Problem 70 - Intermediate Algebra

Question

Translate each English phrase into an algebraic expression and use \(n\) to represent the unknown number. One-third of a number

Step-by-Step Solution

Answer

The expression is \(\frac{1}{3}n\).

Step 1: Understand the Phrase

The phrase 'one-third of a number' indicates that we need to take a fraction of a number. In this case, 'one-third' suggests dividing by 3 or multiplying by the reciprocal of 3.

Step 2: Identify the Unknown

In algebra, it is common to use variables to represent unknown quantities. Here, we use the variable \(n\) to represent the unknown number.

Step 3: Translate into Algebraic Expression

To find one-third of \(n\), you take \(n\) and divide it by 3, which is equivalent to multiplying by \(\frac{1}{3}\). This can be expressed algebraically as \(\frac{1}{3}n\).

Key Concepts

Variable Representation

Understanding how to use variables in algebra can be a real game-changer for students learning math. Variables are essential tools in algebra because they allow us to represent unknown quantities. In our exercise, the phrase "a number" refers to an unknown value. This value is not specified, so we represent it using a variable. The standard practice in algebra is to use letters to denote these unknowns, and in this exercise, we're using the letter \( n \).

Here's how variable representation works:

• Think of variables like placeholders that can stand in for any number in an equation or expression.
• This makes equations flexible, allowing them to apply to many different scenarios.
• By using variables, we can form expressions that represent mathematical relationships or operations on unknown numbers.

Variable representation helps in developing general solutions and in solving equations readily.

Fractional Parts

Taking fractional parts in algebra is simply about understanding how fractions can represent parts of a whole. In this context, a "fractional part" refers to any fraction that identifies a portion of a given number. Our phrase "one-third of a number" suggests that we are interested in one part out of three equal parts of the unknown number \( n \).

Here's how you work with fractional parts in algebra:

• Understand the terminology: "one-third" is a fraction represented by \( \frac{1}{3} \).
• To find "one-third" of a number, you multiply the entire number by \( \frac{1}{3} \) or divide the number by 3.

When translating to an algebraic expression:

• The expression \( \frac{1}{3}n \) tells us that \( n \) is divided by 3 or multiplied by \( \frac{1}{3} \).

Working with fractional parts is crucial because it extends our ability to model real-world situations where things are shared, divided, or considered in parts.

Translating Phrases into Algebra

Translating phrases into algebra is like learning a new language. The key is to recognize the math terms hidden in everyday language. In our example, the English phrase "one-third of a number" needs to be converted into an algebraic expression. Accounting for each part of the phrase, we identify the mathematical operation it implies and express it in terms of variables.

Here's a simple process to follow:

• Identify operations: (*of* usually means multiplication in algebraic terms).
• Interpret fractions: The phrase "one-third" translates to multiplication by \( \frac{1}{3} \).
• Use variables: Use \( n \) for "a number" to generalize the expression.

When all parts are combined correctly, an English phrase becomes a mathematical formula: \( \frac{1}{3}n \).

This practice helps to solve problems more efficiently, emphasizing understanding over memorization. It also demystifies challenging language by breaking down phrases into manageable parts.