Problem 17 - Algebra 1

Question

Write the verbal phrase as an algebraic expression. Use \(x\) for the variable in your expression. Quotient of a number and two tenths

Step-by-Step Solution

Answer

The algebraic expression for the verbal phrase 'quotient of a number and two tenths' is \(\frac{x}{0.2}\).

Step 1: Identify the Variable

Start by identifying the variable in the equation. The question tells us to use \(x\) as a variable. This variable represents the 'a number' part of the verbal phrase.

Step 2: Identify the Operation and Numbers

Next, identify the operation and numbers involved. The term 'quotient' in mathematics is used when one number is divided by another. So, when we see 'quotient', we know we're dealing with division. 'Two tenths' can be written as \(0.2\).

Step 3: Form the Expression

Now we can combine the variable, operation, and number to form our algebraic expression. The quotient of a number (\(x\)) and two tenths (\(0.2\)) will be represented as \(\frac{x}{0.2}\).

Key Concepts

Variables

Variables play an essential role in algebra, acting as placeholders for numbers. They let us write formulas and equations that can solve a wide variety of problems repeatedly with different inputs. In the exercise you were given, the variable was denoted by \(x\). This "\(x\)" stands for "a number" as per the verbal phrase. This means that the number isn't explicitly known yet. It's like a box where any number can be stored.

• A variable can be any letter, but common ones are \(x\), \(y\), and \(z\). This makes the algebraic expression flexible and applicable to numerous situations.
• Try thinking of variables as fill-in-the-blank slots. For instance, in "the quotient of a number and two tenths," \(x\) is the blank representing "a number."

By using variables, you can create equations that describe a problem vividly, paving the way towards understanding and solving it.

Division in Algebra

Division in algebra can be quite different from doing long division with numbers. When dealing with algebraic expressions, it often involves reasoning symbolically. In the provided exercise, you encountered the term "quotient," which is key to understanding that division is required. The "quotient of a number and two tenths" directly translates to dividing one quantity by another.
In mathematical terms, the "quotient" is the result of dividing one number by another. Here, you're finding the quotient of a variable \(x\) and \(0.2\).

• So, the operation becomes \(\frac{x}{0.2}\), where \(x\) is divided by 0.2.
• This expression can also be seen as asking, "How many times does 0.2 fit into \(x\)?"

Understanding division in algebra helps simplify and solve equations or expressions made up of both numbers and variables.

Translation of Verbal Phrases into Algebra

Translating verbal phrases into algebra is like converting a story into a mathematical language. It's about understanding the language of math and identifying keywords that tell us what operations to use. The phrase "Quotient of a number and two tenths" is our verbal phrase here. Let’s break it down:

• "A number" in the phrase is represented by a variable, which we've established as \(x\).
• "Quotient" suggests a division operation.
• "Two tenths" becomes \(0.2\) in numerical terms.

Together, these pieces construct the algebraic expression \(\frac{x}{0.2}\). Translating phrases to algebra can initially seem challenging, but understanding the common terms, like "sum," "difference," "product," and specifically here, "quotient," helps clarify the type of math involved.
Recognizing these terms and their implications in algebra allows you to transform detailed scenarios into straightforward algebraic expressions, which is crucial for problem-solving.