Reduce to lowest terms, convert to mixed numbers, compare, and perform operations, all with full step-by-step solutions.
Simplifying fractions makes comparisons and calculations far easier. 12/16 and 3/4 are the same value, but 3/4 is immediately recognizable as 'three quarters' while 12/16 requires mental processing. In hardware stores, products are labeled in simplified fractions (3/4 inch, not 6/8 inch. The mathematical tool for simplifying is the Greatest Common Factor (GCF)) the largest number dividing evenly into both numerator and denominator. For 18/24, the GCF is 6, giving 3/4. The Euclidean algorithm finds the GCF efficiently for large numbers: divide the larger by the smaller, take the remainder, repeat until remainder is 0. Improper fractions (7/4) and mixed numbers (1¾) represent the same value. 7/4 = 1¾ because 4 goes into 7 once with remainder 3. Conversion between them is frequently needed in construction and cooking measurements.
A fraction is in simplest form (also called lowest terms or reduced form) when the numerator and denominator share no common factors other than 1, their Greatest Common Divisor is 1. Simplifying a fraction means dividing both parts by their GCD until no further reduction is possible. The value of the fraction doesn't change; only its representation becomes more concise. 8/12 and 2/3 are the same quantity, the latter is simply easier to work with and recognize.
The Euclidean algorithm is the standard method for finding the GCD, running efficiently even for very large numbers. For two positive integers a and b (where a > b), the algorithm repeatedly replaces the larger with the remainder of dividing the larger by the smaller: GCD(48, 18) → GCD(18, 12) → GCD(12, 6) → GCD(6, 0) = 6. This calculator uses this exact algorithm, so results are always exact, no rounding, no approximation.
Reduces any fraction to its lowest terms. Works with negative numerators and denominators (always moves the sign to the numerator), large integers, and fractions that are already simplified (GCD = 1). Also shows the equivalent decimal, percentage, and mixed number form if the fraction is improper.
An improper fraction has a numerator larger than its denominator (7/4, 11/3). A mixed number combines a whole number and a proper fraction (1¾, 3⅔). To convert improper to mixed: divide numerator by denominator, the quotient is the whole part, the remainder becomes the new numerator over the original denominator. 7/4 = 1 remainder 3 = 1¾. To convert mixed to improper: multiply whole by denominator, add numerator. 1¾ = (1×4)+3 = 7/4.
Determines whether one fraction is greater than, less than, or equal to another using cross-multiplication: the only exact method that avoids floating-point errors. For a/b vs c/d: compute a×d and b×c. If a×d > b×c then a/b > c/d. This works because multiplying both sides of a/b = c/d by b×d gives the cross products. The calculator also shows both fractions converted to a common denominator to make the comparison visually intuitive.
Adds, subtracts, multiplies, or divides two fractions with full simplification. Addition and subtraction use the LCD (Least Common Denominator) method for cleaner intermediate steps. Multiplication cross-cancels before multiplying for the most efficient path. Division uses the "keep-change-flip" rule (multiply by reciprocal). Every operation shows the unsimplified intermediate result and then the simplified final answer.
| Operation | Rule | Example |
|---|---|---|
| Add | a/b + c/d = (a×d + c×b) / (b×d) | 1/2 + 1/3 = (3+2)/6 = 5/6 |
| Subtract | a/b − c/d = (a×d − c×b) / (b×d) | 3/4 − 1/2 = (6−4)/8 = 2/8 = 1/4 |
| Multiply | a/b × c/d = (a×c) / (b×d) | 2/3 × 3/4 = 6/12 = 1/2 |
| Divide | a/b ÷ c/d = (a×d) / (b×c) | 3/4 ÷ 3/8 = 24/12 = 2 |
| Simplify | a/b = (a÷GCD) / (b÷GCD) | 12/18 → GCD=6 → 2/3 |
| Compare | a/b vs c/d → compare a×d vs b×c | 2/3 vs 3/4: 8 vs 9, so 2/3 < 3/4 |
| Improper→Mixed | n/d: whole = floor(n/d), rem = n mod d | 7/4 = 1 remainder 3 = 1¾ |
| Mixed→Improper | w n/d = (w×d + n) / d | 2⅔ = (2×3+2)/3 = 8/3 |
These are among the most frequently encountered fractions in everyday contexts. Recognizing them by sight speeds up mental math considerably.
| Original | Simplified | Decimal | Percent |
|---|---|---|---|
| 2/4 | 1/2 | 0.5 | 50% |
| 2/6 | 1/3 | 0.3333... | 33.3% |
| 3/6 | 1/2 | 0.5 | 50% |
| 2/8 | 1/4 | 0.25 | 25% |
| 4/8 | 1/2 | 0.5 | 50% |
| 6/8 | 3/4 | 0.75 | 75% |
| 4/12 | 1/3 | 0.3333... | 33.3% |
| 6/10 | 3/5 | 0.6 | 60% |
| 10/15 | 2/3 | 0.6666... | 66.7% |
| 12/16 | 3/4 | 0.75 | 75% |
| 15/20 | 3/4 | 0.75 | 75% |
| 24/36 | 2/3 | 0.6666... | 66.7% |
Color-coded tiles make equivalent fractions and simplification instantly visible.
View on Amazon →Daily practice problems to build fraction fluency including simplification.
View on Amazon →Natural display shows fractions as written and simplifies automatically.
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