Problem 34 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

State the real number property that iustifies the statement $$ \frac{x+2 y}{3 x+y} \div \frac{x}{6 x+2 y}=\frac{x+2 y}{3 x+y} \cdot \frac{2(3 x+y)}{x}=\frac{2(x+2 y)}{x} $$

Step-by-Step Solution

Answer

The real number property that justifies the statement is the \(Division Property of Multiplication\).

Step 1: Identify the given expression

We are given the expression: \[ \frac{x+2 y}{3 x+y} \div \frac{x}{6 x+2 y} \]

Step 2: Rewrite division as multiplication using the reciprocal of the second term

Using the Division Property of Multiplication, rewrite the expression as the multiplication of the first fraction with the reciprocal of the second fraction: \[ \frac{x+2 y}{3 x+y} \cdot \frac{6 x+2 y}{x} \]

Step 3: Simplify

We are given the final expression, so let's directly write the simplified version and compare it to the given expression to ensure our process is justified by the Division Property of Multiplication: \[ \frac{2(x+2 y)}{x}=\frac{2(x+2 y)}{x} \] Our simplified expression matches the given one, so the real number property that justifies this statement is the "Division Property of Multiplication."

Key Concepts

Division Property of Multiplication

In mathematics, division and multiplication are closely related, and understanding how to switch between them can make problem-solving much easier. The **Division Property of Multiplication** states that dividing by a number is the same as multiplying by its reciprocal. This property is incredibly useful when you want to simplify complex expressions.

• Suppose you have an expression in the form of \( a \div b \). According to this property, it can be rewritten as \( a \times \frac{1}{b} \).
• This means that any division operation can be transformed into multiplication, making it easier to handle, especially in algebraic terms.

Say you have the original problem: \( \frac{x+2y}{3x+y} \div \frac{x}{6x+2y} \). By applying the Division Property of Multiplication, you can rewrite this division as:\[\frac{x+2y}{3x+y} \cdot \frac{6x+2y}{x}\] This transformed equation is much easier to work with, especially when simplifying algebraic expressions.

Reciprocal

Understanding the idea of a **reciprocal** is essential when dealing with divisions in algebraic expressions. A reciprocal of a number is essentially what you multiply it by to get 1. Think of it as "flipping" the number.

• If you have a number \( a \), its reciprocal is \( \frac{1}{a} \).
• For a fraction \( \frac{a}{b} \), the reciprocal is \( \frac{b}{a} \).

In the exercise, you take the reciprocal of the fraction \( \frac{x}{6x+2y} \), resulting in \( \frac{6x+2y}{x} \). This reciprocal plays a key role in converting the division into multiplication, thereby simplifying the entire problem. It's a critical tool in algebra that helps in transforming and simplifying equations.

Algebraic Expressions

**Algebraic expressions** consist of numbers, variables, and operations, forming a mathematical phrase. They can represent real-world situations and are a foundational part of understanding algebra.

• An example of an algebraic expression is \( x+2y \) or \( 3x+y \), where \( x \) and \( y \) are variables.
• Algebraic expressions can be simplified, combined, and manipulated using various algebraic properties and operations.

In our example problem, the algebraic expressions \( x+2y \) and \( 3x+y \) are parts of larger fractions. Being able to transform and simplify these expressions through properties like the Division Property of Multiplication can make complex mathematical tasks easier. Simplifying these expressions is critical to solving equations efficiently and understanding the flow of mathematical operations.