Reducing Fraction Calculator

Simplify any fraction to its lowest terms with step-by-step explanations. This calculator finds the greatest common divisor and reduces fractions to their simplest form, showing equivalent representations in multiple formats.

Enter Your Fraction

Fraction to Simplify

What This Calculator Does

Reducing (or simplifying) a fraction means finding an equivalent fraction where the numerator and denominator have no common factors other than 1. This makes the fraction easier to understand and work with while maintaining the same value.

The calculator uses the Euclidean algorithm to find the greatest common divisor (GCD) of the numerator and denominator, then divides both by this value to produce the simplest form.

When You'd Use This

Final Answers: Most math problems require answers in simplest form, whether in homework, tests, or professional calculations
Comparing Fractions: Simplified fractions are easier to compare and understand at a glance (3/4 vs 75/100)
Recipe Scaling: After scaling a recipe, you might get fractions like 10/15 cups, which simplifies to the more practical 2/3 cups
Measurement Conversion: When converting between units, you often get fractions that need simplification for practical use

Step-by-Step Method

Find Common Factors

List the factors of both the numerator and denominator. Look for the largest number that divides evenly into both values. This is the greatest common divisor (GCD).

Example: For 12/18, factors of 12 are 1,2,3,4,6,12 and factors of 18 are 1,2,3,6,9,18. The GCD is 6.

Divide Both by the GCD

Divide both the numerator and denominator by the GCD. This gives you an equivalent fraction in simplest form.

Example: 12 ÷ 6 = 2 and 18 ÷ 6 = 3, giving us 2/3

Verify Simplest Form

Check that the numerator and denominator now share no common factors other than 1. If they do, repeat the process with the new GCD.

Example: 2 and 3 share only the factor 1, so 2/3 is fully simplified

Convert to Other Forms

Once simplified, you can easily convert to mixed number form if the numerator is larger than the denominator, or to decimal and percentage forms by division.

Example: 2/3 = 0.6667 = 66.67%

The Euclidean Algorithm (Advanced)

This calculator uses the Euclidean algorithm, an efficient method for finding the GCD without listing all factors. Here's how it works:

Divide the larger number by the smaller number
Replace the larger number with the smaller number
Replace the smaller number with the remainder from step 1
Repeat until the remainder is 0
The last non-zero remainder is the GCD

Example for 48/18: 48 ÷ 18 = 2 remainder 12 → 18 ÷ 12 = 1 remainder 6 → 12 ÷ 6 = 2 remainder 0. The GCD is 6.

For more on the Euclidean algorithm, see Khan Academy's explanation.

Common Mistakes to Avoid

Dividing by a Non-Common Factor

Make sure the number you divide by goes evenly into both the numerator and denominator. For example, you can't simplify 12/15 by dividing by 4, even though 4 divides 12, because 4 doesn't divide 15 evenly.

Not Finding the Greatest Common Divisor

Always use the largest common factor. Simplifying 12/18 by 2 gives 6/9, which can be simplified further. Using the GCD of 6 immediately gives the fully simplified 2/3.

Changing Only One Part

You must divide (or multiply) both numerator and denominator by the same value to maintain an equivalent fraction. Changing only one part produces a different value.

Frequently Asked Questions

What if the fraction is already simplified?

If the numerator and denominator share no common factors other than 1, the fraction is already in simplest form. The calculator will show that the GCD is 1 and return the same fraction.

Can improper fractions be simplified?

Yes, improper fractions (where the numerator is larger than the denominator) can be simplified using the exact same method. You can also convert them to mixed numbers after simplifying.

What about fractions with large numbers?

The Euclidean algorithm used by this calculator works efficiently even with very large numbers. You don't need to find all the factors; the algorithm finds the GCD through repeated division.

Do I always need to simplify fractions?

In most academic and professional contexts, final answers should be in simplest form unless specifically stated otherwise. However, during intermediate calculation steps, you might work with unsimplified fractions for convenience.

What's the difference between simplifying and reducing?

These terms mean the same thing. Both refer to finding an equivalent fraction with the smallest possible numerator and denominator while maintaining the same value.