## Translating Phrases to Expressions with Fractions

The words *quotient* and *ratio* are often used to describe fractions. In Mathematics 101, we defined quotient as the result of division. The quotient of \(a\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b\) is the result you get from dividing \(a\phantom{\rule{0.2em}{0ex}}\text{by}\phantom{\rule{0.2em}{0ex}}b\phantom{\rule{0.2em}{0ex}},\) or \(\frac{a}{b}.\) Let’s practice translating some phrases into algebraic expressions using these terms.

## Example

Translate the phrase into an algebraic expression: “the quotient of \(3x\) and \(8.”\)

### Solution

The keyword is *quotient*; it tells us that the operation is division. Look for the words *of* and *and* to find the numbers to divide.

This tells us that we need to divide \(3x\) by \(8.\) \(\frac{3x}{8}\)

## Example

Translate the phrase into an algebraic expression: the quotient of the difference of \(m\) and \(n,\) and \(p.\)

### Solution

We are looking for the *quotient* of the *difference* of \(m\) and , and \(p.\) This means we want to divide the difference of *\(m\)* and \(n\) by \(p.\)

\(\cfrac{m-n}{p}\)

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