fraction simplification with gcd
Simplifying a fraction means dividing numerator and denominator by the largest number that fits into both at once, the GCD. This page walks through the Euclidean algorithm that finds that GCD step by step, and how a simplified fraction turns into an exact decimal or a repeating one.
The GCD of 48 and 180 is 12, found in just 3 steps of the Euclidean algorithm; left unsimplified, the fraction would still represent the same value, just with larger numbers to work with.
The GCD of 20 is much larger than the 12 from the previous example because 100 and 40 share more prime factors (2² and 5); the mixed form 2 1/2 is just another way to write the same 5/2.
Since the GCD of 7 and 12 is 1, the fraction was already in lowest terms; the decimal has a non-repeating part (58) followed by the block 3 that repeats forever, because the factor 3 in denominator 12 never divides any power of 10.
Simplifying means dividing both the numerator and denominator by their Greatest Common Divisor (GCD) until no integer greater than 1 divides both. The resulting fraction is equivalent but in its most reduced form.
Because two fractions in irreducible form are only equal if numerator and denominator are identical; comparing 4/15 with another fraction just means looking at two small numbers, while comparing 48/180 with another large fraction requires finding a much bigger common denominator first.
Yes: every repeating decimal represents a rational number and has an equivalent exact fraction; the conversion uses the difference between the full number and its version shifted by the period length, a standard algebraic method, but the reverse direction (fraction to decimal) is what this tool computes directly.
Because the GCD of 100 and 40 is 20, not 40 or 100; dividing both by 20 gives 5/2, and since 5 is not a multiple of 2, the result keeps a denominator other than 1 and never becomes a whole number.
The tool always normalizes the sign onto the final result and keeps the denominator positive internally; -48/180 and 48/-180 both land on the same simplified result, -4/15, because the sign is handled separately from the absolute value before dividing by the GCD.
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