Fractions

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Fractionsobjectives

Objectives

Define Fractions
Know the 3 types of fractions
Add Fractions with like denominators
Add Fractions with Unlike denominators
Add Mixed Fractions with Unlike denominators
Practice Questions

Introductionto 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.

$\frac{numerator}{denominator}$

There are Three Types of Fractions

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

1. Proper Fractions

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

2. Improper Fractions

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

$\frac{12}{5}$ $\frac{5}{3}$ $\frac{27}{23}$

3. Mixed Fractions

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

$3\frac{2}{5}$ $7\frac{2}{3}$ $1\frac{11}{12}$

Addinglike denominators

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

Example 1

$\frac{2}{5} + \frac{2}{5} = \frac{4}{5}$

Example 2

$\frac{4}{9} + \frac{3}{9} = \frac{7}{9}$

Example 3

$\frac{14}{29} + \frac{3}{29} = \frac{17}{29}$

Go ahead and try the questions below.

Practice Q1. $\frac{2}{5} + \frac{1}{5} =$

Enter answer

Practice Q2. $\frac{4}{15} + \frac{1}{15} =$

Enter answer

AddingUnlike Denominators

Example 4

$\frac{1}{2} + \frac{3}{7}$

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.

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

Example 5

$\frac{5}{8} + \frac{2}{9}$

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

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

Example 6

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

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

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

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

Addingunlike denominators

Practice Q3: $\frac{3}{4} + \frac{1}{16} =$

Enter answer

Practice Q4: $\frac{3}{14} + \frac{1}{3} =$

Enter answer

Practice Q5: $\frac{2}{5} + \frac{1}{10} =$

Enter answer

AddingMixed Fractions

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.

Example 7

$5\frac{1}{2} + 4\frac{3}{7}$

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:

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

Addingmixed fractions

Practice Q6: $2\frac{1}{5} + 1\frac{3}{4} =$

Enter answer (e.g. 9 & 13/14)

Practice Q7: $5\frac{1}{2} + 6\frac{1}{4} =$

Enter answer (e.g. 9 & 13/14)

Practice Q8: $7\frac{1}{2} + 2\frac{1}{6} =$

Enter answer (e.g. 9 & 13/14)

Multiplyingfractions

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.

Example 8

$\frac{2}{5} \times \frac{4}{9} = \frac{2 \times 4}{5 \times 9} = \frac{8}{45}$

Example 9

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

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

$\frac{10\div5}{15\div5}=\frac{2}{3}$

Example 10

$2\frac{1}{5} \times 3\frac{1}{5}$

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.)

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

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

$\frac{110\div5}{25\div5}=\frac{21}{5}$ or $4\frac{1}{5}$

Example 11

$1\frac{6}{7} \times 3$

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.

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

Example 12

$\frac{1}{5} \times 3\frac{2}{7} \times 10$

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

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

$\frac{230 \div 5}{35 \div 5} = \frac{46}{7}$ or $6\frac{4}{7}$

Go ahead and try the questions below.

Practice Q9. $\frac{4}{5} \times \frac{2}{3} =$

Answer

Practice Q10. $\frac{5}{6} \times \frac{4}{3} =$

Answer

Practice Q11. $\frac{7}{3} \times 2\frac{1}{5} =$

Answer

Practice Q12. $1\frac{3}{4} \times 4\frac{4}{5} \times \frac{16}{7} \times 5=$

Answer

Dividingfractions

Dividing fractions is actually multiplying but first you must change the sign to multiplication ( $\times$ ) 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.

Example 13

$\frac{2}{7} \div \frac{8}{3}$

Change sign and flip the second fraction

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

Example 14

$1\frac{1}{9} \div 1\frac{2}{3}$

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

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

Example 15

$2\frac{1}{5} \div 22$

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

$\frac{11}{5} \div \frac{22}{1}$