## Tutoring: simplifying fractions - reducing them to the lowest terms

### 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 = 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.

### 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.