How do you convert a decimal to a fraction?

Multiply the decimal by a power of 10 to remove the decimal point, creating an integer numerator. Use 10^n as denominator (n = decimal places). Then simplify by dividing both by GCD. Example: 0.75 = 75/100 → divide by 25 → 3/4.

Can all decimals be converted to fractions?

Only rational numbers can be exact fractions. Terminating decimals (0.5, 0.25, 0.625) and repeating decimals (0.333..., 0.142857...) can. Irrational numbers (π, e, √2) only have approximate fractions — never exact. The decimal expansion goes forever without repeating.

How do I convert a repeating decimal?

For 0.333... let x = 0.333..., then 10x = 3.333..., subtract: 9x = 3, x = 1/3. For 0.142857142857... (6 digits repeating), use 10^6 factor: 999999 × x = 142857, so x = 142857/999999 = 1/7.

What's the fraction for 0.625?

5/8. Calculation: 0.625 = 625/1000. GCD(625, 1000) = 125. Divide: 625/125 = 5, 1000/125 = 8. Result: 5/8. Common in woodworking (5/8 inch is a standard size for drill bits and screws).

How do I write 1.75 as a fraction?

As improper: 175/100 = 7/4 (after simplifying by 25). As mixed number: 1 + 3/4 = 1 3/4 (one and three-quarters). Both represent the same value. Mixed numbers more common in everyday use; improper fractions more common in algebra.

Why use fractions instead of decimals?

Fractions are exact (no rounding); decimals may approximate. Easier for mental arithmetic with small denominators. Standard for imperial measurements (inches, cups). Some calculations are cleaner: 1/3 + 1/6 = 1/2 vs 0.333... + 0.166... = 0.5. Modern usage depends on context.

What does π equal as a fraction?

π is irrational — no exact fraction equals π. Common approximations: 22/7 ≈ 3.1429 (school standard, ~0.04% error). Better: 355/113 ≈ 3.14159 (~0.000008% error, accurate to 6 decimals). For more precision, continued fraction provides infinite best approximations.

How precise should my fraction be?

Depends on application. Cooking: 1/4 cup precision usually enough. Woodworking: 1/32 or 1/64 inch. Engineering: depends on tolerance. For irrational numbers, choose denominator limit balancing precision against simplicity (small fractions = clearer; large = more accurate).

Decimal to Fraction Calculator

Enter a decimal number to convert it to a fraction in simplest form. Also shows the mixed number representation if applicable.

Decimals and fractions are two ways of representing the same kinds of numbers — rational numbers. Decimals use base-10 place value (0.5 means 5/10), while fractions use a numerator-over-denominator format (1/2). Each representation has advantages: decimals are easier for arithmetic with calculators and computers; fractions are exact (no rounding) and often easier to grasp conceptually.

Converting decimals to fractions is a fundamental skill that appears throughout math education, engineering work, cooking measurements, woodworking dimensions, and music theory. While modern calculators handle decimals fluently, fractions still dominate in many practical domains where precision matters or where standard fraction units are used (1/4 inch, 3/8 cup, etc.).

Three categories of decimals convert differently:

**Terminating decimals** (like 0.75, 0.125) have a finite number of digits after the point. These convert directly: 0.75 = 75/100 = 3/4. The denominator is 10^n where n is the number of decimal places, then simplified.

**Repeating decimals** (like 0.333..., 0.142857...) have a pattern that repeats forever. These also have exact fraction equivalents (1/3, 1/7) — found using algebra or pattern recognition.

**Non-repeating, non-terminating decimals** (π = 3.14159..., √2 = 1.41421...) are irrational and have no exact fraction. They can be approximated to any desired precision, but not represented exactly as p/q.

This calculator handles terminating decimals exactly. For repeating decimals, enter enough digits to indicate the pattern. For irrational numbers, it provides a best rational approximation within the specified denominator limit.

Common applications: cooking (decimal to fraction for measuring cups), woodworking and construction (decimal inches to fractional inches), music (decimal ratios to fractional intervals), education (math exercises), engineering specifications, and any context preferring exact rational expression.

Inputs

Decimal Number

Max Denominator

Higher values give more precise fractions

Results

Fraction

3/4

Mixed Number

3/4

Numerator

Denominator

Percentage

75.0000%

Last updated: May 29, 2026

Formula

**Converting terminating decimal to fraction:** 1. Count digits after decimal point: n. 2. Write decimal as integer over 10^n. 3. Simplify by dividing top and bottom by GCD. **Worked example: 0.625** 3 decimal places. 0.625 = 625/1000. GCD(625, 1000) = 125. 625/125 = 5; 1000/125 = 8. Answer: 5/8. **Converting repeating decimal:** For 0.3333... (3 repeating): - Let x = 0.333... - Multiply by 10: 10x = 3.333... - Subtract: 10x - x = 3.333... - 0.333... - 9x = 3 - x = 3/9 = 1/3. For 0.142857142857... (6 digits repeating, "142857"): - Let x = 0.142857142857... - Multiply by 10^6 = 1,000,000: 10^6 × x = 142857.142857... - Subtract: (10^6 - 1)x = 142857 - x = 142857/999,999 = 1/7. **Mixed numbers:** For decimals > 1: convert whole and fractional parts separately. 3.75 = 3 + 0.75 = 3 + 3/4 = 3 3/4 (three and three-quarters) Or as improper: 3.75 = 15/4. **Common decimal-fraction pairs:** | Decimal | Fraction | |---|---| | 0.5 | 1/2 | | 0.25 | 1/4 | | 0.75 | 3/4 | | 0.125 | 1/8 | | 0.375 | 3/8 | | 0.625 | 5/8 | | 0.875 | 7/8 | | 0.1 | 1/10 | | 0.2 | 1/5 | | 0.333... | 1/3 | | 0.666... | 2/3 | | 0.1666... | 1/6 | | 0.142857... | 1/7 | | 0.1111... | 1/9 | | 0.0625 | 1/16 | | 0.03125 | 1/32 | **Sixteenths (woodworking):** | Decimal | Fraction | |---|---| | 0.0625 | 1/16 | | 0.125 | 2/16 = 1/8 | | 0.1875 | 3/16 | | 0.25 | 4/16 = 1/4 | | 0.3125 | 5/16 | | 0.375 | 6/16 = 3/8 | | 0.4375 | 7/16 | | 0.5 | 8/16 = 1/2 | | 0.5625 | 9/16 | | 0.625 | 10/16 = 5/8 | | 0.6875 | 11/16 | | 0.75 | 12/16 = 3/4 | | 0.8125 | 13/16 | | 0.875 | 14/16 = 7/8 | | 0.9375 | 15/16 | **Simplification (GCD):** To simplify p/q to lowest terms: 1. Find GCD of p and q. 2. Divide both by GCD. For 24/36: GCD = 12. 24/12 = 2; 36/12 = 3. Result: 2/3. Euclidean algorithm finds GCD efficiently. **Continued fractions (for irrational approximations):** π ≈ 3.14159265... Continued fraction: 3 + 1/(7 + 1/(15 + 1/(1 + 1/(292 + ...)))) Convergents (truncations): - 3 (whole part) - 22/7 ≈ 3.142857 (good!) - 333/106 ≈ 3.141509 - 355/113 ≈ 3.141593 (very good, accurate to 6 decimal places) 355/113 is famous as a remarkably accurate π approximation. **Decimals that aren't simple fractions:** | Decimal | Approximate fraction | |---|---| | π ≈ 3.14159 | 22/7 (rough), 355/113 (great) | | e ≈ 2.71828 | 19/7 (rough), 2721/1001 (great) | | √2 ≈ 1.41421 | 7/5 (rough), 99/70 (great) | | √3 ≈ 1.73205 | 19/11, 71/41 (better) | | φ (golden) ≈ 1.61803 | 8/5, 13/8, 21/13 (Fibonacci) | These rational approximations are best possible for given denominator size. **Why fractions for measurement:** Imperial measurements use fractions: 1/4 inch, 3/8 inch, 5/8 inch, etc. Reasons: - Easier to subdivide ruler marks visually. - Standard fractions of common units (cup, pound, foot). - Historical (developed before decimal calculators). Metric uses decimals: 0.625 cm easier than 5/8 cm. **Mixed and improper fractions:** Improper: numerator > denominator (like 7/4). Mixed: whole part + proper fraction (like 1 3/4). Same value: 7/4 = 1 + 3/4 = 1.75. **Computer storage of fractions:** Languages with rational types: - **Python**: Fraction class in fractions module. - **Ruby**: Rational class. - **SageMath**: native support. - **Haskell, Scala**: built-in. Floating-point (most languages) uses binary fractions internally, so 0.1 isn't exact (1/3 in binary is 0.0101010101... repeating, like 1/3 in decimal). **Practical examples:** - **Cooking**: "1.5 cups" = 1 1/2 cups (more intuitive). - **Woodworking**: "0.625 inch" = 5/8 inch (matches tools). - **Music**: "1.5 ratio" = 3/2 = perfect fifth. - **Finance**: "1.25 multiplier" = 5/4 = 25% premium. **Common applications:** - **Education**: math homework, fraction-decimal conversions. - **Cooking**: recipe scaling, measuring. - **Woodworking**: rulers, blueprints. - **Engineering**: tolerance specs. - **Programming**: precise rational arithmetic. - **Music theory**: intervals as ratios. - **Statistics**: probabilities as fractions. **Software:** - **Calculators**: most modern scientific calculators have decimal-to-fraction button. - **Python**: from fractions import Fraction; Fraction(0.75).limit_denominator(1000). - **Wolfram Alpha**: converts in either direction. - **Mobile apps**: dedicated converters.

How to use this calculator

Enter a decimal number (positive or negative).
Specify maximum denominator (higher = more precise).
Calculator returns simplest fraction form.
For decimals > 1: also shows mixed number form.
For repeating decimals: enter enough digits to indicate pattern.
For irrational numbers: get best rational approximation within denominator limit.

Worked examples

Cooking measurement

**Scenario:** Recipe calls for 0.667 cup. What fraction? **Calculation:** 0.667 ≈ 2/3 (= 0.6667). The 0.667 is a rounded version of 2/3. **Result:** 2/3 cup — standard cooking measure. Your measuring cups have 1/4, 1/3, 1/2, 2/3, 3/4, and 1 cup marks. Calculator approximates 0.667 to 2/3 because it's the cleanest fraction with small denominator.

Woodworking measurement

**Scenario:** Plans show "0.4375 inch" for a slot width. What fractional inch? **Calculation:** 0.4375 = 4375/10000 = 7/16 (after GCD division by 625). **Result:** 7/16 inch — common drill bit and router bit size. Woodworking tools (rulers, drill indexes) are typically marked in sixteenths or thirty-seconds, so converting from decimal to fraction matches tool sizes.

Approximating pi

**Scenario:** Need rational approximation of π for a quick calculation. **Calculation:** π ≈ 3.14159265... Best approximations: - 3/1 (rough) - 22/7 ≈ 3.1429 (good for school) - 333/106 ≈ 3.1415 (better) - 355/113 ≈ 3.14159 (excellent, 6 decimal places) **Result:** 22/7 is famous and useful for mental calculation. 355/113 is remarkably good — used historically before decimal calculators. Both are infinite-precision rationals approximating an irrational.

When to use this calculator

**Use decimal-to-fraction conversion for:**

- **Math education**: building number sense. - **Cooking**: matching standard measure marks. - **Woodworking**: matching ruler markings and tool sizes. - **Engineering specifications**: imperial unit tolerances. - **Music theory**: ratios as intervals. - **Computer programming**: exact rational arithmetic. - **Probabilities**: communicating odds as fractions. - **Statistics**: exact ratio expressions.

**Decimal types:**

- **Terminating** (0.75, 0.125): exact finite fraction. - **Repeating** (0.333..., 0.6666...): exact fraction with finite denominator. - **Irrational** (π, e, √2): only approximate fractions possible.

**Imperial vs metric:**

Imperial favors fractions (1/4 inch, 1/2 cup); metric uses decimals (5 mm, 0.5 L). For US craftspeople, decimal-to-fraction conversion is essential.

**Standard fractions in tools:**

Drill bits, wrenches, sockets come in standard fractional sizes: - 1/16, 1/8, 3/16, 1/4, 5/16, 3/8, 7/16, 1/2, 9/16, 5/8, 11/16, 3/4, 13/16, 7/8, 15/16, 1.

Decimal equivalents help when ordering parts specified in decimal but tooled in fractions.

**Common conversions:**

- 1/8 = 0.125 (eighths inch) - 1/4 = 0.25 - 1/3 ≈ 0.333 (cup) - 1/2 = 0.5 - 2/3 ≈ 0.667 (cup) - 3/4 = 0.75 - 1/16 = 0.0625 (sixteenths) - 1/32 = 0.03125 - 1/64 = 0.015625

**Continued fractions:**

The "best" rational approximations to irrational numbers come from continued fractions. Each convergent is the best possible rational with that-sized denominator.

**Pitfalls:**

- **Rounding errors**: 0.333 might convert to 333/1000 instead of 1/3. - **Repeating decimals**: enter enough digits or use special syntax. - **Irrational numbers**: only approximations exist. - **Floating-point precision**: tiny errors accumulate. - **Calculator default precision**: may give large unsimplified fractions.

**Common applications:**

- **Cooking recipes**: scale ingredients, convert metric to imperial. - **Construction**: blueprint measurements, materials lists. - **Sewing**: pattern measurements. - **Music**: just intonation ratios. - **Finance**: simple ratios for percentages. - **Programming**: avoiding floating-point errors with rational class.

**Software with rationals:**

- **Python**: from fractions import Fraction; f = Fraction(0.75); print(f) gives 3/4. - **Mathematica**: native rational support. - **SageMath**: exact arithmetic. - **Calculators**: TI-83/84 have "MATH" → "1: ▶Frac" function.

**Mental math tricks:**

- 0.5 = 1/2 (obvious). - 0.25 = 1/4; 0.75 = 3/4. - 0.333... = 1/3; 0.667 = 2/3. - 0.2 = 1/5; 0.4, 0.6, 0.8 = 2/5, 3/5, 4/5. - 0.166... = 1/6.

**Repeating decimal patterns:**

For prime denominators p where 10 is a primitive root mod p: - 1/7 = 0.142857... - 1/17 = 0.0588235294117647... - 1/19 = 0.052631578947368421...

Pattern length always divides (p-1) for prime p.

**Pitfalls:**

- **Rounding before conversion**: 0.333 may convert to 333/1000 if not recognized as 1/3 pattern. - **Maximum denominator too small**: poor approximations for complex decimals. - **Sign**: negative decimals → negative fractions. - **Decimal precision**: more digits = more precise fraction.

Common mistakes to avoid

Not simplifying the fraction (e.g., reporting 50/100 instead of 1/2).
Rounding decimal before conversion (causes wrong fraction).
Missing repeating pattern (e.g., treating 0.333 as terminating).
Trying to convert irrational numbers to exact fractions.
Setting denominator limit too low for complex decimals.
Floating-point precision issues (0.1 + 0.2 ≠ 0.3 exactly in computers).
Confusing terminating with repeating decimals.
For mixed numbers: forgetting whole part.

Frequently Asked Questions

Sources & further reading

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