Problem 76 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

\( \frac{1}{3}n + 6 \)

Step 1: Understanding the Phrase

The phrase given is 'Six more than one-third of a number.' Let's break this down. We need to find one-third of a number and then add six to it.

Step 2: Identifying the Unknown Variable

The problem specifies that we should use the variable \(n\) to represent the unknown number. So our unknown number is \(n\).

Step 3: Calculating One-Third of the Number

To express one-third of the number \(n\), we write: \( \frac{1}{3}n \). This is because dividing by 3 is equivalent to multiplying by \( \frac{1}{3} \).

Step 4: Adding Six to One-Third of the Number

The final part of the phrase is to add six to one-third of \(n\). We take our previous expression \( \frac{1}{3}n \) and add 6, resulting in \( \frac{1}{3}n + 6 \).

Step 5: Final Algebraic Expression

Putting it all together, the algebraic expression that represents 'Six more than one-third of a number' is: \( \frac{1}{3}n + 6 \).

Key Concepts

Translating English Phrases

Translating English phrases into algebraic expressions is like turning words into a mathematical language. It involves identifying key phrases in the English sentence that hint at specific mathematical operations. For example:

• 'More than' typically indicates addition.
• 'Less than' usually signifies subtraction.
• 'Product of' corresponds to multiplication.
• 'Quotient of' suggests division.

To translate the phrase "Six more than one-third of a number," start by recognizing components. The phrase "one-third of a number" uses division or multiplication by a fraction. In this case, it's about taking one-third (\[\frac{1}{3}\]) of an unknown number. Then, with "six more than," you need to add 6 to the expression you've formed—showing addition. Taking these steps ensures you reflect the original relationship described in the sentence. This process lets us craft meaning from English descriptions using algebra.

Unknown Variables

In algebra, we often use symbols to represent numbers that we don't know yet; these are called unknown variables. They are placeholders that we substitute with actual numbers once further information is available. Using a variable like \( n \) makes it easier to write and understand algebraic expressions.

Here’s how to think about unknown variables in English phrases:

• Identify what the unknown represents, such as 'a number.'
• Choose a variable to stand in place of this number, like \( n \).

With the phrase, "one-third of a number", the unknown variable \( n \) refers to the 'number.' Thus, \( \frac{1}{3}n \) represents one-third of whatever number \( n \) might stand for. The variable simplifies the process of solving problems and finding answers once more information or specific conditions are provided.

Fractions in Algebra

Fractions in algebra can easily represent parts of a whole. Understanding them is crucial because they often show up in expressions and equations. Let's break down the concept with our example:

• "One-third of a number" implies a part of that whole number.
• In algebraic terms, this is represented by \( \frac{1}{3}n \), which is equivalent to dividing \( n \) by 3.

Algebra utilizes fractions just like regular numbers—except they express relationships more precisely. In our case, \( \frac{1}{3} \) is the fraction indicating one-third of the unknown \( n \).

By learning to work with fractions in this way, you become capable of handling more complex mathematical relationships. This foundational skill is not only essential for problems involving parts and wholes but also for equations that require balancing different components.