Problem 42 - Introductory and Intermediate Algebra for College Students 4th

Question

Write each English phrase as an algebraic expression. Let the variable \(x\) represent the number. four more than the quotient of 30 and a number

Step-by-Step Solution

Answer

The algebraic expression for the English phrase is \(\frac{30}{x} + 4\).

Step 1: Identifying the English phrases with algebraic expressions

The first part of the phrase 'the quotient of 30 and a number' can be written as \(\frac{30}{x}\) where '30' is the dividend and 'x' is the divisor.

Step 2: Translating the second phrase

Then, 'four more than' means we add 4 to the previous expression. Therefore, the second part can be written as \(\frac{30}{x} + 4\).

Step 3: Writing the final algebraic expression

Combining the two phrases, the final representation for the English phrase 'four more than the quotient of 30 and a number' in the form of algebraic expression is \(\frac{30}{x} + 4\).

Key Concepts

Algebraic Expressions

An algebraic expression consists of numbers, variables, and operations that are combined to define a particular quantity or relationship. These expressions can be as simple as a single term, like a number or a variable, or as complex as a series of terms added, subtracted, multiplied, or divided by one another.

For instance, the algebraic expression for the phrase 'four more than the quotient of 30 and a number' reflects a multi-step relationship between numbers. Initially, we have the quotient, which represents a division between two quantities, followed by an addition ('four more'). In algebra, we visually represent this operational sequence using symbols and a variable to stand in for the 'number' being described, which in this case is represented as 'x'.

To make algebraic expressions easier to interpret, it helps to read them step-by-step, decomposing them into smaller, more understandable parts. This method of translation from verbal language to algebraic symbolism is foundational in solving mathematical problems and allows us to capture complex relationships within a concise mathematical statement.

Variables in Algebra

In the realm of algebra, a variable is a symbol, often a letter, that represents a number that can vary—that is, it can have different values. Variables are fundamental to algebra since they serve as placeholders that can be manipulated in equations and expressions without knowing their exact values.

When translating English phrases into algebraic expressions, variables like 'x' get employed to denote unknown numbers or quantities that we are either solving for or using in a given problem. For example, in the phrase 'the quotient of 30 and a number', 'x' is used to represent the unknown number. Using variables makes it possible to perform operations and solve expressions that would otherwise be impossible without specific numerical values.

Quotient Representation

The term quotient specifically refers to the result of dividing one number by another. In algebra, the quotient is visually represented by placing one number or expression over another, with a division line between them.

In quotient representation, the number on top, called the dividend, is divided by the number on the bottom, known as the divisor. For instance, the expression \(\frac{30}{x}\) denotes the quotient of 30 divided by some variable 'x'. Quotient representation is an efficient way to denote division, especially when dealing with variables and more complex relationships. It is crucial that students recognize the top and bottom parts of the quotient and relate them back to how we think about division in a more basic arithmetic context.