Algebra: Adding and subtracting fractions
Making fractions similar
We can make two fractions \[ similar by multiplying the denominators with each other: \[ Note that both fractions now have the same denominator \(\blue b \green d\) and are therefore similar. |
Example Make #\tfrac{\orange 2}{\blue x}# and #\tfrac{\purple 3}{\green y}# similar: |
Sometimes there are common factors, and we do not have to multiply both denominators by each other to find a new denominator. Then you are able to find a new denominator by multiplying by the missing factors in the denominator. |
Example Make #\tfrac{\orange{2}}{\blue{x y}}# and #\tfrac{\purple{3}}{\green{y z}}# similar: \[\dfrac{\orange{2}}{\blue{x y}}= \dfrac{\orange{2} \green{z}}{\blue{x y} \green{z}} \qqquad \dfrac{\purple{3}}{\green{y z}}= \dfrac{\purple{3} \blue x}{{\blue x \green{y z}}} \] |
After all, when making fractions similar we choose the new denominator as the multiplication of both denominators: #\left(a-8\right)\cdot \left(a-4\right)#.
For #\frac{3\cdot a}{a-4}# we find this new denominator by multiplying numerator and denominator by a factor #a-8#.
This gives: \[\frac{3\cdot a}{a-4}=\frac{3\cdot \left(a-8\right)\cdot a}{\left(a-8\right)\cdot \left(a-4\right)}\]
For #\frac{-4\cdot a}{a-8}# we find this new denominator by multiplying numerator and denominator by a factor #a-4#.
This gives: \[\frac{-4\cdot a}{a-8}=\frac{-4\cdot \left(a-4\right)\cdot a}{\left(a-8\right)\cdot \left(a-4\right)}\]
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