# Simplifying ordinary math fractions (reducing to lower terms): steps to follow and examples

- 1. Get the fraction's numerator and denominator prime factorized (break them into prime factors). Prime factorizations here: numbers prime factorization .
- 2. Calculate fraction's numerator and denominator greatest common factor, GCF (greatest common divisor, GCD). Calculate GCF (GCD), here: numbers greatest common factor (or divisor) GCF, GCD .
- 3. Divide fraction's both numerator and denominator by GCF (GCD). Fraction thus obtained is called a reduced fraction (simplified) to its lowest terms.

### Example 1: reduce fraction ^{24}/_{32} to its lower terms.

- Fraction's numerator, 24, prime factorization is: 24 = 2
^{3}× 3.

Fraction's denominator, 16, prime factorization is: 16 = 2^{5}. - The greatest common factor, GCF (24; 32), is calculated by multiplying all the common factors of both the numerator and denominator, by their lowest powers:

GCF (24; 32) = (2^{3}× 3; 2^{5}) = 2^{3}= 8. - Both fraction's numerator and denominator are divided by the greatest common factor GCF (or divisor GCD):

^{24}/_{32}=^{(24 ÷ 8)}/_{(32 ÷ 8)}=^{(23 × 3 ÷ 23)}/_{(25 ÷ 23)}=^{3}/_{4}.

Fraction thus obtained is called a reduced fraction (simplified) to its lowest terms; in this case this is also an irreducible fraction (it can't be reduced anymore).

### Example 2: reduce fraction ^{130}/_{455} to its lower terms.

- Fraction's numerator, 130, prime factorization is: 130 = 2 × 5 × 13.

Fraction's denominator, 455, prime factorization is: 455 = 5 × 7 × 13. - The greatest common factor, GCF (130; 455), is calculated by multiplying all the common factors of both the numerator and denominator, by their lowest powers:

GCF (130; 455) = (2 × 5 × 13; 5 × 7 × 13) = 5 × 13 = 65. - Both fraction's numerator and denominator are divided by the greatest common factor GCF (or divisor GCD):

^{130}/_{455}=^{(2 × 5 × 13)}/_{(5 × 7 × 13)}=^{((2 × 5 × 13) ÷ (5 × 13))}/_{((5 × 7 × 13) ÷ (5 × 13))}=^{2}/_{7}.

Fraction thus obtained is called a reduced fraction (simplified) to its lowest terms.

### Why reducing fractions to lower terms (simplifying)?

Operations with fractions often involve them being brought to the same denominator and sometimes both the numerators and denominators are large numbers. Doing calculations with such large numbers could be difficult. By simplifying (reducing) a fraction, both the numerator and denominator of the fraction are reduced to smaller values, much easier to work with.