[Arithmetic 6] Operations of addition, subtraction, multiplication, division of fractions

I. ADDITION OF FRACTIONS

❄ Adding two fractions with the same denominator $\boxed{\dfrac{a}{m} + \dfrac{b}{m} = \dfrac{{a + b}}{m}}$
❄ Adding two fractions with different denominators: We just convert fractions to the same denominator and applies the above rule.
Example: Calculate:
    1) $\dfrac{3}{8} + \dfrac{5}{8} = \dfrac{{3 + 5}}{8} = \dfrac{8}{8} = 1$
    2) $\dfrac{1}{7} + \dfrac{{ - 4}}{7} = \dfrac{{1 + ( - 4)}}{7} = \dfrac{{ - 3}}{7}$
    3) $\dfrac{6}{{18}} + \dfrac{{ - 14}}{{21}} = \dfrac{1}{3} + \dfrac{{ - 2}}{3} = \dfrac{{1 + ( - 2)}}{3} = \dfrac{{ - 1}}{3}$
    4) $\dfrac{{ - 2}}{3} + \dfrac{4}{{15}} = \dfrac{{ - 10}}{{15}} + \dfrac{4}{{15}} = \dfrac{{ - 6}}{{15}} = \dfrac{{ - 2}}{5}$
    5) $\dfrac{{11}}{{15}} + \dfrac{9}{{ - 10}} = \dfrac{{11}}{{15}} + \dfrac{{ - 9}}{{10}} = \dfrac{{22}}{{30}} + \dfrac{{ - 27}}{{30}} = \dfrac{{ - 5}}{{30}} = \dfrac{{ - 1}}{6}$
    6) $\dfrac{1}{{ - 7}} + 3 = \dfrac{{ - 1}}{7} + \dfrac{3}{1} = \dfrac{{ - 1}}{7} + \dfrac{{21}}{7} = \dfrac{{20}}{7}$
Similarly as the addition of intergers, the addition of fractions has the following fundamental properties:
a) Commutative property: $\dfrac{a}{b} + \dfrac{c}{d} = \dfrac{c}{d} + \dfrac{a}{b}$
b) Associative property: $\left( {\dfrac{a}{b} + \dfrac{c}{d}} \right) + \dfrac{p}{q} = \dfrac{a}{b} + \left( {\dfrac{c}{d} + \dfrac{p}{q}} \right)$
c) Adding to numer 0: $\dfrac{a}{b} + 0 = 0 + \dfrac{a}{b} = \dfrac{a}{b}$

II. SUBTRACTION OF FRACTIONS

Rule: $\boxed{\dfrac{a}{b} \color{red}{-} \dfrac{c}{d} = \dfrac{a}{b} \color{red}{+} \left( {\dfrac{{ - c}}{d}} \right)}$

Note: $\boxed{\dfrac{a}{{ - b}} =  - \dfrac{a}{b} = \dfrac{{ - a}}{b}}$

Example: Calculate:
    1) $\dfrac{1}{8} - \dfrac{1}{2} = \dfrac{1}{8} + \dfrac{{ - 1}}{2} = \dfrac{1}{8} + \dfrac{{ - 4}}{8} = \dfrac{{ - 3}}{8}$
    2) $\dfrac{{ - 5}}{7} - \dfrac{1}{3} = \dfrac{{ - 5}}{7} + \dfrac{{ - 1}}{3} = \dfrac{{ - 15}}{{21}} + \dfrac{{ - 7}}{{21}} = \dfrac{{ - 22}}{{21}}$
    3) $\dfrac{3}{5} - \dfrac{{ - 1}}{2} = \dfrac{3}{5} + \dfrac{1}{2} = \dfrac{6}{{10}} + \dfrac{5}{{10}} = \dfrac{{11}}{{10}}$
    4) $\dfrac{{ - 2}}{5} - \dfrac{{ - 3}}{4} = \dfrac{{ - 2}}{5} + \dfrac{3}{4} = \dfrac{{ - 8}}{{20}} + \dfrac{{15}}{{20}} = \dfrac{7}{{20}}$
     5) $ - 5 - \dfrac{1}{6} = \dfrac{{ - 5}}{1} + \dfrac{{ - 1}}{6} = \dfrac{{ - 30}}{6} + \dfrac{{ - 1}}{6} = \dfrac{{ - 31}}{6}$

III. MULTIPLICATION OF FRACTIONS
Rule: $\boxed{\dfrac{a}{b} . \dfrac{c}{d} = \dfrac{{a . c}}{c . d}}$
Example: Calculate:
    1) $\dfrac{{ - 3}}{7}.\dfrac{2}{{ - 5}} = \dfrac{{( - 3).2}}{{7.( - 5)}} = \dfrac{{ - 6}}{{ - 35}} = \dfrac{6}{{35}}$
    2) $\dfrac{{ - 28}}{{33}}.\dfrac{{ - 3}}{4} = \dfrac{{( - 28).( - 3)}}{{33.4}} = \dfrac{{\color{red}{28}.\color{blue}{3}}}{{\color{blue}{33}.\color{red}{4}}} = \dfrac{{\color{red}{7}.\color{blue}{1}}}{{\color{blue}{11}.\color{red}{1}}} = \dfrac{7}{{11}}$
    3) ${\left( {\dfrac{{ - 3}}{5}} \right)^2} = \dfrac{{ - 3}}{5}.\dfrac{{ - 3}}{5} = \dfrac{9}{{25}}$
    4) ${\left( { - \dfrac{3}{2}} \right)^5} = \left( { - \dfrac{3}{2}} \right)\left( { - \dfrac{3}{2}} \right)\left( { - \dfrac{3}{2}} \right)\left( { - \dfrac{3}{2}} \right)\left( { - \dfrac{3}{2}} \right) =  - \dfrac{{{3^5}}}{{{2^5}}} =  - \dfrac{{243}}{{32}}$
    5) $( - 2).\dfrac{{ - 3}}{7} = \dfrac{{( - 2).( - 3)}}{7} = \dfrac{6}{7}$               $\left(a.\dfrac{b}{c} \color{orange}{ = \dfrac{a}{1}.\dfrac{b}{c}} = \dfrac{{a.b}}{c}\right)$
    6) $\dfrac{5}{{33}}.(-3) = \dfrac{{5.\color{red}{( - 3)}}}{\color{red}{33}} = \dfrac{{5.\color{red}{( - 1)}}}{\color{red}{11}} = \dfrac{{ - 5}}{{11}}$   $\left(\dfrac{a}{b}.c \color{orange}{ = \dfrac{a}{b}.\dfrac{c}{1}} = \dfrac{{a.c}}{b}\right)$
Similarly as the multiplication of intergers, the multiplication of fractions has the following fundamental properties:
a) Commutative property: $\dfrac{a}{b} . \dfrac{c}{d} = \dfrac{c}{d} . \dfrac{a}{b}$
b) Associative property: $\left( {\dfrac{a}{b} . \dfrac{c}{d}} \right) . \dfrac{p}{q} = \dfrac{a}{b} . \left( {\dfrac{c}{d} . \dfrac{p}{q}} \right)$
c) Multiplying by number 1: $\dfrac{a}{b} . 1 = 1 . \dfrac{a}{b} = \dfrac{a}{b}$
d) Distributive property of multiplication over addition: $\dfrac{a}{b}.\left( {\dfrac{c}{d} \pm \dfrac{p}{q}} \right) = \dfrac{a}{b}.\dfrac{c}{d} \pm \dfrac{a}{b}.\dfrac{p}{q}$

IV. DIVISION OF FRACTIONS
Rule: $\boxed{\dfrac{a}{b}:\dfrac{c}{d} = \dfrac{a}{b}.\dfrac{d}{c} = \dfrac{{a.d}}{{b.c}}\;\left( {c \ne 0} \right)}$
Example: Calculate:
    1) $\dfrac{5}{6}:\dfrac{{ - 7}}{{12}} = \dfrac{5}{6}.\dfrac{{ - 12}}{7} = \dfrac{5}{1}.\dfrac{{ - 2}}{7} = \dfrac{{ - 10}}{7}$
    2) $ - 7:\dfrac{{14}}{3} =  - 7.\dfrac{3}{{14}} = \dfrac{{ - 3}}{2}$
    3) $\dfrac{{ - 3}}{7}:9 = \dfrac{{ - 3}}{7}.\dfrac{1}{9} = \dfrac{{ - 1}}{7}.\dfrac{1}{3} = \dfrac{{ - 1}}{{21}}$