Reduce any fraction or mixed number to its simplest irreducible form. See the step-by-step Euclidean Algorithm for Greatest Common Divisor (GCD), prime factor cancellations, and decimal conversions.
Reducing a fraction to its lowest terms involves identifying the Greatest Common Divisor (GCD) and dividing both the numerator and denominator by that common factor:
Input the numerator and denominator. If simplifying a mixed number like 2 4/8, enter 2 in the optional whole number integer field.
The calculator runs the Euclidean Algorithm and prime factor decomposition to determine the largest integer dividing both numbers.
Examine the reduced irreducible fraction, Euclidean division steps, prime cancellations, and equivalent floating-point decimals.
Simplifying awkward fractions with large numerators and denominators like 462/770 is tedious and error-prone by manual trial and error.
Determining with absolute mathematical certainty whether a fraction like 119/143 is already in simplest form or shares hidden prime factors.
Converting raw experiment outcomes (like 36 successes out of 96 trials) into standardized irreducible probability ratios (3/8).
Mechanical drawings require converting measurement ratios into standard nominal fractions for machining drills and reamers.
Provides a complete step-by-step breakdown showing how repeated remainder divisions yield the exact Greatest Common Divisor.
Breaks down both the numerator and denominator into prime factors, demonstrating which factors cancel out.
Visual badge immediately alerts the user whether the fraction was successfully reduced or was already coprime.
Simplifies the fractional portion of mixed numbers directly while preserving the whole integer prefix.
To simplify a fraction \(\frac{N}{D}\) to lowest terms, divide both the numerator and denominator by their Greatest Common Divisor:
A fraction is irreducible if and only if \(\gcd(N, D) = 1\), meaning \(N\) and \(D\) are coprime integers.
Find the greatest common divisor using Euclidean division: gcd(84, 126) = 42. Divide both terms by 42:
Keep the whole number 3 intact. Find gcd(36, 48) = 12 and simplify the fractional component:
| Starting Fraction | Greatest Common Divisor (GCD) | Lowest Terms | Decimal Equivalent |
|---|---|---|---|
| 12/16 | 4 | 3/4 | 0.75 |
| 18/24 | 6 | 3/4 | 0.75 |
| 24/36 | 12 | 2/3 | 0.6667 |
| 45/60 | 15 | 3/4 | 0.75 |
| 32/40 | 8 | 4/5 | 0.8 |
If finding the GCD is difficult in your head, divide by obvious prime factors in steps: if both are even, divide by 2; if digits sum to a multiple of 3, divide by 3; if they end in 0 or 5, divide by 5 until no common factors remain.
When measuring on imperial scales graduated in 16ths or 32nds, raw measurements like 10/16 or 24/32 must be simplified to 5/8 and 3/4 before matching standard tool sizes, drill bits, and lumber cut marks.
A fraction is in lowest terms (also known as simplest form or an irreducible fraction) when its numerator and denominator are coprime integers, meaning their Greatest Common Divisor (GCD) is exactly 1. No integer greater than 1 can divide both numbers evenly.
You can find the GCD using the Euclidean Algorithm: repeatedly divide the larger number by the smaller number and take the remainder until the remainder is zero. The last non-zero divisor is the GCD. Alternatively, you can write out the prime factorizations of both numbers and multiply their common prime factors.
A fraction is already in simplest form if the numerator and denominator share no common factors other than 1. Common indicators include: when the numerator is 1 (such as 1/7), when numerator and denominator are consecutive integers (like 14/15), or when both numbers are prime numbers.
Yes. Improper fractions (where numerator is greater than denominator, such as 36/24) reduce to simplest form exactly like proper fractions (36/24 reduces by GCD 12 to 3/2). For negative fractions, simplify the absolute magnitude first, keeping the single negative sign attached to the numerator or placed before the fraction bar.
No. Dividing both numerator and denominator by the same non-zero integer is equivalent to dividing the fraction by 1 (since k/k = 1). The numerical magnitude, decimal equivalent, and position on the real number line remain identical.
