Problem 9 - Prealgebra

Question

Translate each phrase into an algebraic expression. the quotient of a number and four, minus five

Step-by-Step Solution

Answer

\( \frac{x}{4} - 5 \)

Step 1: Identify the mathematical terms in the phrase

The phrase 'the quotient of a number and four' suggests a division operation involving an unknown number and the number four.

Step 2: Define the variables

Let \( x \) represent the unknown number. This is a common practice in algebra to use a letter to represent an unknown quantity.

Step 3: Write the expression for the quotient

The phrase 'the quotient of a number and four' can be translated into the expression \( \frac{x}{4} \), where \( x \) is the unknown number.

Step 4: Incorporate subtraction

The phrase 'minus five' indicates that we need to subtract 5 from the previous expression. This leads to the algebraic expression: \( \frac{x}{4} - 5 \).

Key Concepts

Understanding the Quotient

When we encounter the term "quotient" in algebra, it refers to the result of dividing one number by another. In general terms, if you have two numbers, say 'a' and 'b,' the quotient is represented as \( \frac{a}{b} \). This tells us how many times 'b' is contained within 'a'.
In your exercise, the phrase "the quotient of a number and four" directs us to divide an unknown number, typically represented by a variable such as 'x,' by the number four. Thus, the expression becomes \( \frac{x}{4} \).
Remember that the order in which numbers appear in quotient matters. \( \frac{a}{b} \) is not the same as \( \frac{b}{a} \), so make sure to maintain the correct order based on the phrase given.

Variables in Algebra

Variables are fundamental tools in algebra used to represent unknown quantities. They are typically represented by letters such as 'x', 'y', or 'z'. These letters stand in place for numbers that are not yet known or defined but can be found by solving equations or expressions.

• Variables allow us to generalize mathematical concepts and solve problems that involve unknowns.
• They provide a way to describe patterns and relationships in numbers.

In the context of the exercise, we choose 'x' as our variable to signify the unknown number. This variable is essential for converting verbal expressions into algebraic expressions, providing a clear and straightforward way to manipulate and solve the given problem.

Subtraction in Algebra

Subtraction is one of the core operations in algebra used to find the difference between numbers or expressions. In a subtraction sentence, the number that is taken away is called the "subtrahend," and the number from which it is subtracted is called the "minuend."
In the exercise, we see the phrase "minus five." This clearly indicates that after finding the 'quotient of a number and four', we must subtract 5 from our result. The subtraction operation is symbolized by the minus sign '-'.

• The algebraic operation becomes: \( \frac{x}{4} - 5 \).
• Ensure the terms are in the right order to accurately reflect the subtraction.

Subtraction in algebra operates under the same principles as basic arithmetic but allows for more flexibility and abstraction through the use of variables.