Fraction Calculator | Add, Subtract & More
Add, subtract, multiply, and divide fractions and mixed numbers with LCM common denominators, cross-simplification, and full step-by-step working.
Quick Examples
Input Type
Operation
What Is the Fraction Calculator | Add, Subtract & More?
Fractions represent a ratio between two integers, the numerator (top) divided by the denominator (bottom). They appear in every field from cooking and construction to music theory and quantum mechanics. Despite being introduced in elementary school, fraction arithmetic remains a persistent source of errors because each operation follows its own distinct rule.
Why the Four Rules Differ
- ›Addition & Subtraction, fractions must share a common denominator before their numerators can be combined. The safest choice is the least common multiple (LCM) of the two denominators, which keeps numbers smaller and avoids extra simplification work at the end.
- ›Multiplication, multiply numerator × numerator and denominator × denominator. Cross-simplification (cancelling a factor between a numerator and the opposite denominator before multiplying) keeps intermediate values small.
- ›Division, multiply by the reciprocal of the second fraction. Flipping c/d to d/c converts division into a multiplication problem, which is then handled the same way.
GCD and the Euclidean Algorithm
Every fraction result is reduced to lowest terms by dividing both numerator and denominator by their GCD. The Euclidean algorithm finds the GCD in O(log min(a,b)) steps, making it fast even for large numbers. For example: GCD(48, 36) → GCD(36, 12) → GCD(12, 0) = 12, so 48/36 reduces to 4/3.
LCM vs. Product as Common Denominator
Using the product b×d as a common denominator always works but often produces an unnecessarily large intermediate fraction. Using LCM(b,d) gives the minimal denominator, for example, adding 1/6 + 1/4 with the product gives denominator 24, whereas LCM(6,4) = 12 is the true least common denominator, keeping all numbers smaller and the result already in lowest terms.
Mixed Numbers
A mixed number like 3¼ represents 3 + 1/4. To compute with mixed numbers the calculator first converts each to an improper fraction: 3¼ → 13/4. After the operation the result is converted back if a whole part exists. For negative mixed numbers the sign belongs to the whole part (−1½ = −3/2, not −1+½).
Formula
| Symbol | Name | Description |
|---|---|---|
| a | First numerator | The top number of the first fraction; can be negative |
| b | First denominator | The bottom number of the first fraction; must be non-zero |
| c | Second numerator | The top number of the second fraction; can be negative |
| d | Second denominator | The bottom number of the second fraction; must be non-zero |
| GCD | Greatest Common Divisor | Largest integer that divides both numerator and denominator exactly |
| LCM | Least Common Multiple | Smallest positive integer divisible by both denominators; used as common denominator |
| w | Whole part | Integer portion of a mixed number (e.g. 3 in 3¼); carries the sign |
| n/d | Fractional part | Remainder portion of a mixed number; numerator is always non-negative |
How to Use
- 1Choose mode: Select Simple Fractions for standard a/b inputs, or Mixed Numbers if you have whole parts alongside fractions.
- 2Select operation: Click +, −, ×, or ÷ to choose the arithmetic operation.
- 3Enter first fraction: Type the numerator and denominator (or whole + numerator + denominator for mixed mode).
- 4Enter second fraction: Fill in the same fields for the second fraction.
- 5Try a preset: Use one of the six preset buttons to load a worked example immediately, great for checking your own calculation.
- 6Calculate: Press Calculate or hit Enter. The result appears as a simplified fraction, mixed number, and decimal instantly.
- 7Read the steps: Expand the Step-by-Step section to see exactly how the common denominator was found, how the operation was applied, and how the result was simplified.
- 8Copy or reset: Copy the result to the clipboard with one click, or press Esc / Reset to clear all fields and start over.
Example Calculation
Example 1, Adding Unlike Fractions: 3/8 + 5/12
The denominators 8 and 12 are different, so we need a common denominator. LCM(8,12) = 24.
Example 2, Multiplication with Cross-Simplification: 6/7 × 14/9
Before multiplying, cancel common factors between numerators and opposite denominators.
Example 3, Mixed Number Subtraction: 3¼ − 1⅔
Convert each mixed number to an improper fraction, then subtract using LCM.
Understanding Fraction | Add, Subtract & More
Types of Fractions
| Type | Definition | Example |
|---|---|---|
| Proper | Numerator < denominator | 3/4, 7/12 |
| Improper | Numerator ≥ denominator | 5/3, 11/4 |
| Mixed | Whole + proper fraction | 1⅔, 3¼ |
| Unit | Numerator = 1 | 1/2, 1/7 |
| Equivalent | Same value, different form | 2/4 = 1/2 |
| Negative | Numerator or denominator < 0 | −3/4 = 3/(−4) |
Common Denominators in Depth
When adding or subtracting fractions, the denominators must be equal because you can only add like parts, eighths to eighths, thirds to thirds. Two strategies exist:
- ›Least Common Denominator (LCD), the LCM of the denominators. Minimises the size of all intermediate numbers and usually produces a result already in lowest terms.
- ›Product method, multiply the denominators together (b×d). Always works but may introduce large numbers that need extra simplification afterwards.
This calculator always uses the LCD method. For example, 1/6 + 1/10 uses LCD = LCM(6,10) = 30, not the product 60, giving 5/30 + 3/30 = 8/30 = 4/15 directly.
Practical Applications
- ›Cooking & baking, scaling recipes up or down (2/3 cup × 3/2)
- ›Carpentry & construction, adding measurements in fractional inches (5/8″ + 3/16″)
- ›Financial ratios, P/E ratios, debt-to-equity, and interest rate fractions
- ›Music theory, note durations (half note + dotted quarter = 3/4 bar)
- ›Statistics, probability calculations (P(A∪B) = P(A) + P(B) − P(A∩B))
- ›Chemistry, molarity and dilution ratios (C₁V₁ = C₂V₂)
- ›Map scales, converting distances using representative fractions (1:25,000)
Fraction Simplification Rules
- ›A fraction is in lowest terms when GCD(numerator, denominator) = 1
- ›The sign of a fraction belongs conventionally to the numerator: −3/4, not 3/−4
- ›If the result is a whole number (d=1), it is displayed without a denominator
- ›Zero in the numerator always gives 0 regardless of the denominator
- ›Zero in the denominator is undefined, the calculator flags this as an error
All calculations run entirely in your browser using JavaScript integer arithmetic. No data is sent to any server. Fraction arithmetic is exact for all integer inputs within JavaScript's safe integer range (±2⁵³ − 1).
Frequently Asked Questions
How does the calculator find a common denominator?
It uses the Least Common Multiple (LCM) of the two denominators, which gives the smallest possible common denominator, also called the Least Common Denominator (LCD).
The LCM is computed from the GCD via LCM(b, d) = b / GCD(b, d) × d. Using the LCM instead of the product b×d keeps numbers smaller and avoids unnecessary simplification afterwards.
- • For 1/6 + 1/10: LCM(6,10) = 30, giving 5/30 + 3/30 = 8/30 = 4/15
- • Product method would give 60 and require one extra reduction step
What is a least common denominator and why does it matter?
The Least Common Denominator (LCD) is the smallest number that is a multiple of both denominators. It matters because:
- • It keeps intermediate numbers as small as possible, reducing the chance of arithmetic errors
- • Results are often already in lowest terms, so no extra simplification step is needed
- • It is the standard taught in mathematics curricula worldwide
For denominators that share no common factors (coprime), the LCD equals their product anyway, the distinction only matters when GCD > 1.
How does the calculator simplify a fraction to lowest terms?
It divides both numerator and denominator by their Greatest Common Divisor (GCD), found using the Euclidean algorithm:
- • GCD(a, b) = GCD(b, a mod b), repeated until the remainder is 0
- • Example: GCD(48, 36) → GCD(36, 12) → GCD(12, 0) = 12
- • So 48/36 ÷ 12/12 = 4/3 (fully reduced)
The sign is normalised so the denominator is always positive, any negative sign moves to the numerator.
How do mixed numbers work in fraction arithmetic?
Mixed numbers like 3¼ are converted to improper fractions before any arithmetic:
- • For positive: w n/d → (w×d + n) / d
- • For negative: −w n/d → (w×d + n) / (−d) so the fractional part stays non-negative
- • Example: 3¼ → (3×4+1)/4 = 13/4; −2½ → −(2×2+1)/2 = −5/2
After the operation, if the result has a magnitude ≥ 1, it is converted back to mixed number form. The whole part carries the sign; the fractional remainder is always non-negative.
Can I enter negative fractions?
Yes. Type a negative number in the numerator field. The calculator normalises the sign convention so the denominator is always positive, but the computed value is identical.
- • −3/4 is the same as 3/−4, both equal −0.75
- • −3/−4 = 3/4 (double-negative simplifies to positive)
- • For mixed numbers, enter a negative whole part to represent negative mixed numbers
What is cross-simplification and when is it used?
Cross-simplification (also called cross-cancellation) applies only during multiplication. Before computing a/b × c/d, the calculator checks for common factors between:
- • The first numerator (a) and second denominator (d)
- • The second numerator (c) and first denominator (b)
Dividing by those common factors reduces the intermediate numbers. For example, 6/7 × 14/9: GCD(6,9)=3 and GCD(14,7)=7, so it becomes (2/1) × (2/3) = 4/3, avoiding the large intermediate 84/63.
Why does dividing by a fraction flip the denominator?
Dividing by a number is the same as multiplying by its reciprocal, this is a fundamental property of multiplicative inverses:
- • The reciprocal of c/d is d/c (swap numerator and denominator)
- • So a/b ÷ c/d = a/b × d/c = (a×d) / (b×c)
- • Example: 3/4 ÷ 3/8 = 3/4 × 8/3 = 24/12 = 2
This works because (c/d) × (d/c) = 1, multiplying a number by its reciprocal always gives 1, the multiplicative identity.
What is the difference between GCD and LCM?
These are complementary concepts that serve different purposes in fraction arithmetic:
- • GCD (Greatest Common Divisor), the largest number that divides both integers evenly. Used to simplify fractions: divide num and denom by GCD to reach lowest terms.
- • LCM (Least Common Multiple), the smallest number that is a multiple of both integers. Used to add/subtract fractions: it provides the LCD for combining unlike denominators.
They are related by GCD(a,b) × LCM(a,b) = a × b, so knowing one lets you compute the other. For GCD(8,12) = 4: LCM = 8×12/4 = 24.