Addition and Subtraction of Fractions

Two or more fractions with the same denominator are said to have a common denominator. The following rules for adding and subtracting fractions with a common denominator can be derived from the basic algebraic properties of real numbers and the definition of a quotient.

Rules for Addition Subtraction of Fractions with Common Denominator:

If p, q and r are real numbers, q \ne 0, then \frac{p}{q} + \frac{r}{q} = \frac{{p + r}}{q}   and   \frac{p}{q} - \frac{r}{q} = \frac{{p - r}}{q}

For instance, \frac{3}{7} + \frac{2}{7}  = \frac{{3 + 2}}{7} = \frac{5}{7}   and   \frac{3}{7} - \frac{2}{7} = \frac{{3 - 2}}{7} = \frac{1}{7}
Again, the same rules apply to adding and subtracting fractional expressions.

Example:

Perform each operation
            (a) \frac{{5x}}{{2x - 1}} +  \frac{{3x}}{{2x - 1}}   (b) \frac{{5x}}{{{{\left( {3x  - 2} \right)}^2}}} - \frac{{3x}}{{{{\left( {3x - 2} \right)}^2}}}
Solution:
            (a) \frac{{5x}}{{2x - 1}} + \frac{{3x}}{{2x - 1}} = \frac{{5x + 3x}}{{2x  - 1}} = \frac{{8x}}{{2x - 1}}
            (b) \frac{{5x}}{{{{\left( {3x - 2} \right)}^2}}} - \frac{{3x}}{{{{\left(  {3x - 2} \right)}^2}}} = \frac{{5x - 3x}}{{{{\left( {3x - 2} \right)}^2}}} =  \frac{{2x}}{{{{\left( {3x - 2} \right)}^2}}}
           

To add or subtract fractions that do not have a common denominator, you must rewrite the fractions so they do have denominator. To do this, multiply the numerator and denominator of each fraction by an appropriate. For instance,

\frac{2}{3} + \frac{4}{5} = \frac{{2 \cdot 5}}{{3  \cdot 5}} + \frac{{3 \cdot 4}}{{3 \cdot 5}} = \frac{{10}}{{15}} +  \frac{{12}}{{15}} = \frac{{10 + 12}}{{15}} = \frac{{22}}{{15}}


More generally, if q \ne s, you can always add p/qand r/s as follows:

\frac{p}{q} + \frac{r}{s} = \frac{{p \cdot s}}{{q \cdot s}} +  \frac{{q \cdot r}}{{q \cdot s}} = \frac{{ps + qr}}{{qs}}


Example:

Add the expression \frac{{3x}}{{4x - 1}} and \frac{{2x}}{{3x - 5}}
Solution:
            \frac{{3x}}{{4x - 1}} + \frac{{2x}}{{3x - 5}} = \frac{{3x\left( {3x -  5} \right)}}{{\left( {4x - 1} \right)\left( {3x - 5} \right)}} + \frac{{\left(  {4x - 1} \right)2x}}{{\left( {4x - 1} \right)\left( {3x - 5} \right)}}
                                           = \frac{{3x\left( {3x -  5} \right) + \left( {4x - 1} \right)2x}}{{\left( {4x - 1} \right)\left( {3x -  5} \right)}} = \frac{{9{x^2} - 15x + 8{x^2} - 2x}}{{\left( {4x - 1}  \right)\left( {3x - 5} \right)}}
                                           = \frac{{17{x^2} -  17x}}{{\left( {4x - 1} \right)\left( {3x - 5} \right)}} = \frac{{17x\left( {x -  1} \right)}}{{\left( {4x - 1} \right)\left( {3x - 5} \right)}}

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