How do you add fractions?

Find a common denominator (LCM of the two denominators), convert both fractions to use that denominator, then add the numerators. For example, 3/4 + 1/2 = 3/4 + 2/4 = 5/4. Always simplify the result if possible.

How do you multiply fractions?

Multiply the numerators together and the denominators together. For example, 3/4 × 1/2 = (3×1)/(4×2) = 3/8. Then simplify by dividing both by their GCD if needed. No common denominator needed.

How do you divide fractions?

Multiply by the reciprocal of the second fraction. So a/b ÷ c/d = a/b × d/c. For example, 3/4 ÷ 1/2 = 3/4 × 2/1 = 6/4 = 3/2 = 1.5. The mnemonic "keep, change, flip" reminds you: keep first fraction, change to multiplication, flip second.

How do I simplify a fraction?

Find the GCD (greatest common divisor) of numerator and denominator, then divide both by it. For 12/18: GCD = 6. 12/6 = 2; 18/6 = 3. Simplified: 2/3. Always present fractions in simplest form (GCD of numerator and denominator = 1).

How do I convert a fraction to a decimal?

Divide numerator by denominator. 3/4 = 0.75. 1/3 = 0.333... (repeating). 5/8 = 0.625. Some fractions terminate (with denominators 2 or 5 in lowest terms); others repeat. To convert decimals back to fractions: use 10^n denominator for n decimal places, then simplify.

What is a mixed number?

A mixed number combines a whole number and a proper fraction, like 2 1/3 (two and one-third). Equivalent to an improper fraction: 2 1/3 = 7/3. For arithmetic, often convert to improper form first, then convert back to mixed if needed for the answer.

How do I subtract mixed numbers?

Method 1: convert to improper fractions. 3 1/4 - 1 3/4 = 13/4 - 7/4 = 6/4 = 3/2 = 1 1/2. Method 2: subtract whole parts and fraction parts separately, borrowing if needed. 3 1/4 - 1 3/4: can't subtract 3/4 from 1/4, so borrow: 2 5/4 - 1 3/4 = 1 2/4 = 1 1/2.

When should I use fractions vs decimals?

Fractions: exact (no rounding), good for ratios and probability, standard in imperial measurements. Decimals: easier for calculators and computers, more familiar in metric and scientific contexts, easier to compare. Convert between as needed for the context.

Fraction Calculator

Perform arithmetic on fractions: addition, subtraction, multiplication, and division. Enter two fractions and choose an operation to get the result as both a fraction and a decimal, automatically simplified.

Fractions represent rational numbers — values that can be expressed as one integer divided by another. Written as a numerator over a denominator (3/4), they're one of the oldest mathematical concepts, used by ancient Egyptians, Babylonians, and Greeks. Fractions remain essential in cooking, woodworking, music, finance, probability, and many areas of mathematics where exact values matter more than decimal approximations.

The four basic operations on fractions follow consistent rules: - **Addition/Subtraction**: find common denominator, then add/subtract numerators. - **Multiplication**: multiply numerators, multiply denominators. - **Division**: multiply by reciprocal (flip second fraction).

Always simplify the result by dividing both numerator and denominator by their greatest common divisor (GCD). For example, 6/8 simplifies to 3/4 (both divided by 2).

Fractions can also be expressed as: - **Decimals**: 3/4 = 0.75. - **Percentages**: 3/4 = 75%. - **Mixed numbers**: improper fractions like 5/4 = 1 1/4.

Each form has advantages: decimals for calculators and computers; percentages for comparisons; mixed numbers for everyday measurements; fractions for exact arithmetic and clear ratios.

Common applications: cooking (recipe scaling), construction (imperial measurements), music (rhythm and intervals), finance (interest rates as fractions), probability (likelihoods), education (mathematical foundations), and any context requiring exact rational expression.

Inputs

Numerator 1

Denominator 1

Operation

Numerator 2

Denominator 2

Results

Result (Fraction)

5/4

Result (Decimal)

1.25

Last updated: May 29, 2026

Formula

**Fraction operations:** **Addition:** a/b + c/d = (a×d + c×b) / (b×d) Simplified version (with LCD): Find LCD = LCM(b, d), convert both, add. **Subtraction:** a/b - c/d = (a×d - c×b) / (b×d) **Multiplication:** a/b × c/d = (a×c) / (b×d) **Division:** a/b ÷ c/d = a/b × d/c = (a×d) / (b×c) (Multiply by reciprocal of second fraction.) **Worked examples:** 3/4 + 1/2: = (3×2 + 1×4) / (4×2) = (6 + 4) / 8 = 10/8 = 5/4 (simplified). Or: 3/4 + 2/4 = 5/4 (using LCD = 4). 3/4 - 1/2: = 3/4 - 2/4 = 1/4. 3/4 × 1/2: = 3/8. 3/4 ÷ 1/2: = 3/4 × 2/1 = 6/4 = 3/2 = 1.5. **Simplifying fractions:** Divide numerator and denominator by their GCD. For 12/18: GCD(12, 18) = 6. 12/6 = 2; 18/6 = 3. Simplified: 2/3. **Common fractions and decimals:** | Fraction | Decimal | Percentage | |---|---|---| | 1/2 | 0.5 | 50% | | 1/3 | 0.333... | 33.3% | | 2/3 | 0.667 | 66.7% | | 1/4 | 0.25 | 25% | | 3/4 | 0.75 | 75% | | 1/5 | 0.2 | 20% | | 1/6 | 0.167 | 16.7% | | 1/8 | 0.125 | 12.5% | | 1/10 | 0.1 | 10% | **Mixed numbers:** Improper fraction → mixed number: Divide numerator by denominator. Quotient is whole part; remainder over denominator is fraction part. 5/4 = 1 (remainder 1) → 1 1/4 (one and one-fourth). Mixed → improper: Whole × denominator + numerator, over denominator. 2 1/3 = (2×3 + 1) / 3 = 7/3. **Like vs unlike fractions:** Like: same denominators (1/4 + 3/4). Unlike: different denominators (1/4 + 1/3). For unlike, find common denominator first. **Finding common denominators:** **LCM (Least Common Multiple) of denominators**: - 1/4 and 1/3: LCM(4,3) = 12. - 1/4 = 3/12, 1/3 = 4/12, sum = 7/12. **Quick method (cross-multiplication)**: For a/b + c/d: Result = (a×d + c×b) / (b×d) Always works but may need simplification at the end. **Negative fractions:** Same operations, but track signs: -1/2 + 1/4 = -2/4 + 1/4 = -1/4. -1/2 × -1/3 = 1/6 (negatives cancel). **Properties of fractions:** - 0/n = 0 for any n ≠ 0. - n/n = 1 for any n ≠ 0. - 1/0, n/0 undefined (division by zero). - a/1 = a (any integer over 1). - Reciprocal of a/b is b/a. **Multiplying mixed numbers:** Convert to improper first. 1 1/2 × 2 1/3 = 3/2 × 7/3 = 21/6 = 7/2 = 3 1/2. **Adding mixed numbers:** Method 1: Convert to improper, add, convert back. Method 2: Add whole parts and fraction parts separately. 1 1/4 + 2 1/2 = 3 + 1/4 + 2/4 = 3 + 3/4 = 3 3/4. **Subtracting with borrowing:** 3 1/4 - 1 3/4: can't subtract 3/4 from 1/4. Borrow: = 2 5/4 - 1 3/4 = 1 2/4 = 1 1/2. **Comparing fractions:** Convert to common denominator or convert to decimals. 3/8 vs 5/12: LCD = 24. 3/8 = 9/24; 5/12 = 10/24. 3/8 < 5/12. **Reciprocals:** Reciprocal of a/b = b/a. Multiplying by reciprocal is same as dividing: 3 ÷ (1/4) = 3 × (4/1) = 12. **Continued fractions:** Any real number can be approximated by continued fractions: π = 3 + 1/(7 + 1/(15 + 1/(1 + ...))) Best rational approximations come from truncating: 22/7, 333/106, 355/113, ... **Common applications:** - **Cooking**: 1/2 cup, 1/4 teaspoon, 3/4 pound. - **Woodworking**: 1/4 inch, 3/8 inch, 5/8 inch. - **Music**: 1/4 note, 1/8 note, time signatures (3/4, 4/4). - **Finance**: percentage = fraction × 100; interest as fractions. - **Probability**: chances expressed as fractions. - **Mathematics**: algebra, calculus involve fractions everywhere. - **Science**: ratio formulas, dimensional analysis. **Imperial vs metric:** Imperial measurements heavily use fractions: - Inches: 1/16, 1/8, 1/4, 3/8, 1/2, 5/8, 3/4, 7/8. - Cooking: 1/4 cup, 1/3 cup, 1/2 cup. Metric uses decimals more naturally: - Centimeters: 0.5 cm, 1.5 cm. - Liters: 0.25 L, 0.5 L. **Programming:** Most languages have rational/fraction libraries: - Python: from fractions import Fraction - Ruby: built-in Rational class. - Java: BigInteger for arbitrary precision. - C++: Boost rational library. Most languages default to floating-point — which can introduce tiny errors. **Software:** - **Calculators**: most scientific calculators have fraction mode. - **Wolfram Alpha**: fraction arithmetic and conversion. - **Python**: Fraction class for exact arithmetic. - **Spreadsheets**: format cells as fractions. **Pitfalls:** - **Division by zero**: denominator must be nonzero. - **Not simplifying**: 6/8 should reduce to 3/4. - **Forgetting common denominator**: can't add 1/4 + 1/3 directly. - **Reciprocal confusion**: when dividing, flip second fraction. - **Mixed number subtraction**: may need to borrow. - **Negative signs**: place consistently (top, bottom, or in front). - **Floating-point errors**: 0.1 + 0.2 != 0.3 exactly in binary.

How to use this calculator

Enter numerator and denominator of first fraction.
Select operation: add, subtract, multiply, divide.
Enter numerator and denominator of second fraction.
Calculator returns result as simplified fraction and decimal.
For mixed numbers: convert to improper first (whole × denom + num)/denom.
For multiple operations: do step by step or use grouping.

Worked examples

Recipe scaling

**Scenario:** Recipe calls for 3/4 cup of sugar, you want to make 2/3 of the recipe. How much sugar? **Calculation:** 3/4 × 2/3 = (3 × 2) / (4 × 3) = 6/12 = 1/2 cup. **Result:** Use 1/2 cup of sugar. Notice the elegant cancellation: 3/4 × 2/3 simplifies cleanly. Practical recipe math.

Combining time fractions

**Scenario:** Spent 1/3 hour on task A and 1/4 hour on task B. Total time in hours? **Calculation:** 1/3 + 1/4 = (1×4 + 1×3)/(3×4) = 7/12 hour. As decimal: 0.583 hours. As minutes: 7/12 × 60 = 35 minutes. **Result:** 7/12 hour = 35 minutes. LCD of 3 and 4 is 12. Useful for combining time durations expressed in fractions.

Dividing by a fraction

**Scenario:** You have 5 cups of flour. Each batch uses 2/3 cup. How many batches? **Calculation:** 5 ÷ 2/3 = 5 × 3/2 = 15/2 = 7.5 batches. **Result:** 7 full batches (7.5 means 7 complete batches with half-batch leftover material). Division by fraction is equivalent to multiplying by its reciprocal. Dividing always makes the answer larger when divisor is less than 1.

When to use this calculator

**Use fraction arithmetic for:**

- **Cooking and recipes**: scaling ingredients exactly. - **Construction**: imperial measurements (inches, feet). - **Music**: time signatures, note values. - **Finance**: interest, percentages as fractions. - **Math education**: foundational arithmetic. - **Probability**: exact likelihoods. - **Engineering**: ratios and proportions. - **Programming**: exact rational arithmetic.

**Key operations summary:**

- **Add/Subtract**: same denominator, then add/subtract numerators. - **Multiply**: numerators × numerators, denominators × denominators. - **Divide**: flip second, then multiply. - **Simplify**: divide by GCD.

**Mixed vs improper:**

Use whichever is more useful: - **Mixed (1 3/4)**: more intuitive for everyday measurement. - **Improper (7/4)**: easier for arithmetic operations.

Convert between as needed.

**Common imperial fractions:**

- **Inches**: 1/16, 1/8, 1/4, 1/2 (sixteenths most precise common). - **Cups**: 1/4, 1/3, 1/2, 2/3, 3/4, 1 (with 1/8 for spices). - **Tablespoons**: 1/4, 1/2, 3/4, 1. - **Tooth size in dentistry**: 1/4 mm precision.

**Common applications:**

- **Recipe scaling**: 1.5× or 2/3 batch. - **Sewing**: yardage fractions. - **Woodworking**: blueprint dimensions in 16ths. - **Music**: 4/4 vs 3/4 time signatures. - **Probability**: card hand chances as fractions. - **Finance**: interest rates (e.g., 5/8% = 0.625%). - **Gear ratios**: 5:3 ratio = 5/3.

**Exact vs decimal arithmetic:**

Decimals can lose precision: 1/3 = 0.333... (truncated). Fractions are exact: 1/3 is exactly 1/3.

For exact calculations (finance, probability), fractions can be better. For practical engineering: decimals usually sufficient.

**Programming benefits:**

Languages with rational types (Python's Fraction, Ruby's Rational): - No floating-point errors. - Exact representation. - Good for symbolic math, accurate finance.

For most performance-critical work: floating-point is faster.

**Common errors students make:**

- Adding numerators and denominators separately: 1/2 + 1/3 ≠ 2/5 (must find common denom). - Forgetting to invert when dividing: a/b ÷ c/d = a/b × d/c (not a/b × c/d). - Not simplifying: 4/8 should be 1/2. - Sign mistakes: -1/2 - (-1/3) = -1/2 + 1/3.

**Simplification:**

Always simplify final answer: 1. Find GCD of numerator and denominator. 2. Divide both by GCD.

For 12/18: GCD = 6. 12/6 = 2; 18/6 = 3. Simplified: 2/3.

**Conversion to other forms:**

- **To decimal**: divide numerator by denominator. 3/4 = 0.75. - **To percentage**: multiply by 100. 3/4 = 75%. - **To mixed**: divide for whole and remainder. 7/3 = 2 1/3.

**Software:**

- **Calculators**: most modern have fraction modes. - **Wolfram Alpha**: comprehensive fraction support. - **Python (Fraction)**: arbitrary precision rationals. - **Excel**: fraction format for display. - **CAD software**: often supports fractional dimensions.

**Pitfalls:**

- **Division by zero**: denominator can't be 0. - **Not simplifying**: standard form has GCD = 1. - **Mixed number arithmetic**: convert to improper for multiplication/division. - **Sign placement**: -a/b = a/(-b) = -(a/b). - **Decimal approximations**: lose precision for some operations. - **Common denominators**: must find before adding/subtracting. - **Reciprocals**: only flip when dividing.

Common mistakes to avoid

Adding numerators AND denominators separately (1/2 + 1/3 ≠ 2/5).
Not finding common denominator before adding/subtracting.
Forgetting to invert when dividing (a/b ÷ c/d ≠ a/b × c/d).
Not simplifying the final answer.
For mixed numbers: not converting to improper before multiplying.
Treating division by zero as defined (it's not).
Sign errors with negative fractions.
Confusing reciprocals (1/x) with inverses (-x).

Frequently Asked Questions

Sources & further reading

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