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Translating Phrases to Expressions with Fractions

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.

$$\text{The quotient}\phantom{\rule{0.2em}{0ex}}\text{of}\phantom{\rule{0.2em}{0ex}}3x\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}8\text{.}$$

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 $$p.$$

$$\cfrac{m-n}{p}$$