Addition and Subtraction of Rational Expressions

Adding and Subtracting Rational Expressions

Tip: This isn't the place to ask a question because the teacher can't reply.

Key Questions

• $\frac{A}{B} + \frac{C}{D} = \frac{A D + B C}{B D}$

and

$\frac{A}{B} - \frac{C}{D} = \frac{A D - B C}{B D}$

• You can keep the same denominator and add or subtract the numerators.

$\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}$ and $\frac{a}{c} - \frac{b}{c} = \frac{a - b}{c}$

I hope that this was helpful.

• The least common denominator is the least common multiple of denominators.

Example
Find the least common denominator to compute:

$\frac{1}{3} + \frac{1}{4} - \frac{1}{6}$

Let us observe:

Multiples of $3$: $3 , 6 , 9$, 12,$15 , \ldots$
Multiples of $4$: $4 , 8$,12, $16 , \ldots$
Multiples of $6$: $6$,12,$18 , \ldots$,

So, the least common multiple of $3$,$4$, and $6$ is $12$.

Hence,

$\frac{1}{3} + \frac{1}{4} - \frac{1}{6} = \frac{4}{12} + \frac{3}{12} - \frac{2}{6} = \frac{4 + 3 - 2}{12} = \frac{5}{12}$
...
I hope that this was helpful.

• Imagine your fractions are $\frac{a}{b}$ and $\frac{c}{d}$.

Obviously $\frac{a}{b} = \frac{a d}{b d}$ and $\frac{c}{d} = \frac{c b}{\mathrm{db}}$,

so $\frac{a}{b} + \frac{c}{d} = \frac{a d}{b d} + \frac{c b}{\mathrm{db}} = \frac{a d + c b}{b d}$.

Also, $\frac{a}{b} - \frac{c}{d} = \frac{a d}{b d} - \frac{c b}{\mathrm{db}} = \frac{a d - c b}{b d}$.

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