Problem 58 - Intermediate Algebra

Question

Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen. a. the quotient of a number and 6 increased by the number b. the quotient of a number and \(6,\) increased by the number

Step-by-Step Solution

Answer

The algebraic expression is \( \frac{x}{6} + x \).

Step 1: Identifying the variable

Choose a variable to represent "a number". Let's use the variable \( x \) to denote the unknown number in both phrases.

Step 2: Understanding the phrase 'the quotient of a number and 6'

A quotient represents division. Therefore, 'the quotient of a number and 6' means dividing the number \( x \) by \( 6 \). This can be represented as \( \frac{x}{6} \).

Step 3: Interpreting 'increased by the number'

The phrase 'increased by the number' implies adding the number \( x \) to the previous expression. So we add \( x \) to \( \frac{x}{6} \), which results in the algebraic expression \( \frac{x}{6} + x \).

Key Concepts

Quotient

When dealing with algebraic expressions, one of the core concepts to understand is the term "quotient". A "quotient" is simply the result of division. In arithmetic, you might readily think of division producing a straightforward number. However, in algebra, variables are involved, making expressions more dynamic.

For example, if we want to express "the quotient of a number and 6", this means we take an unknown number (let's say it's represented by the variable \( x \)) and divide it by 6. In algebraic terms, this is written as \( \frac{x}{6} \). It showcases the division relationship between \( x \) and 6. Understanding how to translate word phrases like this into algebraic expressions helps in solving complex math problems and aids in the understanding of the underlying mathematical principles.

Variable Representation

Variables are a powerful and foundational tool in algebra. They are used to represent unknown or changing quantities in an expression or an equation. When facing a problem, such as translating phrases to algebraic expressions, selecting a variable to represent an unknown number is crucial.

In the given exercise, the variable \( x \) is chosen to represent "a number". This choice is arbitrary—you could use \( y \), \( z \), or any other letter. However, \( x \) is commonly used and widely accepted. The whole idea of variables is to have them act as placeholders.

• They allow for flexibility in expressions and equations.
• They can be manipulated algebraically to solve for their value.
• They stand in for numbers whose values may not initially be known but can be uncovered through problem-solving methods.

Recognizing the significance of variable representation helps you translate complex statements, as in this exercise, where "a number" transforms smoothly into the variable \( x \).

Step-by-Step Solution

Providing a thorough, step-by-step solution is immensely helpful, especially when dealing with algebraic expressions that might at first be confusing. Breaking down a problem, as done with the original exercise, makes it easier to comprehend and solve.

The first step is to identify the unknown element that's often stated in words. Here, we chose \( x \) to denote "a number".
Following that, we interpreted the word "quotient", translating it into a division operation, yielding \( \frac{x}{6} \).

Finally, deciphering the phrase "increased by the number" led us to add \( x \) to our expression, resulting in \( \frac{x}{6} + x \). Each step builds on the previous, allowing for an increasing understanding of how word phrases are structured as algebraic expressions.

• Breaking down each phrase makes the transition from words to symbols smooth.
• Each step adds clarity, helping to prevent errors in interpretation.
• Understanding this process enhances problem-solving skills in algebra and builds confidence.

Using a clear, step-by-step method not only helps solve the specific problem at hand but also improves comprehension of algebra as a whole.