How To Completely Reduce (Simplify) the Fraction ²¹⁹/₁₀₀ to the Lowest Terms, Its Simplest Equivalent Form, Irreducible, With the Smallest Possible Numerator and Denominator, That Are Prime to Each Other? Write the Result as an Improper Fraction, as a Mixed Number, as a Decimal Number and as a Percentage

Other similar operations with fractions reducing (simplification):

Reduce (simplify) the fraction ²¹¹/₈₅ to the lowest terms (to its simplest form, the smallest possible numerator and denominator)

Reduce (simplify) the fraction ²²²/₁₀₄ to the lowest terms (to its simplest form, the smallest possible numerator and denominator)

To fully reduce (simplify) fractions to the lowest terms:

Divide the numerator and denominator of the fraction by their greatest (highest) common factor (divisor), gcf (hcf, gcd).

Result written as a proper or an improper fraction, as a mixed number, as a decimal or an integer and as a percentage.

Steps to simplify a fraction, to reduce it to its lowest terms:

• A fraction fully simplified, a fraction reduced to its lowest terms is a fraction that can no longer be simplified, it has been reduced to its simplest equivalent fraction, the one having the smallest numerator and denominator possible - prime to each other.
• 1) Run the prime factorization of both the numerator and the denominator of the fraction.
• 2) Calculate the greatest common factor, GCF (or the greatest common divisor, GCD) of the fraction's numerator and denominator.
• 3) Divide both the numerator and the denominator of the fraction by their greatest common factor, GCF (GCD).
• The fraction thus obtained is called a reduced fraction or a fraction simplified to its lowest terms.
• A fraction reduced to its lowest terms may no longer be reduced and it is called an irreducible fraction.

Example: reduce the fraction ³¹⁵/_1,155 as much as possible, simplify it to the lowest terms.

• 1) Run the prime factorization of both the numerator and the denominator of the fraction.

• The numerator of the fraction is 315, its breaking down into prime factors is:
  315 = 3 × 3 × 5 × 7 = 3² × 5 × 7
• The denominator of the fraction is 1,155, its breaking down into prime factors is:
  1,155 = 3 × 5 × 7 × 11.

• 2) Calculate the greatest common factor, GCF (or the greatest common divisor, GCD) of the fraction's numerator and denominator.

• The greatest common factor, gcf (315; 1,155), is calculated by multiplying all the common prime factors of the numerator and the denominator, taken by their lowest powers (their lowest exponents):
• GCF (315; 1,155) = (3² × 5 × 7; 3 × 5 × 7 × 11) = 3 × 5 × 7 = 105

• 3) Divide both the numerator and the denominator of the fraction by their greatest common factor, GCF (GCD).

• The numerator and denominator of the fraction are divided by their greatest common factor, GCF:
• ³¹⁵/_1,155 =

  ---

  ^{(32 × 5 × 7)}/_{(3 × 5 × 7 × 11)} =

  ---

  ^{((32 × 5 × 7) ÷ (3 × 5 × 7))} / _{((3 × 5 × 7 × 11) ÷ (3 × 5 × 7))} =

  ---

  ³/₁₁
• The fraction thus obtained is called a fraction reduced to the lowest terms.

Why reducing (simplifying) fractions to lower terms?

• When running operations with fractions we are often required to bring them to the same denominator, for example when adding, subtracting or comparing.
• Sometimes both the numerators and the denominators of those fractions are large numbers and doing calculations with such numbers could be difficult.
• By reducing (simplifying) a fraction, both its numerator and denominator are reduced to smaller values, much easier to work with, which will decrease the overall effort of working with that fraction.

What is a prime number? Definition, examples

What is a composite number? Definition, examples

The prime numbers up to 1,000

The prime numbers up to 10,000

The Sieve of Eratosthenes

The Euclidean Algorithm

Completely reduce (simplify) fractions to the lowest terms: Steps and Examples