# Mathematics I (Math 1132) Durham College, Mathematics
Free • 42 lessons
• 0 quizzes
• 10 week duration
• ##### Measurements

An introduction to numerical computation. Emphasis is placed on scientific and engineering notation, the rule of significant figures, and converting between SI and Imperial units.

• ##### Fractions, Percentage, Ratios and Proportion

Emphasis here is placed on understanding fractions, percent, and using ratios to compare quantities and set up proportions to solve problems.

• ##### Geometry

This unit focuses on analyzing and understand the characteristics of various shapes, both 2D and 3D.

## Mathematics I (Math 1132)

### Multiply and Divide Algebraic Fractions

A key component to multiplying and dividing algebraic fractions is knowing how to do it to ordinary fractions. That being said, you’re first expected to review how it’s done before continuing. If after a few examples you feel confident enough, you may skip it.

# Multiplying Algebraic Fractions

Just as you would multiply ordinary fractions as discussed above, algebraic fractions are no different. Take, for example, the expression:

$\frac{2a}{3b}·\frac{5c}{4a}\phantom{\rule{0ex}{0ex}}$

To find the product of these monomials, you multiply the numerators separately from the denominators.

Numerator product:   2a × 5c = 10ac   (confused? review this lesson)

Numerator product:   3b × 4a = 12ab

$\frac{10ac}{12ab}=\frac{5\overline{)a}c}{6\overline{)a}b}=\frac{5c}{6b}\phantom{\rule{0ex}{0ex}}$

Let’s try another example:

$\frac{x}{x+2}·\frac{{x}^{2}–4}{{x}^{3}}\phantom{\rule{0ex}{0ex}}$

After multiplying the numerators and denominators, you should end up with the following expression ↓. You might be tempted to expand x(x² − 4), but instead, you could cancel out this x with 1 of 3 x‘s found in the denominator (x³).

# $=\frac{x\left({x}^{2}–4\right)}{\left(x+2\right){x}^{3}}\phantom{\rule{0ex}{0ex}}=\frac{\overline{)x}\left({x}^{2}–4\right)}{\left(x+2\right){x}^{\overline{)3}2}}\phantom{\rule{0ex}{0ex}}=\frac{{x}^{2}–4}{x+2}$

You may think you’re done at this point, but recall that x² − 4 is a difference of squares. Therefore:

$\frac{\left(x–2\right)\left(x+2\right)}{x+2}\phantom{\rule{0ex}{0ex}}=\frac{\left(x–2\right)\overline{)\left(x+2\right)}}{\overline{)\left(x+2\right)}}\phantom{\rule{0ex}{0ex}}=x–2\phantom{\rule{0ex}{0ex}}$

Try this final example on your own:

$\frac{{x}^{2}+x–2}{{x}^{2}–4x+3}·\frac{2{x}^{2}–3x–9}{2{x}^{2}+7x+6}\phantom{\rule{0ex}{0ex}}$

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# Dividing Algebraic Fractions

As with any fractions being divided, the second fraction in the expression is always reciprocated (flipped) and the sign changed to multiplication.

$\frac{a}{b}÷\frac{c}{d}\phantom{\rule{0ex}{0ex}}=\frac{a}{b}×\frac{d}{c}\phantom{\rule{0ex}{0ex}}=\frac{ad}{bc}\phantom{\rule{0ex}{0ex}}$

Let’s look at an easy example containing algebraic terms.

$\frac{x}{y}÷\frac{x+2}{y–1}\phantom{\rule{0ex}{0ex}}$

Here we’re expected to divide and simplify. We first reciprocate the second fraction, then change the symbol to multiply. You’ll also notice that further simplification isn’t needed.

$\frac{x}{y}×\frac{y–1}{x+2}\phantom{\rule{0ex}{0ex}}=\frac{x\left(y–1\right)}{y\left(x+2\right)}\phantom{\rule{0ex}{0ex}}$

Here’s another example:

$\frac{{x}^{2}+x–2}{x}÷\frac{x+2}{{x}^{2}}\phantom{\rule{0ex}{0ex}}$

You might notice that x² + x – 2 can be factored by trial-and-error into (x + 2)(x – 1). Also, after flipping the second equation, x² cancels out with x and (x + 2) in the denominator with (x + 2) in the numerator:

$\frac{\left(x+2\right)\left(x–1\right)}{x}×\frac{{x}^{2}}{x+2}\phantom{\rule{0ex}{0ex}}=\frac{\left(x+2\right)\left(x–1\right){x}^{2}}{x\left(x+2\right)}\phantom{\rule{0ex}{0ex}}=\frac{\overline{)\left(x+2\right)}\left(x–1\right){x}^{\overline{)2}1}}{\overline{)x}\overline{)\left(x+2\right)}}\phantom{\rule{0ex}{0ex}}=x\left(x–1\right)\phantom{\rule{0ex}{0ex}}$

# Simplify Complex Fractions

Fractions that have only one fraction line are called simple fractions. Fractions with more than one fraction line are called complex fractions. Examples are shown below:

The general method to tackling these types of fractions is to first convert them into a format that’s understandable to you. Example 1 can be rewritten as:

$\frac{a}{b}÷\frac{c}{d}\phantom{\rule{0ex}{0ex}}$

and example 2 as:

$\left(\frac{a}{b}+\frac{c}{d}\right)÷\frac{e}{f}\phantom{\rule{0ex}{0ex}}$

Notice how the numerator was placed in parentheses. This suggests that you have to combine the terms in the numerator before you can start dividing. Let’s look at examples with a thorough step-by-step explanation: