### 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. |
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. |
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(z-1\right)\cdot z#.

For #\frac{4\cdot z}{z-1}# we find this new denominator by multiplying numerator and denominator by a factor #z#.

This gives: \[\frac{4\cdot z}{z-1}=\frac{4\cdot z^2}{\left(z-1\right)\cdot z}\]

For #\frac{2\cdot z}{z}# we find this new denominator by multiplying numerator and denominator by a factor #z-1#.

This gives: \[\frac{2\cdot z}{z}=\frac{2\cdot \left(z-1\right)\cdot z}{\left(z-1\right)\cdot z}\]

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