Simplify Fractions Calculator
Reduce any fraction to its simplest form using the greatest common divisor (GCD), with the Euclidean algorithm shown step by step, mixed number and decimal conversions, and a visual bar model showing why the reduced fraction is the exact same amount. It also handles decimals inside a fraction (like 1.5/3), a fraction on top of a fraction (like (1/2)/(3/4)), and powers (like 2^3/2^5).
Background
A fraction is in simplest form (also called lowest terms) when its numerator and denominator share no common factor greater than 1. To simplify, find the greatest common divisor (GCD) of the numerator and denominator, then divide both by it — the value doesn't change, only how it's written. This calculator accepts a plain numerator/denominator pair, or a fraction written as text (including negative fractions, mixed numbers, decimals, and complex fractions), and shows exactly how the reduction happens.
How to use this calculator
- Choose Numerator / Denominator for a fraction like 42/56. Each box can also hold a decimal (1.5) or a power (2^3).
- Choose Type it in for a mixed number (-2 1/4), a decimal (0.75), decimals inside a fraction (1.5/3), or a fraction over a fraction ((1/2)/(3/4)).
- Click Simplify to see the reduced fraction first, then short steps you can follow by hand. Open How was it found? for Euclid's method written out, and More detail for the decimal value and prime factors.
- Turn on equivalent fractions to see a few other ways to write the same value (useful for checking your own reduction by hand).
- Want to add, subtract, multiply or divide fractions instead? Use the Fraction Calculator.
How simplifying fractions works
A fraction n/d represents a single value. Multiplying or dividing both the numerator and denominator by the same nonzero number never changes that value — it only changes how the fraction looks.
The greatest common divisor (GCD) of the numerator and denominator is the largest number that divides both evenly. Dividing both by the GCD removes every shared factor at once, landing on lowest terms in a single step.
The Euclidean algorithm finds the GCD without factoring: repeatedly replace the larger number with the remainder of dividing it by the smaller, until the remainder is 0 — the last nonzero number is the GCD. The calculator writes these divisions out for you.
By convention, the denominator of a simplified fraction is kept positive: any negative sign is moved to the numerator, so β10/β20 and 10/β20 both end up written as a signed numerator over a positive denominator.
A fraction's decimal form terminates (ends) exactly when the denominator, once fully reduced, has no prime factors other than 2 and 5. Otherwise the decimal repeats forever, and any displayed value is a rounded approximation.
An improper fraction (numerator's absolute value at least the denominator) can always be rewritten as a mixed number — a whole number plus a proper fraction — without changing its value.
To simplify a fraction that has decimals in it, multiply the top and the bottom by 10, 100, 1000… (one zero for each decimal place) so both become whole numbers. The value stays the same, because you multiplied top and bottom by the same number.
A fraction on top of a fraction (a complex fraction) means divide: (a/b) ÷ (c/d). Dividing by a fraction is the same as multiplying by its reciprocal, so it equals (a/b) × (d/c) = ad/bc, which can then be reduced.
Formulas & Equations Used
Simplify: n/d = (n ÷ gcd(n,d)) / (d ÷ gcd(n,d))
Euclidean algorithm: gcd(a,b) = gcd(b, a mod b), repeated until the remainder is 0
Decimal to fraction: 0.abc = abc / 10ⁿ, where n is the number of decimal digits, then simplified
Decimals inside a fraction: x/y = (x × 10ⁿ) / (y × 10ⁿ), choosing n so both become whole numbers
Complex fraction: (a/b) / (c/d) = (a × d) / (b × c)
Mixed number: n/d = whole + remainder/d, where whole = ⌊|n|/d⌋
Example Problems & Step-by-Step Solutions
Example 1 — A basic reduction
Simplify 42/56.
Step: gcd(42, 56) = 14. Divide both by 14: 42÷14 = 3, 56÷14 = 4.
Result: 42/56 = 3/4.
Example 2 — A double negative
Simplify −10/−20.
Step: A negative over a negative is positive: this is the same value as 10/20. gcd(10,20) = 10, giving 1/2.
Result: −10/−20 = 1/2 (not −1/2 — the two negatives cancel).
Example 3 — Mixed number input
Simplify the mixed number −2 1/4.
Step: Convert to an improper fraction first: −2 1/4 = −(2×4 + 1)/4 = −9/4. gcd(9,4) = 1.
Result: Already simplest form: −9/4 (equivalently −2 1/4).
Example 4 — A terminating decimal
Simplify the decimal 0.125.
Step: 0.125 = 125/1000. gcd(125, 1000) = 125, giving 1/8.
Result: 0.125 = 1/8 exactly — 8's only prime factor is 2, so the decimal terminates.
Example 5 — A fraction that's already simplest form
Simplify 7/11.
Step: 7 and 11 are both prime and different, so gcd(7,11) = 1.
Result: 7/11 stays 7/11 — dividing by 1 changes nothing.
Example 6 — A repeating decimal
What does 1/3 look like as a decimal?
Step: 1/3 is already simplest form (gcd(1,3)=1). Since 3 isn't a power of 2 or 5, the division never ends.
Result: 1/3 ≈ 0.3333333333... — any decimal shown is necessarily rounded, never exact.
Example 7 — A decimal inside a fraction
Simplify 1.5/3.
Step: Remove the decimal by multiplying the top and the bottom by 10: 1.5 × 10 = 15 and 3 × 10 = 30. Then gcd(15, 30) = 15, so divide both by 15.
Result: 1.5/3 = 15/30 = 1/2.
Example 8 — A fraction over a fraction
Simplify (1/2)/(3/4).
Step: The fraction bar means divide: 1/2 ÷ 3/4. Multiply by the reciprocal: 1/2 × 4/3 = 4/6. Then gcd(4, 6) = 2.
Result: (1/2)/(3/4) = 4/6 = 2/3.
Frequently Asked Questions
What does "simplest form" actually mean?
It means the numerator and denominator have no common factor bigger than 1 — equivalently, their GCD is 1. The fraction's value hasn't changed, only its written form has been reduced as far as it can go.
What happens if the denominator I enter is negative?
The calculator moves the negative sign onto the numerator so the denominator is always shown as positive — this is the standard convention, and it doesn't change the fraction's value.
Can I enter decimals or mixed numbers?
Yes. A decimal like 0.75 is converted to an exact fraction (75/100) before simplifying; a mixed number like 2 1/4 is converted to an improper fraction (9/4) first. You can also put decimals inside a fraction, like 1.5/3 — the calculator multiplies the top and bottom by 10 to remove the decimal.
Why does the decimal value sometimes look rounded?
Some simplified fractions (any with a prime factor other than 2 or 5 in the denominator, like thirds or sevenths) never terminate as a decimal. The calculator flags these and shows a rounded approximation rather than implying an exact value.
What's the difference between the GCD method and prime factorization?
Both reach the same simplified fraction. Prime factorization breaks each number into its prime building blocks and cancels the shared ones; the Euclidean algorithm (used here, and written out under "How was it found?") finds the GCD directly through repeated division, which is faster for large numbers and doesn't require factoring at all.
Are equivalent fractions the same as simplified fractions?
No — equivalent fractions are any fractions with the same value (like 2/4, 3/6, and 50/100), while "the simplified fraction" specifically means the one equivalent fraction with the smallest possible whole-number numerator and denominator.
Can I add, subtract, multiply or divide fractions here?
No — this calculator simplifies one fraction at a time. To add, subtract, multiply or divide fractions and decimals with steps, use the Fraction Calculator, which also reduces the answer to lowest terms.
How do I enter a fraction on top of a fraction?
Use "Type it in" and put the top and the bottom in parentheses, like (1/2)/(3/4). Without parentheses, a string like 1/2/3/4 is ambiguous, so the calculator asks you to add them.
Can I use powers or letters like x?
Whole-number powers work: 2^3/2^5 is worked out as 8/32 and reduced to 1/4. Letters are not supported here; for expressions like x/2 or x²/x⁴, use the Simplify Expression Calculator.
Why can't the denominator be zero?
A fraction means "split into this many equal parts." You can't split something into zero parts, so dividing by 0 is undefined and no fraction with a denominator of 0 can be simplified.
All calculators