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.
**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.
**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.
**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.
**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.
Convert a fraction to its decimal and percentage equivalent.
Add, subtract, multiply, and divide fractions with step-by-step results.
Convert between mixed numbers and improper fractions.
Solve a/b = c/d for any missing value in a proportion.
Simplify ratios and scale them to a desired total.
Calculate golden ratio relationships using phi (1.618...).
Higher values give more precise fractions
Fraction
3/4
Mixed Number
3/4
Numerator
3
Denominator
4
Percentage
75.0000%