Problem 18 - Intermediate Algebra

Question

Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen. one-third of the sum of the length and width

Step-by-Step Solution

Answer

The algebraic expression is \( \frac{1}{3}(l + w) \).

Step 1: Identify Key Phrases

The phrase provided is "one-third of the sum of the length and width." The key components here are "one-third," "sum," and "length and width."

Step 2: Define Variables

Let's assign variables for the quantities involved. Let \( l \) represent the length and \( w \) represent the width. These variables are placeholders for any real numbers that represent the dimensions.

Step 3: Formulate the Sum

The phrase "the sum of the length and width" indicates that we need to add the variables \( l \) and \( w \). This can be expressed as an algebraic expression: \( l + w \).

Step 4: Apply One-third

The phrase "one-third of" tells us to multiply the result from Step 3 by one-third. Mathematically, this is expressed as \( \frac{1}{3} \). Therefore, the expression becomes \( \frac{1}{3}(l + w) \).

Key Concepts

Key Phrases in Algebra

Algebra often involves translating phrases from everyday language into mathematical expressions. Recognizing key words and phrases is the first step in this process. Let's break down the exercise here to understand these phrases better:

• "One-third": This phrase indicates a fraction, specifically \( \frac{1}{3} \), of a quantity.
• "The sum of": This means you should add the numbers or variables mentioned.
• "Length and width": These likely refer to components of a geometric shape, such as a rectangle, and should be added together.

Understanding these phrases is crucial, as they guide you in forming the correct algebraic expression. Once familiar with these terms, interpreting similar expressions becomes much easier. This knowledge is foundational in algebra and helps in solving various mathematical problems.

Defining Variables in Algebra

In algebra, variables act as symbols representing unknown values. Assigning variables allows you to simplify expressions and solve equations. In our exercise, the phrase mentions length and width, which are measurable quantities.

Thus, we define variables:

• Let \( l \) represent the length.
• Let \( w \) represent the width.

These variables are placeholders that can be replaced by any numerical values. For example, if the actual length was 5 units, then \( l \) would be 5. This flexibility is one of the strengths of using algebra. Clearly defining your variables at the start makes the entire problem-solving process more streamlined and avoids confusion later on.

Formulating Expressions in Algebra

Once you have identified key phrases and defined your variables, you move on to formulating expressions. This involves stating the relationship between the variables using algebraic operations.

In our specific exercise, we start by interpreting "the sum of the length and width," which we denote as \( l + w \). This expression represents adding the two dimensions of a shape.

Next, we incorporate the phrase "one-third of." This tells us to multiply the sum by \( \frac{1}{3} \). Therefore, the complete algebraic expression becomes \(rac{1}{3}(l + w)\), which compactly represents one-third of the combined measures of length and width.

Formulating expressions accurately is an essential skill in algebra that allows you to construct mathematical models of real-world scenarios, making the abstract more tangible.