# 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?

• It is advisable to reduce fractions to the lowest terms, since by this operation the value of both the numerator and of the denominator are reduced, facilitating the calculations in which the respective fractions will be used.