Simplify Fractions Calculator

This free simplify fractions calculator reduces any fraction to its simplest form by dividing the numerator and denominator by their greatest common divisor. Example: 84/126 has a GCD of 42, so it reduces to 2/3, which is 0.6667 as a decimal. The tool shows every step of the Euclidean algorithm, converts improper fractions to mixed numbers, and accepts mixed numbers as input.

Two sister tools cover the neighboring jobs: the Fraction Calculator adds, subtracts, multiplies and divides two fractions and then reduces the answer, and the Decimal to Fraction Calculator turns a decimal such as 0.375 into the fraction 3/8. Use this page when you already have a fraction and only want it in lowest terms, with the working shown.

How simplifying a fraction works

A fraction is in lowest terms when its numerator and denominator share no factor other than 1. To get there, divide both parts by their greatest common divisor:

Dividing top and bottom by the same number does not change the value, because you are multiplying by g/g = 1. What changes is the way the fraction is written. Because g is the greatest common divisor, the result cannot be reduced any further; that is what makes it the simplest form, and every fraction has exactly one.

The calculator finds g with the Euclidean algorithm: replace the larger number with the remainder of dividing it by the smaller, and repeat until the remainder is 0. The last non-zero divisor is the GCD. This takes only a handful of steps even for numbers in the millions, which is why it beats listing every factor by hand.

Worked example: 84/126 in simplest form

Take 84/126. Run the Euclidean algorithm on 126 and 84:

1. 126 mod 84 = 42 (126 = 1 × 84 + 42)
2. 84 mod 42 = 0 (84 = 2 × 42 exactly)

The remainder hit 0 when the divisor was 42, so gcd(84, 126) = 42. Now divide both parts: 84 ÷ 42 = 2 and 126 ÷ 42 = 3. The simplest form is 2/3. As a check, 2 and 3 share no factor other than 1, and 84 ÷ 126 = 0.6667, exactly what 2 ÷ 3 gives. In percent terms the fraction is 66.67%.

The prime-factor route reaches the same answer: 84 = 2 × 2 × 3 × 7 and 126 = 2 × 3 × 3 × 7. The shared factors are one 2, one 3 and one 7, whose product is 2 × 3 × 7 = 42. Cancel them and 2/3 is left. If you instead divide by small common factors one at a time, you go 84/126 → 42/63 (divide by 2) → 14/21 (divide by 3) → 2/3 (divide by 7); three steps instead of one, but the same destination.

Second example: an improper fraction to a mixed number

Now take 45/12, where the numerator is larger than the denominator. First reduce: gcd(45, 12) = 3, so 45 ÷ 3 = 15 and 12 ÷ 3 = 4, giving 15/4. Then convert: 15 ÷ 4 = 3 remainder 3, so the mixed number is 3 3/4, and the decimal is 15 ÷ 4 = 3.75. Reducing before converting keeps the numbers small; converting 45/12 directly gives 3 9/12, which you would still have to reduce to 3 3/4.

Common fractions reduced to lowest terms

The fractions that come up most in recipes, tape measures and school worksheets reduce as follows. Each row was computed by dividing both parts by the GCD shown.

Notice how many different-looking fractions collapse to the same value: 6/8, 9/12, 12/16, 18/24, 36/48, 45/60 and 75/100 are all 3/4. That is the point of simplest form: it gives every value one standard name, so two fractions are equal exactly when their reduced forms match.

Euclidean algorithm step by step

The table shows the full remainder chain for a few fractions. The GCD is the divisor in the row where the remainder first becomes 0. Even the largest example needs only three steps.

The last row is a classic: 1071/462 reduces to 51/22, an improper fraction that the calculator would also write as 2 7/22 (51 ÷ 22 = 2 remainder 7) and as the decimal 2.3182.

Improper fractions and mixed numbers

An improper fraction has a numerator at least as large as its denominator (7/4, 45/12). A mixed number writes the same value as a whole part plus a proper fraction (1 3/4, 3 3/4). Simplest form applies to both: reduce the fraction first, then divide to split off the whole part. The table shows the reduced improper fraction, the mixed number, and the decimal.

Going the other way, a mixed number becomes an improper fraction with whole × denominator + numerator over the denominator: 3 3/4 = (3 × 4 + 3)/4 = 15/4. The calculator's Mixed number mode does this conversion for you before reducing, so an input like 2 12/16 comes back as 2 3/4 and 11/4.

How to use this calculator

1. Pick the input type: Simple fraction for a plain a/b, or Mixed number when you have a whole part as well.
2. Type the numerator in the top box and the denominator in the bottom box. Whole numbers only; negatives are fine.
3. Read the simplest form in the blue result card, along with the GCD, the mixed-number form for improper fractions, the decimal and the percent.
4. Follow the steps: the step-by-step card lists every Euclidean remainder, the division by the GCD, and the final rewrite.
5. Check with prime factors if your class uses the factor-tree method; the shared primes multiply to the same GCD.
6. Try the quick buttons for common fractions such as 6/8 or 25/100 to see the pattern.

Everything updates as you type, and no data leaves your browser.

Who this calculator is for

• Students in grades 4 through 8 learning to write fractions in simplest form, and anyone checking homework where the steps must be shown.
• Parents and tutors who want a worked example to explain alongside, rather than just an answer.
• Cooks and bakers scaling recipes: 6/8 cup is 3/4 cup, and 10/16 of a cup is 5/8.
• Woodworkers and DIYers reading tape measures, where 12/16 inch is 3/4 inch and 10/16 inch is 5/8 inch.
• Anyone converting ratios or survey counts to a clean fraction: 12 out of 18 is 2/3.

Key terms

• Numerator / denominator: the top and bottom numbers of a fraction. The denominator says how many equal parts make a whole; the numerator says how many of them you have.
• Simplest form (lowest terms, reduced form): the version of a fraction in which the numerator and denominator have no common factor other than 1.
• GCD / GCF / HCF: greatest common divisor, greatest common factor and highest common factor are three names for the same number, the largest integer dividing both parts. See the GCF Calculator.
• Coprime (relatively prime): two numbers whose GCD is 1. A fraction is in lowest terms exactly when its numerator and denominator are coprime.
• Equivalent fractions: fractions with the same value, such as 6/8 and 3/4. All equivalent fractions share one simplest form.
• Proper / improper fraction: proper when the numerator is smaller than the denominator (3/4), improper when it is not (7/4).
• Mixed number: a whole number plus a proper fraction, such as 1 3/4.

What changes the result

• The GCD: the only thing that matters. A GCD of 1 means no reduction at all; a large GCD such as 500 in 1000/3500 collapses the fraction dramatically, to 2/7.
• Shared prime factors: the GCD is the product of every prime the two numbers have in common, counted with multiplicity. 84 and 126 share 2, 3 and 7 once each, so the GCD is 42 rather than, say, 84.
• Sign: has no effect on the digits. -18/24 reduces to -3/4 exactly as 18/24 reduces to 3/4.
• Scaling both parts: multiplying numerator and denominator by the same number changes the look but never the reduced form. 2/3, 14/21 and 40/60 all reduce to 2/3.
• Denominators of 10, 100, 1000: these reduce whenever the numerator shares a factor of 2 or 5, which is why 75/100 becomes 3/4 and 125/1000 becomes 1/8.

Tips for simplifying by hand

• Try 2, 3 and 5 first. Both even: divide by 2. Both digit sums divisible by 3: divide by 3. Both ending in 0 or 5: divide by 5. Repeat until none applies.
• Use the Euclidean algorithm for big numbers. Finding gcd(1071, 462) by listing factors is slow; three remainder steps give 21 immediately.
• Reduce before you convert. Turning 45/12 into 15/4 first makes the mixed number 3 3/4 fall out directly.
• Check your answer by cross-multiplying: 84 × 3 = 252 and 126 × 2 = 252, so 84/126 and 2/3 are equal.
• Watch for GCD = 1. If nothing divides both parts, the fraction was already in simplest form. 17/23 is one such case.

Limitations

• Inputs must be whole numbers. For a decimal numerator or denominator, multiply both parts by 10, 100 or 1000 first, or start from the Decimal to Fraction Calculator.
• The tool simplifies one fraction. It does not add, subtract, multiply or divide fractions; that is the job of the Fraction Calculator.
• Decimal and percent outputs are rounded for display (to 6 decimal places and 2 decimal places respectively). The fraction itself is exact.
• Values are limited to JavaScript's safe integer range (about 9 × 10¹⁵). Anything longer than 15 digits should be reduced by hand or with a computer algebra system.

Related fraction and factor tools

Use this page when you already have a single fraction and want it in lowest terms with the working shown. Reach for the Fraction Calculator when you need to combine two fractions with an operation, the Decimal to Fraction Calculator when your starting point is a decimal like 0.625 (which becomes 5/8), the Mixed Number Calculator when you are doing arithmetic on mixed numbers, the GCF Calculator when the greatest common factor itself is the question, the LCM Calculator when you need a common denominator, and the Ratio Calculator when the numbers are a ratio such as 12:18 (which simplifies to 2:3 by the same GCD method).