What is a fraction in mathematics? 
In this lecture, we are going to discuss addition and subtraction of fractions. 
Fractions are rational numbers with the form $\frac{a}{b}$ for every $a,b\in\mathbb{Q}$, where $\mathbb{Q} $ is the set of rational number and $a$ is called the numerator, $b$  the denominator. There are three types of fractions namely:
  1. Proper fraction
  2.  Improper fraction
  3. Mixed fraction 
Proper fractions are fractions whose denominator is greater than the numerator e.g 
$\frac{2}{7},\frac{1}{3},\frac{7}{13}$.

Improper fractions are fractions whose denominator is less than the numerator e.g $\frac{7}{2},\frac{3}{2},\frac{13}{7}$.

Mixed fractions are fractions that consist of a whole number and a fraction e.g $3\frac{1}{2},7\frac{2}{5},1\frac{8}{9}$.
Mixed fractions can be converted to improper fraction by simply multiplying the whole number by the denominator and adding the result to the numerator. 

Example
  1.  $3\frac{1}{2}=\frac{3\times{2}+1}{2}=\frac{7}{2}$.
  2.  $7\frac{3}{4}=\frac{7\times{4}+3}{4}=\frac{31}{4}$
  3. $5\frac{8}{9}=\frac{5\times{9}+8}{9}=\frac{53}{9}$

Addition and Subtraction of Mixed Fraction


In this post, I will be showing you two methods for solving questions that involves addition and subtraction of mixed fractions. 

  1.  Evaluate $2\frac{2}{3}+1\frac{1}{2}$
Method I 
This method involves adding the whole numbers first before adding the fractions.
$2\frac{2}{3}+1\frac{1}{2}$
$=2+1(\frac{2}{3}+\frac{1}{2})$.
Whole number=$2+1=3$. 
To add the fraction, first find the LCM of the denominators $3$ and $2$ which is $6$. The LCM is obtained by multiplying the denominators of both fractions since $3$ cannot be divided by $2$, therefore, LCM=$3\times{2}=6$. Next divide each denominator by $6$ and multiply by the numerator of each fraction.
$3(\frac{2}{3}+\frac{1}{2})=3\frac{(2\times{2})+(3\times{1})}{6}$
$=3\frac{4+3}{6}=3\frac{7}{6}$.

Method II
This method involves converting the mixed fractions to improper fraction.
$2\frac{2}{3}+1\frac{1}{2}$
$2\frac{2}{3}=\frac{2\times{3}+2}{3}=\frac{8}{3}$
$1\frac{1}{2}=\frac{1\times{2}+1}{2}=\frac{3}{2}$
Now add the converted fractions
$\frac{8}{3}+\frac{3}{2}=\frac{16+9}{6}=\frac{25}{6}$

  1. Evaluate $5\frac{1}{3}-2\frac{7}{8}$
Method I
Subtract the whole numbers before adding the fractions 
$5\frac{1}{3}-2\frac{7}{8}$
$=5-2(\frac{1}{3}-\frac{7}{8})$
$=3\frac{1}{3}-\frac{7}{8}$
LCM=$3\times{8}=24$
$=3(\frac{(8\times{1})-(3\times{7})}{24})$
$=3\frac{8-21}{24}=3\frac{-13}{24}$
Convert the result to an improper fraction
$=\frac{3\times{24}-13}{24}$
$=\frac{59}{24}$. 

Method II
Convert $5\frac{1}{3}-2\frac{7}{8}$ to improper fraction before subtracting the result.
$5\frac{1}{3}=\frac{5\times{3}+1}{3}$
$=\frac{16}{3}$
$2\frac{7}{8}=\frac{2\times{8}+7}{8}$
$=\frac{23}{8}$
Subtract the resulting fractions
$=\frac{16}{3}-\frac{23}{8}$
$=\frac{8\times{16}-3\times{23}}{24}$
$=\frac{128-69}{24}=\frac{59}{24}$.

  1. Evaluate $1\frac{5}{8}+\frac{1}{2}$
Method I
Add the whole numbers and then the fractions.
$1+0=(\frac{5}{8}+\frac{1}{2})$
$=1\frac{(1\times{5})+(4\times{1})}{8}$
$=1\frac{5+4}{8}=1\frac{9}{8}=\frac{17}{8}$.

Method II
$1\frac{5}{8}+\frac{1}{2}$
$=\frac{1\times{8}+5}{8}+\frac{1}{2}$
$=\frac{13}{8}+\frac{1}{2}$
$=\frac{1\times{13}+4\times{1}}{8}$
$=\frac{13+4}{8}=\frac{17}{8}$