Problem 37 - Intermediate Algebra

Question

Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen. one-hundredth of the distance

Step-by-Step Solution

Answer

\(\frac{d}{100}\)

Step 1: Understanding the Phrase

The phrase 'one-hundredth of the distance' implies we need to represent 1/100 of a given distance. We can choose any letter to represent the distance, commonly represented by variable names such as \(d\), \(x\), or any other suitable letter.

Step 2: Selecting a Variable

Let's choose the letter \(d\) to represent the distance. This makes it clear which variable stands for the distance we are working with.

Step 3: Formulating the Expression

To express "one-hundredth of the distance" mathematically, we find 1/100 of \(d\). This is mathematically represented as \(\frac{1}{100} \times d\) or simply \(\frac{d}{100}\).

Step 4: Verifying the Expression

Check that the expression \(\frac{d}{100}\) correctly represents the phrase. This is the distance divided by 100, matching the concept of one-hundredth of the distance.

Key Concepts

Algebraic Translation

Algebraic translation is the process of converting written statements or real-world situations into algebraic expressions. This is often a critical first step when solving mathematical problems.

• The aim is to move from verbal descriptions to a form that can be analyzed mathematically.
• Keywords play a crucial role. For instance, words like "of" typically indicate multiplication in math.

In our exercise, the phrase "one-hundredth of the distance" needs to be converted into an algebraic expression. The keyword here is "one-hundredth," which suggests taking 1/100 of a quantity. Once we identify these clues, they guide us in constructing the correct algebraic expression.

Variable Selection

Choosing the right variable is an important step when translating phrases into algebraic expressions. Variables are symbols, often letters, that represent unknown or varying quantities.

• Common choices for variables are letters like \(x\), \(y\), or \(d\), but any letter can be used.
• Choose a variable that makes the context of your problem easy to understand.

For our exercise, the variable \(d\) was chosen to represent distance. This choice helps clarify the meaning of the expression. Effective variable selection simplifies the problem-solving process and ensures a smooth transition from description to expression.

Mathematical Representation

Once you have translated a phrase and chosen your variables, it's time for mathematical representation. This involves crafting a precise algebraic expression that accurately represents the given statement.

• Mathematical operations are used based on the keywords identified in the phrase.
• In our example, "one-hundredth of the distance" means taking \(\frac{1}{100}\) times \(d\).

This can be mathematically represented as \(\frac{1}{100} \times d\) or \(\frac{d}{100}\). Both forms are valid and communicate the same idea. Mathematical representation is about capturing the essence of the statement in numerical terms, ensuring it aligns with the original phrase's meaning.