Fractions, theory: how to simplifying reducing ordinary math fractions to lower terms and the role of the greatest common factor GCF

Changing form. Multiplying numerator and denominator by the same non-zero number. Reducing or simplifying fractions.

If we divide a number into 3 equal parts and only take a part, we have the same amount as if we divide the number into 6 equal parts and take 2 parts, so 1/3 = 2/6. As stated, we can write: 2/5 = 4/10, 5/3 = 20/12, 2/3 = 4/6 = 6/9 = ... = 24/36 = ...

Multiplying fraction's numerator and denominator by the same non-zero number and reducing or simplifying it:

If the numerator and denominator of a fraction are multiples of the numerator and denominator of another fraction, we say that the fraction was created by multiplying numerator and denominator of the other fraction by the same non-zero number. For example, 8/9 = (8 * 5) / (9 * 5) = 40/45.

Multiplying fraction's numerator and denominator by the same non-zero number will yield an equivalent fraction: a/b = (a * c) / (b * c).

Dividing the numerator and denominator of a fraction by the same non-zero number will yield an equivalent fraction and is called reducing or simplifying the fraction: a/b = (a : c) / (b : c).

When and why do we reduce or simplify a fraction:

A simple fraction, in which the numerator and denominator are coprime (the only factor that goes into both the numerator and denominator is 1) is said to be irreductible, in lowest terms, or in simplest terms and cannot be reduced or simplified anyfurther. For example, fraction 4/16 is not in lowest terms because both 4 and 16 can be exactly divided by 4. In contrast, fraction 4/5 is in lowest terms, the only factor that goes into both 4 and 5 is 1.

Any fraction with both the numerator and denominator containing common factors can be reduced to lower terms (or simplified).

Operation 2/7 = (2 * 3) / (7 * 3) = 6/21 is from left to right, an operation of multiplying both the numerator and denominator of the fraction by the same number, 3, and from right to left, a reducing or simplifying of the fraction to the lowest terms.

Reducing or simplifying fractions is very helpfull since this operation lowers both denominator and the numerator values, making further calculations on that fraction easier.

More on ordinary math fractions theory:

Fractions operations that can be run automatically, with explanations: