# Fractions: changing the 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 away one part, we have the same quantity as when we divide the whole into 6 equal parts and take away 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 kind of fractions can be reduced? Irreducible fractions.

- A common 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) anymore. - 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 recommended to reduce fractions to the lowest terms, since by this operation the value of both the numerator and the denominator are reduced, making easier the calculations in which the respective fractions will be used.

## More on ordinary (common) fractions / theory:

### (2) Changing the form of fractions, by expanding or reducing (simplifying)

## Fractions operations that can be run automatically, with explanations:

### Mathematical operations with fractions, plus the theory behind: addition, subtraction, multiplication, division, fractions reducing (simplifying), comparing, sorting in ascending order, converting numbers to fractions and percentages.