Problem 70 - Algebra for College Students

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 algebraic expression is \(\frac{n}{3}\).

Step 1: Define the unknown

Let the unknown number be represented by the variable \(n\). This allows us to create an algebraic expression with \(n\) as the placeholder for the number we are trying to find.

Step 2: Understand the phrase 'One-third of a number'

The phrase 'one-third of a number' can be interpreted mathematically as dividing the unknown number by 3. In algebra, this is equivalent to multiplying the number by the fraction \(\frac{1}{3}\).

Step 3: Construct the algebraic expression

Now that we understand 'one-third' refers to multiplying by \(\frac{1}{3}\), we can construct the algebraic expression as \(\frac{1}{3} \times n\), which is written algebraically as \(\frac{n}{3}\).

Key Concepts

Algebra

Algebra is like a language in mathematics where we use symbols and letters to represent numbers and express relationships. This is particularly helpful when trying to find unknown values or when creating formulas. In algebra:

• We often use letters like \(n\), \(x\), or \(y\) to stand for unknown numbers. These are called variables.
• Algebraic expressions combine variables, numbers, and operations such as addition, subtraction, multiplication, and division.

Think of algebra as a tool to translate real-world problems into mathematical expressions, allowing you to find solutions easily. When you read a problem, look for key phrases that hint at mathematical operations: words like 'sum', 'difference', 'product', and 'quotient' can guide you on what to do mathematically.

In this exercise, we use algebra to translate the phrase 'one-third of a number' into a symbolic expression. Here, \(n\) stands for the unknown number we are investigating.

Fractions

Fractions are a way of expressing numbers that are not whole, indicating a part of a whole. A fraction consists of two numbers: the numerator and the denominator.

• The numerator is the top part of the fraction and indicates how many parts we have.
• The denominator is the bottom part and tells us into how many parts the whole is divided.

For example, the fraction \(\frac{1}{3}\) means one part out of three equal parts. When we say 'one-third of a number,' we imply finding a fraction of that number. In mathematical terms, multiplying by \(\frac{1}{3}\) changes the size of the whole (or the number in question) to one-third of its original size.

In our exercise, this concept helps us to understand how 'one-third' can be expressed as an algebraic operation, leading us to write \(\frac{n}{3}\). This operation signifies that we are considering one part out of three equal parts of \(n\).

Mathematical Translation

Mathematical translation involves converting a verbal statement into a mathematical expression. This skill is crucial for solving word problems in mathematics. Let's break down how this is done:

• Identify key numbers and variables: Look for numbers mentioned in the statement and decide what the unknown variable should be. In our case, \(n\) represents a number we haven't identified yet.
• Select the correct operation: Words like 'of', 'sum', and 'times' guide us to choose multiplication, addition, or another operation.
• Create the expression: Once you've identified the operation and variable, write the expression. In this task, 'one-third of a number' prompts us to multiply \(n\) by \(\frac{1}{3}\), resulting in the expression \(\frac{n}{3}\).

This process helps bridge the gap between everyday language and structured mathematical reasoning. Mastering it makes solving problems much more approachable, as it turns complex sentences into simple mathematical operations.