Problem 74 - Algebra for College Students

Question

Translate each English phrase into an algebraic expression and use \(n\) to represent the unknown number. The quotient of 50 and a number

Step-by-Step Solution

Answer

The algebraic expression is \(\frac{50}{n}\).

Step 1: Identify Key Terms

The English phrase is "The quotient of 50 and a number". The key term here is "quotient", which refers to the result of division.

Step 2: Determine the Components

Identify the components involved in the division. In this case, the components are 50 (the dividend) and an unknown number (the divisor), represented by the variable \(n\).

Step 3: Set Up the Algebraic Expression

Since the quotient is the result of dividing one number by another, the algebraic expression will be \(\frac{50}{n}\). This represents the division of 50 by the unknown number \(n\).

Step 4: Review the Expression

Verify that the expression \(\frac{50}{n}\) accurately represents the phrase "The quotient of 50 and a number". Ensure each element of the phrase has been correctly translated into the algebraic expression.

Key Concepts

Translating Phrases into Expressions

Understanding how to translate phrases into algebraic expressions is a fundamental skill in algebra. It involves converting words into mathematical symbols to form an equation or expression. When you come across phrases like "the quotient of 50 and a number," you need to break down the words systematically. Here are some steps to guide you:

• Identify mathematical terms: Words like sum, difference, product, and quotient often indicate specific operations - in this case, 'quotient' signals division.
• Determine the components: Recognize the numbers and variables involved. In the example phrase, 50 is one component, and the unknown number is another.
• Use variables: Assign a variable to represent the unknown quantity, so it becomes part of your expression.

Using this approach helps you set up the expression \( \frac{50}{n} \), interpreting 'the quotient of 50 and a number' effectively. By mastering this skill, you can easily navigate between verbal statements and mathematical representations.

Division in Algebra

Division in algebra may seem complex at first, but it boils down to understanding how numbers and variables interact. You'll often translate division phrases into expressions using variables.Let's take a closer look at the expression \( \frac{50}{n} \):

• Dividend: This is the number being divided, which is 50 in the given example.
• Divisor: This represents the number by which the dividend is divided. In our expression, \( n \) acts as the divisor, indicating a variable.
• Quotient: The result of the division, represented by the entire expression \( \frac{50}{n} \).

When you set up your expressions, remember that the order matters. The dividend is typically placed on the top or in front in a division scenario. Grasping this structure aids in understanding how division works with variables and numbers in algebraic settings.

Representing Unknowns with Variables

In algebra, representing unknowns with variables like \( n \) is a powerful tool. It allows you to express statements involving unknown quantities and solve problems systematically.To represent an unknown:

• Assign a variable: Common choices include \( x, y, \) or \( n \). These act as placeholders for the unknown numbers.
• Integrate into expressions: In the expression \( \frac{50}{n} \), \( n \) indicates the divisor whose value we don't know yet.
• Manipulation of expressions: Variables enable you to perform operations and transformations on the expressions to solve for unknowns.

Using variables turns complex word problems into understandable expressions or equations. This technique provides clarity and simplicity, especially when handling unknown quantities in algebra.