Fraction decomposition

Algebra: Adding and subtracting fractions

Fraction decomposition

Until now we have taken fractions together to compose one fraction, but sometimes it can help to decompose fractions.

A fraction with multiple terms in the numerator can be decomposed by using the terms in the numerator as the numerator of a new fraction. The denominator does not change.

\[\frac{{\blue a+ \green b}}{{\purple c}}=\frac{{\blue a}}{{\purple c}}+ \frac{{\green b}}{{\purple c}}\]

Example

\[\begin{array}{rcl} \dfrac{\blue{x^2}+\green{5 \cdot x}}{\purple{x \cdot y}}&=&\dfrac{\blue{x^2}}{\purple{x \cdot y}}+ \dfrac{\green{5 \cdot x}}{\purple{x \cdot y}} \\ &=& \dfrac{{x}}{{y}}+ \dfrac{{5}}{{y}} \end{array}\]

Minus signs

Above we have seen fraction decomposition when adding, and this will also work with subtracting.

\[\frac{\blue{a}-\green{b}}{\purple{c}}=\frac{\blue{a}}{\purple{c}}- \frac{\green{b}}{\purple{c}}\]

Example

\[\begin{array}{rcl} \dfrac{\blue{x^2}-\green{3\cdot x}}{\purple{x \cdot y}}&=&\dfrac{\blue{x^2}}{\purple{x \cdot y}}- \dfrac{\green{3 \cdot x}}{\purple{x \cdot y}} \\ &=& \dfrac{{x}}{{y}}- \dfrac{{3}}{{y}} \end{array}\]

Do not decompose denominators

When there are multiple terms in the denominator, it can be tempting to decompose the denominator, but this is not always allowed.

Examples

\[\begin{array}{rcl} \dfrac{{x}}{{x+5}}&\red \ne&\dfrac{{x}}{{x}}+ \dfrac{{x}}{{5}} \\ \\ \dfrac{{x+3}}{{x+4}}&\red\ne &\dfrac{{x}}{{x }}+ \dfrac{{3}}{{4}} \end{array}\]

Split \({{-a-3\cdot b}\over{2-2\cdot b}}\) in fractions with only one term in the numerator.

#{{-a-3\cdot b}\over{2-2\cdot b}}=# #-{{a}\over{2-2\cdot b}}-{{3\cdot b}\over{2-2\cdot b}}#

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