Fractions

A fraction is a representation of a part of a whole.
Fractions have a top part called a Numerator and a bottom part called a Denominator.

            $latex \frac{numerator}{denominator}&s=3$

Proper, Improper, and Mixed Fractions. Please note that Mixed Fractions are also known as Mixed Numbers.

In proper fractions, the numerator must be smaller than the denominator as shown below.
$latex \frac{2}{5}&s=3$            $latex \frac{5}{7}&s=3$            $latex \frac{21}{23}&s=3$

In improper fractions, the numerator must be bigger than the denominator as shown below.


            $latex \frac{12}{5}&s=3$             $latex \frac{5}{3}&s=3$            $latex \frac{27}{23}&s=3$

Mixed numbers have a whole number attached to a proper fraction as shown below.


            $latex 3\frac{2}{5}&s=3$            $latex 7\frac{2}{3}&s=3$            $latex 1\frac{11}{12}&s=3$

Fractions can only be added if (when) the denominators are the same. For example:

            $latex \frac{2}{5} + \frac{2}{5} = \frac{4}{5}&s=3$

            $latex \frac{4}{9} + \frac{3}{9} = \frac{7}{9}&s=3$

            $latex \frac{14}{29} + \frac{3}{29} = \frac{17}{29}&s=3$

            $latex \frac{1}{2} + \frac{3}{7}&s=3$


These fractions cannot be added as is because they have different denominators. We must find the Least Common Multiple (LCM) of 2 and 7. The LCM is the smallest number divisible by both 2 and 7. That number is 14; therefore the LCM is 14.

            $latex \frac{1 \cdot 7}{2 \cdot 7} + \frac{3 \cdot 2}{7 \cdot 2} = \frac{7}{14} + \frac{6}{14} = \frac{13}{14}&s=3$

            $latex \frac{5}{8} + \frac{2}{9}&s=3$


Once again, we must find the LCM of 8 and 9 so that we can be able to add the fractions.

            $latex \frac{5 \cdot 9}{8 \cdot 9} + \frac{2 \cdot 8}{9 \cdot 8} = \frac{45}{72} + \frac{16}{72} = \frac{61}{72}&s=3$

            $latex \frac{1}{2} + \frac{1}{6} = \frac{1 \cdot 3}{2 \cdot 3} + \frac{1 \cdot 1}{6 \cdot 1} = \frac{4}{6}&s=3$

$latex \frac{4}{6}&s=3$ can be reduced (simplified) because both the numerator (4), and denominator (6) are divisible by their Greatest Common Factor (2).

            So, $latex \frac{4 \div 2}{6\div 2}&s=3$   =   $latex \frac{2}{3}&s=3$ Therefore the final answer is   $latex \frac{2}{3}&s=3$


**Remember to always reduce your answer to simplest form as show in this example.**

When adding mixed fractions, you first add the whole numbers and then add the fraction components. Keep in mind that, fractions can only be added or subtracted if (when) they have a common denominator. The common denominator is the Least Common Multiple (LCM) of the denominators.

$latex 5\frac{1}{2} + 4\frac{3}{7}&s=3$

We begin by adding the whole numbers (5 + 4). We then find the LCM of 2 and 7. We learnt in the previous lesson that the LCM for 2 and 7 is 14. Therefore, we give each numerator a denominator of 14 as shown:

$latex 5\frac{1 \cdot 7}{2 \cdot 7} + 4\frac{3 \cdot 2}{7 \cdot 2} = 5\frac{7}{14} + 4\frac{6}{14} = 9\frac{13}{14}&s=3$

Multiplying fractions is quite easy. You simply multiply the top (numerators) as well as the bottom (denominators).

Remember: Fractions must ALWAYS be simplified to lowest terms whenever possible.

Let’s take a look at a few examples.

            $latex \frac{2}{5} \times \frac{4}{9} = \frac{2 \times 4}{5 \times 9} = \frac{8}{45}&s=3$

            $latex \frac{5}{3} \times \frac{2}{5} = \frac{5 \times 2}{3 \times 5} = \frac{10}{15}&s=3$    Remember to simplify.

**Simplifying the fraction by the Greatest Common Factor (GCF = 5)**

           $latex \frac{10\div5}{15\div5}=\frac{2}{3}&s=3$

            $latex 2\frac{1}{5} \times 3\frac{1}{5} &s=3$

Fractions can only be added when in Proper or Improper form. We must then convert the Mixed Fractions to Improper Fractions. (Watch the video if you need a reminder on how to convert Mixed Fractions to Improper Fractions.)

           $latex \frac{11}{5} \times \frac{10}{5} = \frac{110}{25}&s=3$    Remember to simplify.


**Simplifying the answer by the Greatest Common Factor (GCF = 5)**


           $latex \frac{110\div5}{25\div5}=\frac{21}{5}&s=3$    or   $latex 4\frac{1}{5}&s=3$

            $latex 1\frac{6}{7} \times 3&s=3$

Fractions can be multiplied with whole numbers because ALL whole numbers have the denominator 1. Remember to convert Mixed Fractions to Improper Fractions before multiplying.

           $latex \frac{13}{7} \times \frac{3}{1} = \frac{13 \times 3}{7 \times 1} = \frac{39}{7}&s=3$     or   $latex 5\frac{4}{7}&s=3$

            $latex \frac{1}{5} \times 3\frac{2}{7} \times 10&s=3$

Remember: Change the Mixed Fraction to Improper Fraction and the whole number (10) has a denominator of 1

           $latex \frac{1}{5} \times \frac{23}{7} \times \frac{10}{1} = \frac{1 \times 23 \times 10}{5 \times 7 \times 1} = \frac{230}{35}&s=3$    Simplify…

           $latex \frac{230 \div 5}{35 \div 5} = \frac{46}{7}&s=3$    or   $latex 6\frac{4}{7}&s=3$

Dividing fractions is actually multiplying but first you must change the sign to multiplication ( $latex \times &s=3$ ) and ‘flip’ the second fraction.

Remember: Once you’ve completed the first two rules, you simply multiply the top (numerators) and also multiply the bottom (denominators).
Remember: Fractions must ALWAYS be simplified to lowest terms whenever possible.
Let’s take a look at a few examples.

            $latex \frac{2}{7} \div \frac{8}{3}&s=3$

Change sign and flip the second fraction


           $latex \frac{2}{7} \times \frac{3}{8} = \frac{2 \times 3}{7 \times 8} = \frac{6}{56}&s=3$     …simplify to    $latex \frac{3}{28} &s=3$

            $latex 1\frac{1}{9} \div 1\frac{2}{3} &s=3$


Fractions can only be divided (or multiplied) when in Proper or Improper form as shown below.


            $latex \frac{19}{9} \div \frac{5}{3}= \frac{19 \times 3}{9 \times 5} = \frac{57}{45}&s=3$    …simplify to    $latex \frac{19}{15}&s=3$

            $latex 2\frac{1}{5} \div 22 &s=3$


Remember: All whole numbers have a denominator of 1. So change the mixed number to improper form, and give 22 a denominator of 1.


            $latex \frac{11}{5} \div \frac{22}{1}&s=3$

Change the sign to $latex ( \times )&s=3$, and ‘flip’ the second fraction then solve.

           $latex \frac{11}{5} \times \frac{1}{22} = \frac{11}{110}&s=3$    …simplify to   $latex \frac{1}{10}&s=3$