Simplify Calculator — Fractions Step by Step

What Is a Simplified Fraction?

A fraction is in its simplest form (lowest terms) when the only positive integer that divides both the numerator and the denominator is 1. In other words, they share no common factors greater than 1.

• 24/36 → not simplified (GCD = 12) → simplifies to 2/3
• 7/9 → already simplified (GCD = 1)
• 100/200 → GCD = 100 → simplifies to 1/2

The GCD Method (Standard Approach)

The Greatest Common Divisor is found using the Euclidean algorithm, one of the oldest known algorithms (circa 300 BC):

GCD(a, b):
  while b ≠ 0:
    a, b = b, a mod b
  return a

For GCD(36, 24):

Step	a	b	a mod b
1	36	24	12
2	24	12	0
Done	GCD = 12

Then: 36/24 = (36÷12)/(24÷12) = 3/2

When the Result Is a Mixed Number

If |numerator| ≥ denominator, the fraction is “improper” and can be expressed as a mixed number:

• 7/3: whole part = floor(7/3) = 2, remainder = 7 − 2×3 = 1 → 2 and 1/3
• 9/4 = 2 and 1/4
• Mixed numbers are common in cooking (2½ cups) and construction (3¾ inches)

Common Mistakes When Simplifying

1. Dividing by a common factor but not the GCD: 8/12 → some students simplify by 2 to get 4/6, then by 2 again to get 2/3. This works, but finding the GCD first is more efficient.
2. Changing the fraction’s value: Simplification does NOT change the value — 24/36 and 2/3 represent exactly the same part of a whole.
3. Applying simplification to mixed-operation expressions: You can only cancel common factors that appear in both numerator and denominator, not terms that are added or subtracted. (x+3)/(x+3) simplifies to 1, but (x+3)/x does not simplify further.

Decimal and Percentage Equivalents

Once simplified, the decimal form is straightforward: divide the numerator by the denominator. Multiply by 100 for the percentage:

Fraction	Simplified	Decimal	Percent
2/4	1/2	0.5	50%
6/9	2/3	0.6̄	66.67%
15/20	3/4	0.75	75%
12/16	3/4	0.75	75%

Why Simplification Matters

• Test scores: Unsimplified answers are typically marked wrong in school exams.
• Cooking and recipes: 8/16 cups is confusing; 1/2 cup is clear.
• Engineering ratios: Gear ratios and mix ratios are always expressed in lowest terms (1:3, not 2:6).
• Probability: A probability of 12/48 is immediately read as 1/4 (25%) when simplified.