- 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. Get the fraction's numerator and denominator prime factorized (break them down 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, the simplest equivalent fraction, irreducible (the one with the smallest possible numerator and denominator). The GCF, HCF, GCD of the numerator and the denominator of an irreducible fraction is equal to 1.

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

**A proper fraction:**a fraction in which the denominator is larger than the numerator. Example:^{1}/_{3},^{2}/_{6},^{24}/_{32}**An improper fraction:**a fraction in which the denominator is equal to or smaller than the numerator. Example:^{3}/_{2},^{2}/_{2},^{36}/_{34}#### 1) Run the prime factorization of both the numerator and the denominator of the fraction.

- Fraction's numerator, 24, its prime factorization is:

24 = 2^{3}× 3. - Fraction's denominator, 32, its prime factorization is: 32 = 2
^{5}. #### 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 (24; 32), is calculated by multiplying all the common factors of both the numerator and denominator, taken by their lowest powers (exponents):
- GCF (24; 32) = (2
^{3}× 3; 2^{5}) = 2^{3}= 8. #### 3) Divide both the numerator and the denominator of the fraction by their greatest common factor, GCF (GCD).

- 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, it has the smallest possible numerator and denominator).

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

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

- Fraction's numerator, 130, its prime factorization is:

130 = 2 × 5 × 13. - Fraction's denominator, 455, its prime factorization is:

455 = 5 × 7 × 13. #### 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 (130; 455), is calculated by multiplying all the common factors of both the numerator and denominator, taken by their lowest powers (exponents):
- GCF (130; 455) = (2 × 5 × 13; 5 × 7 × 13) = 5 × 13 = 65.
#### 3) Divide both the numerator and the denominator of the fraction by their greatest common factor, GCF (GCD).

- 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.

### Example 3: reduce the proper 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 = 3^{2}× 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) = (3
^{2}× 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.

### Example 4: reduce the improper fraction ^{455}/_{130} to its lower terms.

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

- Fraction's numerator, 455, its prime factorization is:

455 = 5 × 7 × 13 - Fraction's denominator, 130, its prime factorization is:

130 = 2 × 5 × 13 - The greatest common factor, GCF (455; 130), is calculated by multiplying all the common factors of both the numerator and denominator, taken by their lowest powers (exponents):
- GCF (455; 130) = (5 × 7 × 13; 2 × 5 × 13) = 5 × 13 = 65
- Both fraction's numerator and denominator are divided by the greatest common factor GCF (or divisor GCD):
^{455}/_{130}=^{(5 × 7 × 13)}/_{(2 × 5 × 13)}=^{((5 × 7 × 13) ÷ (5 × 13))}/_{((2 × 5 × 13) ÷ (5 × 13))}=^{7}/_{2}- Fraction thus obtained is called a reduced fraction, simplified to its lowest terms.
- But there is more: any improper fraction can be written as the sum of an integer and a proper fraction.
- In our case:
^{7}/_{2}=^{(2 × 3 + 1)}/_{2}=^{(2 × 3)}/_{2}+^{1}/_{2}=^{3}/_{1}+^{1}/_{2}=

3 +^{1}/_{2}=

3^{1}/_{2}- 3
^{1}/_{2}is called a mixed number (a mixed fraction).

### 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.