How do you simplify a fraction to its lowest terms?

To simplify a fraction, divide both the numerator and denominator by their Greatest Common Divisor (GCD). The GCD is the largest whole number that divides both evenly. The result is the fraction in lowest terms — where the only common factor of numerator and denominator is 1. Step-by-step for 12/18: • Find GCD(12, 18) using the Euclidean algorithm: 18 = 1×12 + 6, then 12 = 2×6 + 0 → GCD = 6 • Divide numerator: 12 ÷ 6 = 2 • Divide denominator: 18 ÷ 6 = 3 • Result: 2/3 — confirmed in lowest terms since GCD(2,3) = 1 Quick checks before calculating: • Both even → divide by 2 first • Both end in 0 or 5 → divide by 5 first • Digit sum divisible by 3 → divide by 3 • Keep dividing until no common factor remains

What is the Euclidean algorithm for finding GCD?

The Euclidean algorithm is the fastest method for finding the GCD of two numbers. It works by repeatedly dividing and taking the remainder until the remainder is 0. The last non-zero remainder is the GCD. Algorithm: GCD(a, b) — divide a by b, take remainder r. Then GCD(b, r). Repeat until remainder = 0. Example — GCD(48, 64): • 64 = 1 × 48 + 16 • 48 = 3 × 16 + 0 • GCD = 16 • Simplify 48/64: 48÷16 = 3, 64÷16 = 4 → 3/4 ✓ Example — GCD(100, 75): • 100 = 1 × 75 + 25 • 75 = 3 × 25 + 0 • GCD = 25 → 100/75 = 4/3 = 1⅓ Why it works: GCD(a,b) = GCD(b, a mod b) — the GCD of two numbers equals the GCD of the smaller and the remainder. This calculator shows all Euclidean steps.

What are equivalent fractions?

Equivalent fractions are different fractions that represent exactly the same value. You can create equivalent fractions by multiplying or dividing both the numerator and denominator by the same non-zero number. For example, 1/2 = 2/4 = 3/6 = 4/8 — all equal 0.5. Key facts about equivalent fractions: • Multiplying both top and bottom by the same number gives an equivalent fraction • Dividing both by their GCD gives the simplest equivalent fraction (lowest terms) • All equivalent fractions have the same decimal value • The simplest form is unique — there is only one fraction in lowest terms for any value • To check if two fractions are equivalent: cross-multiply and compare (a/b = c/d if a×d = b×c) • This calculator shows a table of equivalent fractions by multiplying the simplified form by 2, 3, 4, 5...

How do you simplify a negative fraction?

A negative fraction is simplified the same way — find the GCD of the absolute values of numerator and denominator, then divide both by it. The negative sign belongs to the numerator by convention. For example, −15/25: GCD(15,25) = 5, so −15/25 = −3/5. Sign rules for fractions: • −a/b = a/(−b) = −(a/b) — all three mean the same negative fraction • Convention: keep the negative sign on the numerator: −3/5 not 3/−5 • (−a)/(−b) = a/b — negative divided by negative is positive • When simplifying, work with absolute values for the GCD, then apply the sign • −15/25 → GCD(15,25)=5 → −(15/5)/(25/5) = −3/5 • This calculator automatically handles the sign correctly

When is a fraction already in its simplest form?

A fraction is in its simplest (lowest) terms when the GCD of the numerator and denominator is 1 — meaning they share no common factors other than 1. Such numbers are called coprime or relatively prime. Quick ways to check: • GCD = 1 → already simplified • Both odd and don't share obvious factors: likely already simple • One is prime: if the prime doesn't divide the other, already simplified (e.g. 7/13 — 7 is prime, doesn't divide 13 → GCD=1) • Examples already in lowest terms: 1/2, 3/4, 2/3, 5/7, 7/11, 3/8 • Examples NOT in lowest terms: 2/4 (÷2), 3/9 (÷3), 4/6 (÷2), 6/10 (÷2) • If you apply the Euclidean algorithm and get GCD=1, the fraction was already simplified • This calculator shows 'Already in lowest terms ✓' when no simplification is needed

Fraction Simplifier — Reduce to Lowest Terms Instantly

The Fraction Simplifier reduces any fraction to its lowest terms by finding the Greatest Common Divisor (GCD) of the numerator and denominator using the Euclidean algorithm. Enter any numerator and denominator — including negative fractions and improper fractions — and the simplified result appears ...

SIMPLIFY FRACTION

Enter your fraction

SIMPLIFIED (÷ 6)

2/3

MIXED

2/3

DECIMAL

0.6666666667

PERCENT

66.6667%

STEP-BY-STEP (EUCLIDEAN ALGORITHM)

Use the Euclidean algorithm to find GCD(12, 18):

12 = 0 × 18 + 12

18 = 1 × 12 + 6

12 = 2 × 6 + 0

GCD(12, 18) = 6

Divide numerator by GCD: 12 ÷ 6 = 2

Divide denominator by GCD: 18 ÷ 6 = 3

12/18 = 2/3

QUICK EXAMPLES

LIVE DIAGRAM

ORIGINAL

12/18

→

SIMPLIFIED

2/3

Divided by GCD = 6

MIXED

2/3

DECIMAL

0.6666666667

PERCENT

66.6667%

FRACTION COMPARISON

Original: 12/1866.67%

Simplified: 2/366.67%

Both bars are the same length — equal values, different notation

PRIME FACTORISATION

12 = 2 × 2 × 3

18 = 2 × 3 × 3

Common factors cancelled: 6 = 2 × 3

Live diagram · updates as you type

Created with❤️byeaglecalculator.com

EQUIVALENT FRACTIONS

12/18 = 2/3 = all of these fractions

FRACTION	MULTIPLY BY	DECIMAL	PERCENT
2/3 ← simplified	1	0.6666666667	66.6667%
4/6	×2	0.6666666667	66.6667%
6/9	×3	0.6666666667	66.6667%
8/12	×4	0.6666666667	66.6667%
10/15	×5	0.6666666667	66.6667%
12/18 ← original	×6	0.6666666667	66.6667%
14/21	×7	0.6666666667	66.6667%
16/24	×8	0.6666666667	66.6667%

All fractions in this table are equivalent — same value, different notation. The original is highlighted in yellow.

HOW TO USE

1
Enter the numerator (top number) in the first box and the denominator (bottom number) in the second box. Negative fractions are supported — enter a negative numerator.
2
The simplified fraction appears instantly in the result box. If the fraction is already in lowest terms, you will see 'Already in lowest terms ✓'.
3
Read the step-by-step Euclidean algorithm to see exactly how the GCD was found and how each number was divided.
4
Check the equivalent fractions table — your original fraction is highlighted, and the table shows all equivalent forms from the simplified base.
5
Use the quick example buttons to load common fractions and see how simplification works across different cases.

WORKED EXAMPLE

Example 1: 12/18 → GCD(12,18)=6 → 2/3. Euclidean: 18=1×12+6, 12=2×6+0, GCD=6. Example 2: 48/64 → GCD=16 → 3/4. Example 3: 100/75 → GCD=25 → 4/3 = 1⅓. Example 4: 7/13 → GCD=1 → already in lowest terms. Example 5: −15/25 → GCD(15,25)=5 → −3/5.

REFERENCE FORMULAS

SIMPLIFICATION REFERENCE

NAME	FORMULA	DESCRIPTION
Simplify	(n/d) ÷ GCD(n,d) = simpN/simpD	Divide both by GCD to get lowest terms
Euclidean GCD	GCD(a,b) = GCD(b, a mod b) until b=0	Repeated division until remainder is 0
Lowest terms test	GCD(simpN, simpD) = 1	Fraction is simplified when GCD of result = 1
Equivalent fractions	n/d = (n×k)/(d×k) for any k≠0	Multiply both by same number to get equivalent fraction
Mixed number	n/d = whole remainder/d	Convert improper fraction: divide, use remainder as numerator

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Last updated: April 28, 2026 · Formula verified by EagleCalculator team · Eagle-eyed accuracy for every calculation.