Mathematics, Statistics & Geometry

Simplifying Fractions Calculator

Instantly simplify any fraction to its lowest terms using the greatest common factor. See every step of the reduction process clearly.

Simplified Fraction
2/3
Greatest Common Factor (GCF)12
Decimal Equivalent0.667
Calculation StepsOriginal Fraction: 24/36 GCF of 24 and 36 = 12 Divide both by GCF: 24 / 12 = 2, 36 / 12 = 3 Simplified: 2/3

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The Complete Guide to Simplifying Fractions

A simplifying fractions calculator reduces any fraction to its lowest terms by finding the greatest common factor (GCF) of the numerator and denominator. This is one of the most fundamental skills in arithmetic and algebra, essential for homework, exam preparation, and everyday problem-solving.

ab=a÷gcd⁡(a,b)b÷gcd⁡(a,b)\begin{aligned} \frac{a}{b} = \frac{a \div \gcd(a,b)}{b \div \gcd(a,b)} \end{aligned}

Where:
a=
The top number of the fraction
b=
The bottom number of the fraction
gcd⁡(a,b)\gcd(a,b)=
The largest number that divides both a and b evenly

How to Simplify a Fraction Step by Step

  1. Identify the numerator and denominator: The numerator is the top number; the denominator is the bottom number.
  2. Find the GCF: Determine the greatest common factor of the two numbers.
  3. Divide both: Divide the numerator and the denominator by the GCF.
  4. Write the result: The new fraction is the simplest form.

Worked Example: 24/36

  • Step 1: Numerator = 24, Denominator = 36
  • Step 2: Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36. GCF = 12
  • Step 3: 24 / 12 = 2, 36 / 12 = 3
  • Step 4: 24/36 = 2/3

Worked Example: 45/60

  • Step 1: Numerator = 45, Denominator = 60
  • Step 2: GCF(45, 60) = 15
  • Step 3: 45 / 15 = 3, 60 / 15 = 4
  • Step 4: 45/60 = 3/4

The Euclidean Algorithm

Our calculator uses the Euclidean algorithm to find the GCF, which is far more efficient than listing all factors manually. The algorithm works by repeatedly dividing the larger number by the smaller one and taking the remainder until the remainder reaches zero. The last non-zero remainder is the GCF.

For example, to find GCF(48, 18):

  • 48 / 18 = 2 remainder 12
  • 18 / 12 = 1 remainder 6
  • 12 / 6 = 2 remainder 0
  • GCF = 6

Why Simplify?

  • Clarity: 2/3 is much easier to understand and work with than 178/267.
  • Comparison: Simplified fractions make it far simpler to compare two quantities. Is 15/20 larger or smaller than 9/12? Once simplified (3/4 vs. 3/4), the answer is obvious: they are equal.
  • Further calculations: Adding, subtracting, multiplying, or dividing fractions is significantly easier when they are already in their simplest form.

Real-World Applications

  • Cooking: Scaling recipes (half of 3/4 cup = 3/8 cup).
  • Construction: Simplifying measurements in inches and feet.
  • Finance: Expressing odds, proportions, and market share in reduced form.
  • Education: A core skill tested from primary school through to university-level mathematics.

Frequently Asked Questions

Simplifying (or reducing) a fraction means dividing both the numerator and denominator by their greatest common factor (GCF) until no number other than 1 divides both evenly. The simplified fraction represents the same value in its most compact form. For example, 12/18 simplifies to 2/3 because the GCF of 12 and 18 is 6.

List all the factors of both numbers and find the largest one they share. For example, the factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24 and the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36. The largest common factor is 12, so GCF(24, 36) = 12. Our calculator uses the Euclidean algorithm to compute this instantly.

If the GCF of the numerator and denominator is 1, the fraction cannot be simplified further. The calculator will clearly indicate this. For example, 7/13 is already in its simplest form because 7 and 13 share no common factor other than 1.

Yes. An improper fraction (where the numerator is larger than the denominator, like 15/6) can be simplified just like any other fraction. 15/6 simplifies to 5/2, which can also be expressed as the mixed number 2 and 1/2.

No. Simplifying a fraction never changes its mathematical value. 12/18 and 2/3 are exactly equal; they represent the same point on the number line. Simplifying only changes the representation, not the quantity.

Yes. Our calculator handles negative numerators and denominators correctly. The sign is always placed on the numerator in the simplified result. For example, -8/12 simplifies to -2/3 and 8/-12 also simplifies to -2/3.

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