Simplify Fractions Calculator
Reduce any fraction to its simplest form (lowest terms) using the GCD. Shows the GCD used and the decimal equivalent.
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How to use this calculator
Dividing numerator and denominator by their Greatest Common Divisor produces the equivalent fraction in lowest terms.
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Enter the numerator and denominator of the fraction you want to simplify.
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The GCD of the two numbers is found and both are divided by it.
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The simplified fraction and decimal equivalent are displayed.
Frequently asked questions
What does "lowest terms" mean?
A fraction is in lowest terms (or simplest form) when the numerator and denominator share no common factor other than 1 — their GCD is 1. For 12/16: GCD(12,16)=4, so 12/16 = 3/4 (in lowest terms). For 7/9: GCD(7,9)=1, so it is already in lowest terms.
What is the GCD and how do you find it?
The Greatest Common Divisor (GCD) of two integers is the largest integer that divides both without remainder. Find it by listing factors: factors of 12 are {1,2,3,4,6,12}; factors of 16 are {1,2,4,8,16}; GCD = 4. Or use the Euclidean algorithm: GCD(16,12) = GCD(12,4) = GCD(4,0) = 4. The Euclidean algorithm is much faster for large numbers.
Does simplifying a fraction change its value?
No. 12/16 and 3/4 represent exactly the same quantity — three quarters. Simplifying only changes the notation, making it more readable and easier to work with. Equivalent fractions (12/16, 6/8, 3/4) all describe the same point on the number line.
Can negative fractions be simplified?
Yes. −12/16 = −3/4. By convention, the negative sign is placed in the numerator (or in front of the fraction), not the denominator. This calculator normalizes the sign so the denominator is always positive.
Simplify fractions — GCD, lowest terms, and mixed numbers
The Euclidean algorithm for GCD
The most efficient way to find the GCD is Euclid's algorithm (c. 300 BC): repeatedly replace the larger number with the remainder when it is divided by the smaller, until the remainder is zero. The last non-zero remainder is the GCD. For GCD(48, 18): 48 = 2×18 + 12; 18 = 1×12 + 6; 12 = 2×6 + 0. GCD = 6. So 48/18 = 8/3. This algorithm is O(log min(a,b)) — extremely fast even for numbers with thousands of digits.
Why simplify fractions?
Simplified fractions are easier to compare (is 7/12 or 5/9 larger?), easier to use in further calculations (adding 3/4 is simpler than adding 12/16), and required in most exam contexts. Many real-world quantities have natural simplified forms: 50 minutes out of 60 = 5/6 of an hour; 3 correct out of 12 questions = 1/4 score. The simplified form communicates the ratio more clearly.
Fractions, ratios, and proportional thinking
A fraction p/q is identical to the ratio p:q. Simplifying a ratio is the same as simplifying a fraction. In scale drawings, a 1:50 scale means 1 cm on paper = 50 cm in reality — equivalent to the fraction 1/50, already in simplest form. In cooking, if a recipe for 6 servings calls for 9 cups of flour, the flour-per-serving ratio is 9/6 = 3/2, meaning 1.5 cups per serving. Proportional thinking is fraction thinking.
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