4.6Algebraic Fractions

A fraction coefficient obeys the ordinary fraction rules: combine like terms by adding or subtracting the coefficients over a common denominator, found from the LCM when the denominators differ.

Same denominator

Simplify $\displaystyle \frac{5}{8}y - \frac{1}{8}y$ .

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\displaystyle \frac{5}{8}y - \frac{1}{8}y

\displaystyle = \frac{5 - 1}{8}y

\displaystyle = \frac{4}{8}y

\displaystyle = \frac{1}{2}y

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Simplify $\frac{1}{2}x + \frac{1}{4}x$ .

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LCM of

2

and

4

4

\displaystyle \frac{1}{2}x + \frac{1}{4}x

\displaystyle = \frac{2}{4}x + \frac{1}{4}x

\displaystyle = \frac{3}{4}x

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Different denominators

Express $\displaystyle \frac{3a + 1}{4} - \frac{a - 2}{6}$ as a single fraction.

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\frac{3a + 1}{4} - \frac{a - 2}{6}

= \frac{3(3a + 1)}{12} - \frac{2(a - 2)}{12} \quad \scriptsize\textit{LCM is 12}

= \frac{9a + 3 - 2a + 4}{12}

= \frac{7a + 7}{12}

Now you try

Express $\frac{x}{5} + \frac{2x - 2}{3}$ as a single fraction in its simplest form.

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LCM of

5

and

3

15

\displaystyle \frac{x}{5} + \frac{2x - 2}{3}

\displaystyle = \frac{3x}{15} + \frac{5(2x - 2)}{15}

\displaystyle = \frac{3x + 10x - 10}{15}

\displaystyle = \frac{13x - 10}{15}

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