Simplify Fractions Calculator
Reduce fractions to their simplest form
How It Works
Overview
Simplifying a fraction — also called reducing it to lowest terms — means dividing both the numerator and the denominator by their largest common factor so that no further reduction is possible. The result represents the same numerical value, just expressed with the smallest whole numbers. For example, 8/12, 6/9, and 2/3 are all equal, but 2/3 is the simplest form.
The key tool is the greatest common divisor (GCD), the largest positive integer that divides two numbers exactly. Once you find GCD(numerator, denominator), divide both by it once and the fraction is fully reduced. This calculator uses the Euclidean algorithm to find the GCD efficiently and then returns the simplified fraction along with its decimal and percentage equivalents.
The Formula
The Euclidean algorithm for GCD:
- Start with two positive integers a and b.
- Replace (a, b) with (b, a mod b).
- Repeat until b = 0; the remaining a is the GCD.
Once the GCD is known, divide numerator and denominator by it. The resulting fraction is in lowest terms, and dividing again by anything other than 1 would not produce integers.
Worked Example
Simplify 24/36:
- Find GCD(24, 36): 36 mod 24 = 12; 24 mod 12 = 0; GCD = 12.
- Divide: 24 ÷ 12 = 2, 36 ÷ 12 = 3.
- Simplified: 2/3 (decimal ≈ 0.6667, percent 66.67%).
Simplify 45/100:
- GCD(45, 100) = 5.
- 45 ÷ 5 = 9, 100 ÷ 5 = 20.
- Simplified: 9/20 (decimal 0.45, percent 45%).
Simplify 17/19: GCD = 1, so the fraction is already in lowest terms. Decimal ≈ 0.8947, percent ≈ 89.47%.
When to Use This
- Homework and exams: answers are conventionally expected in lowest terms.
- Recipes & measurement: 8/16 cup is just 1/2 cup — simpler to scoop.
- Probability: probabilities like 30/50 read more clearly as 3/5.
- Comparing fractions: simplified forms make like-denominator comparisons faster.
- Engineering ratios: gear ratios and aspect ratios (like 16:9) are always quoted in lowest terms.
Common Mistakes to Avoid
- Dividing only one term. You must divide numerator and denominator by the same factor; otherwise the value changes.
- Stopping early. Reducing 24/36 to 12/18 is correct but not done; keep going to 2/3, or just divide once by the GCD.
- Subtracting instead of dividing. 6/8 ≠ 5/7. Reduction is by division, never subtraction of the same number from each part.
- Forgetting the negative sign. Convention puts a single minus sign in front (or on the numerator); -6/-8 simplifies to 3/4, not -3/4.
- Trying to simplify when the denominator is 0. Always check first — division by zero is undefined.
Frequently Asked Questions
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