Fractions, theory: how to simplifying (reducing) or expanding ordinary math fractions and the role of the greatest common factor GCF

Changing form. Expanding a fraction: multiplying fraction's numerator and denominator by the same non-zero number. Reducing (simplifying) fractions to lower terms.

If we divide an integer into 3 equal parts and only take a part, we have the same amount as if we divide the integer into 6 equal parts and take 2 parts, so 1/3 = 2/6

According to the statement above, we can write:

  • 2/5 = 4/10
  • 5/3 = 20/12
  • 2/3 = 4/6 = 6/9 = ... = 24/36 = ...

Expanding fraction: multiplying a fraction's numerator and denominator by the same non-zero number. Reducing (simplifying) fraction to lower terms.

  • If a fraction's numerator and denominator are multiples of other fraction's numerator and denominator respectively, we say that the fraction was created by expanding that other fraction (multiplying the other fraction's numerator and denominator by the same non-zero number).
  • For example, 8/9 = (8 * 5) / (9 * 5) = 40/45
  • Expanding a fraction is multiplying fraction's numerator and denominator by the same non-zero number and it will always yield an equivalent fraction: a/b = (a * c) / (b * c)

  • By reversing the above process of expanding a fraction we are saying that we are reducing or simplifying a fraction: we divide a fraction's numerator and denominator by the same non-zero number and this operation will also yield an equivalent fraction: a/b = (a : c) / (b : c)

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

What fractions can be reduced? Irreducible fractions.

  • An ordinary 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 irreducible, in its lowest terms, or in its simplest terms and cannot be reduced or simplified anyfurther.
  • For example, fraction 4/16 is not in its lowest terms because both 4 and 16 can be exactly divided by 4.
  • On the contrary, fraction 4/5 is in its lowest terms, the only factor that goes into both 4 and 5 is 1.
  • Any fraction in which the numerator and denominator have common factors can be reduced to lower terms (or simplified).

Why do we reduce or simplify a fraction?

  • Reducing (simplifying) fractions is very helpfull since this operation lowers both the denominator and the numerator values, easing further calculations on that fraction.

More on ordinary math fractions theory:

Fractions operations that can be run automatically, with explanations: