create terminating decimals—these conversions are fundamental in algebra, geometry, probability, and real-world applications where exact fractional values matter more than decimal approximations, such as measuring ingredients, calculating probabilities, or working with musical intervals.
📚 Types of Decimals
Terminating Decimals
Decimals that end after a finite number of digits—examples include 0.5, 0.75, 2.125, 0.0625. These result from fractions whose denominators contain only factors of 2 and 5 (factors of 10).
Conversion: Multiply by 10n where n = decimal places
Repeating Decimals
Decimals with one or more digits that repeat infinitely—written with a bar over repeating digits: 0.3̅ (0.333...), 0.1̅4̅2̅8̅5̅7̅ (1/7). Result from fractions with denominators containing factors other than 2 and 5.
Conversion: Use algebraic method with multiplication and subtraction
Mixed Repeating Decimals
Decimals with some non-repeating digits followed by repeating pattern—like 0.16̅ (0.1666...) or 0.583̅ (0.58333...). Require combined approach: handle non-repeating and repeating parts separately.
Conversion: Algebraic method accounting for non-repeating prefix
Common Decimal to Fraction Conversions
0.5
= 1/2
0.25
= 1/4
0.75
= 3/4
0.333...
= 1/3
0.666...
= 2/3
0.125
= 1/8
0.2
= 1/5
0.6
= 3/5
🌐 Real-World Applications
Cooking & Baking
Recipes require precise fractional measurements—converting 0.375 cups to 3/8 cup allows accurate measuring with standard kitchen tools. Scaling recipes by 1.5× means understanding 1.5 = 3/2 for proper ingredient ratios.
Construction & Carpentry
Measurements like 2.625 inches convert to 2 5/8 inches for tape measure readings—architects and carpenters work exclusively in fractions (1/16", 1/8", 1/4") making decimal conversion essential for precision work.
Mathematics & Education
Understanding that 0.5 = 1/2 and 0.333... = 1/3 builds number sense—fraction conversion helps students see relationships between decimal and fractional representations, essential for algebra, probability, and higher mathematics.
Finance & Investing
Stock prices like $45.375 per share equal $45 3/8—bond prices are quoted in 32nds (1/32) requiring decimal conversion. Understanding 0.0625 = 1/16 helps interpret financial data and calculate precise investment values.
Music & Rhythm
Musical time signatures use fractions (4/4, 3/4, 6/8)—tempo ratios like 1.5 beats per second = 3/2 ratio help musicians understand rhythmic relationships. Note durations (whole, 1/2, 1/4, 1/8) are fundamentally fractional.
Science & Engineering
Gear ratios (2.5:1 = 5/2), chemical concentrations (0.85 molar = 17/20), and mechanical advantage calculations require converting between decimals and fractions—exact fractional values prevent compounding errors in multi-step engineering calculations.
⚠️ Important Considerations
🔄 Repeating Decimal Detection:
Not all decimals that appear to repeat actually do—0.142857142857... (from 1/7) repeats with 6-digit cycle, but 0.12345678... might not repeat at all. When manually converting, verify the pattern continues before applying repeating decimal methods. Our calculator detects patterns automatically but be cautious with manual calculations.
📏 Simplification Is Essential:
Always reduce fractions to lowest terms by finding the Greatest Common Divisor (GCD)—0.75 initially converts to 75/100 but simplifies to 3/4. The simplified form is standard in mathematics and more meaningful in real-world contexts. Unsimplified fractions like 75/100 are technically correct but not in standard form.
➗ Mixed Numbers vs. Improper Fractions:
Decimals greater than 1 can be expressed as mixed numbers (2 3/4) or improper fractions (11/4)—both are mathematically equivalent but serve different purposes. Mixed numbers are intuitive for measurements and everyday use, while improper fractions are often preferred in algebra and calculus for easier manipulation and calculation.
∞ Irrational Numbers Cannot Be Converted:
Numbers like π (3.14159...), √2 (1.41421...), and e (2.71828...) are irrational—their decimal expansions never terminate or repeat, so they cannot be expressed as exact fractions. You can approximate them (π ≈ 22/7 or 355/113) but no fraction exactly equals an irrational number, making these approximations useful but not precise.
About the Author
Adam
Co-Founder @RevisionTown
Math Expert specializing in diverse international curricula including IB (International Baccalaureate), AP (Advanced Placement), GCSE, IGCSE, and various other educational programs worldwide.
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