### Learning Outcomes

- Translate word phrases into mathematical expressions representing division
- Solve word problems by using division
- Convert math expressions and word problems between division and multiplication

## Translate Word Phrases to Math Notation

Earlier in this section, we translated math notation for division into words. Now we’ll translate word phrases into math notation. Some of the words that indicate division are given in the table below.

Operation | Word Phrase | Example | Expression |
---|---|---|---|

Division | divided by
quotient of divided into |
[latex]12[/latex] divided by [latex]4[/latex]
the quotient of [latex]12[/latex] and [latex]4[/latex] [latex]4[/latex] divided into [latex]12[/latex] |
[latex]12\div 4[/latex]
[latex]\frac{12}{4}[/latex] [latex]12\text{/}4[/latex] [latex]4\overline{)12}[/latex] |

### example

Translate and simplify: the quotient of [latex]51[/latex] and [latex]17[/latex].

Solution:

the quotient of 51 and 17: The word *quotient* tells us to divide

translate: [latex]51\div 17[/latex]

divide: [latex]3[/latex]

We could just as correctly have translated *the quotient of* [latex]51[/latex] *and* [latex]17[/latex] using the notation

[latex]17\overline{)51}[/latex] or [latex]\frac{51}{17}[/latex].

### try it

## Divide Whole Numbers in Applications

We will use the same strategy we used in previous sections to solve applications. First, we determine what we are looking for. Then we write a phrase that gives the information to find it. We then translate the phrase into math notation and simplify it to get the answer. Finally, we write a sentence to answer the question.

### example

Cecelia bought a [latex]\text{160-ounce}[/latex] box of oatmeal at the big box store. She wants to divide the [latex]160[/latex] ounces of oatmeal into [latex]\text{8-ounce}[/latex] servings. She will put each serving into a plastic bag so she can take one bag to work each day. How many servings will she get from the big box?

Solution

We are asked to find how many servings she will get from the big box.

Write a phrase. | [latex]160[/latex] ounces divided by [latex]8[/latex] ounces |

Translate to math notation. | [latex]160\div 8[/latex] |

Simplify by dividing. | [latex]20[/latex] |

Write a sentence to answer the question. | Cecelia will get [latex]20[/latex] servings from the big box. |

In the following video, we show another example of how to translate a situation into an algebraic expression.

Now that we have practiced using both multiplication and division to solve application problems, let’s combine both skills to answer the next question.

### Example

To save money and reduce waste, Frances bought a giant box of crackers with the plan to divide them up into small bags to take to school with her every day. The box contains [latex]48[/latex] oz. of crackers. Answer the following questions about Frances and her crackers:

1) How many [latex]2[/latex] oz. bags of crackers can Frances make from the big bag?

2) If there are [latex]150[/latex] calories in a [latex]1[/latex] oz serving of crackers, how many calories are each 2 oz. bag?

In the following video, we show another example of how to translate a situation that contains both multiplication and division into an algebraic expression.