Problem 118 - Introductory Algebra for College Students

Question

Write each phrase as an algebraic expression. a loss of \(\frac{1}{3}\) of an investment of \(d\) dollars

Step-by-Step Solution

Answer

The algebraic expression for the phrase 'a loss of 1/3 of an investment of d dollars' is \(d - \frac{1}{3}d\).

Step 1: Identify the variables

In the given phrase, the variable is \(d\), which represents the amount of investment in dollars.

Step 2: Determine the operation

The phrase speaks about a 'loss' of an amount, which will correspond to a subtraction in the algebraic expression.

Step 3: Write the algebraic expression

Since it's a loss of 1/3 of the investment, we represent 'a third of \(d\)' as \(\frac{1}{3}d\). The loss would then mean subtracting this from the initial investment \(d\). So the resulting expression is \(d - \frac{1}{3}d\).

Key Concepts

Writing Algebraic Expressions

When it comes to translating phrases into algebra, writing algebraic expressions is a fundamental skill that one must acquire. The process involves taking a verbal description of a mathematical relationship and expressing it using numbers, variables, and operational symbols. For instance, consider the phrase 'a loss of \(\frac{1}{3}\) of an investment of \(d\) dollars.' Here, the goal is to encapsulate this financial situation in a concise mathematical snippet.

A solid grasp of how to convert words into mathematical symbols is crucial for this task. You start by identifying the variable involved—here we have \(d\), signifying the total investment. The operation indicated by the word 'loss' implies a subtraction. Finally, to write the expression, we formulate the scenario as \(d - \frac{1}{3}d\), which succinctly illustrates the reduction in the initial amount due to the loss.

Remember, effective algebraic expressions omit unnecessary information and include only the variables and operations directly related to the situation. Brevity and precision are the keys.

Variables in Algebra

The concept of variables is at the heart of algebra. Essentially, a variable is a symbol, typically a letter, used to represent an unknown or a quantity that can change. In our exercise, \(d\) is the variable that stands for the amount of the investment in dollars. It’s essential to understand that variables aren't just placeholders; they're the bridge between real-world quantities and the abstract world of mathematics.

Variables allow us to create formulas that are universally applicable. For example, whether your investment is \(100 or \)1,000, the expression for the loss \(d - \frac{1}{3}d\) remains valid because \(d\) adapts to represent the specific investment amount. They provide flexibility and scalability to algebraic expressions, making them powerful tools for solving a wide range of problems.

Basic Algebra Operations

In algebra, we use a set of standard operations to work with numbers and variables. These operations include addition, subtraction, multiplication, division, and exponentiation. Let's focus on the exercise at hand, which involves subtraction, and more subtly, multiplication (considering that \(\frac{1}{3}\) of \(d\) is indeed \(\frac{1}{3}\) multiplied by \(d\)).

To properly convey the 'loss of a third,' you combine subtraction with multiplication as embellished by the expression \(d - \frac{1}{3}d\). This might appear simple, but understanding and applying these basic operations is essential for constructing and manipulating algebraic expressions and equations. Mastery of these operations is what allows individuals to build the foundation necessary to tackle more complex problems in algebra and beyond.