# Question #82869

Jan 9, 2018

I shall attempt to do so. :)

#### Explanation:

There are three types of fractions: proper, improper, and mixed.

In a Proper Fraction the numerator is less than the denominator. Examples: $\frac{2}{3}$, $\frac{3}{4}$, and $\frac{9}{21}$.

In an Improper Fraction the numerator is more than the denominator. Examples: $\frac{3}{2}$, $\frac{4}{3}$, and $\frac{21}{9}$.

In a Mixed Fraction there is a whole number followed by a proper fraction in which the numerator is less than the denominator. Examples: $1 \frac{1}{2}$, $1 \frac{1}{3}$, and $2 \frac{3}{9}$.

I have used the same numbers in the fractions to make it easier.

Equal or Equivalent Fractions:
These are fractions where the numerators and denominators have the same value in relation to each other. Examples: $\frac{2}{3} = \frac{4}{6} = \frac{6}{9} = \frac{20}{30}$.

If you examine these fractions, they have the same value. This can be determined by factorising the numerators and denominators, cancelling the common factors and seeing whether the fractions are the same. So in the above examples, the smallest fraction is $\frac{2}{3}$. Factorising the second example, we get:
$\frac{4}{6} = \frac{2 \times 2}{3 \times 2} = \frac{2 \times \cancel{2}}{3 \times \cancel{2}} = \frac{2}{3}$.

The same can be done with the others, and we will get the same result. This means that all four examples are equivalent fractions.

Simplifying Fractions:
Fractions can be simplified in two ways.

1. If it is a proper or improper fraction, check if the numerator and denominator can be factorised, cancel the common factors, and the result is a simplified fraction. Example: $\frac{9}{21} = \frac{3 \times 3}{7 \times 3} = \frac{3 \times \cancel{3}}{7 \times \cancel{3}} = \frac{3}{7}$

2. If it is an improper fraction, first check if it can be simplified using the method above. Then you could convert it into a mixed fraction. This is done by dividing the numerator by the denominator and writing the result as a whole number. If there is a remainder, that is written after the whole number as the numerator over the same denominator. Example: $\frac{21}{9} = \frac{7 \times 3}{3 \times 3} = \frac{7 \times \cancel{3}}{3 \times \cancel{3}} = \frac{7}{3} = 2 \frac{1}{3}$ (The denominator $3$ goes into the numerator $7$ twice, so we get $2$ as the whole number; the remainder, $1$, is written over the denominator).

I do hope this helps. :)