Fractions: changing form. Expanding and reducing (simplifying) to equivalent fractions, examples

Changing form. Expanding, reducing (simplifying) fractions to lower terms

Changing form:

  • If we divide a whole into 3 equal parts and then take one part, we have the same quantity as when we divide the whole into 6 equal parts and take two parts.
    • This way:
    • 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 = ...

Expanding and reducing (simplifying) a fraction:

  • If the numerator and the denominator of a fraction A are multiples of the numerator and of the denominator of another fraction, B, we say that fraction A was calculated by expanding the fraction B.
    • For example:
    • 8/9 = (8 × 5) / (9 × 5) = 40/45
    • In this case we say that the fraction 40/45 was calculated by expanding the fraction 8/9 - more precisely, by multiplying both the numerator and the denominator by the number 5.
  • Expanding a fraction means to multiply both the numerator and the denominator of the fraction by the same non-zero number - this operation is generating an equivalent fraction:
    • a/b = (a × c) / (b × c)
  • The reverse operation of expanding a fraction is called reducing or simplifying a fraction.
  • Reducing or simplifying a fraction to lower terms means to divide both the numerator and the denominator of the fraction by the same non-zero number - this operation is generating an equivalent fraction:
    • a/b = (a ÷ c) / (b ÷ c)
  • The operation:
  • 2/7 = (2 × 3) / (7 × 3) = 6/21
  • represents, from left to right, an expanding, and from right to left, a simplification.

What fractions can be reduced? Irreducible fractions.

  • An ordinary fraction in which the numerator and the denominator are coprime numbers (their only common factor is 1) is called an irreducible fraction and cannot be reduced (simplified).
  • For example, fraction 4/16 is not in its lowest terms and can be reduced since both 4 and 16 can be evenly divided by 4.
  • On the contrary, fraction 4/5 is in its lowest terms and cannot be reduced anymore, since the only factor that goes into both 4 and 5 is 1.
  • Any fraction in which the numerator and the denominator have common factors others than 1 can be reduced (simplified).

Why do we reduce fractions?

More on ordinary (common) math fractions theory: