Problem 42 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Indicate whether the statement is true or false. $$ a \div(b \div c)=(a \div b) \div c $$

Step-by-Step Solution

Answer

The statement \(a \div (b \div c) = (a \div b) \div c\) is False.

Step 1: Understand the given expression

We have the expression \(a \div (b \div c) = (a \div b) \div c\), and we need to determine whether it's true or false.

Step 2: Use examples to test the expression

Now, let's select some real numbers for \(a\), \(b\), and \(c\) and substitute them into the expression to see if the statement holds true. For this example, let \(a = 6\), \(b = 3\), and \(c = 2\). Now we will evaluate both sides of the equation separately.

Step 3: Evaluate the left side of the equation

The left side of the equation is given as \(a \div (b \div c)\). By substituting the values, we get: \(6 \div (3 \div 2)\) First, we evaluate inner parentheses: \(6 \div (1.5)\) Finally, we perform the division: \(4\) So the left side of the equation is equal to \(4\).

Step 4: Evaluate the right side of the equation

The right side of the equation is given as \((a \div b) \div c\). By substituting the values, we get: \((6 \div 3) \div 2\) First, we evaluate inner parentheses: \(2 \div 2\) Finally, we perform the division: \(1\) So the right side of the equation is equal to \(1\).

Step 5: Compare the results

Comparing the results of both sides, we see that the left-side result \(4\) is not equal to the right-side result \(1\). Left side: \(a \div (b \div c) = 4\) Right side: \((a \div b) \div c = 1\)

Step 6: Determine if the statement is true or false

Since the results obtained for both sides of the equation are not equal, the given statement: \(a \div (b \div c) = (a \div b) \div c\) is False.

Key Concepts

Division Properties

Division, like multiplication, follows specific properties that govern how it can be manipulated and evaluated. Understanding these properties helps us assess equations, such as the one provided in the exercise.

• Non-Associative Nature: Unlike addition or multiplication, division is non-associative. This means the grouping of numbers matters when you perform division. In simpler terms, \((a \div b) \div c\) is not necessarily the same as \(a \div (b \div c)\). Changes in the parentheses change the order in which operations are carried out.
• Order of Evaluation: In an expression involving multiple divisions, you must evaluate them in the order they appear from left to right, unless parentheses dictate a different grouping.

The exercise illustrates this principle, showing that when division is performed in a different order (due to changed grouping with parentheses), the results differ significantly. Being aware of these properties is crucial to solving and understanding complex algebraic expressions involving division.

Order of Operations

To correctly solve mathematical expressions, it's important to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). This ensures consistency in solving any mathematical expression. In the given exercise, the order of operations plays a critical role.

• First, solve any operations inside the parentheses. For our expression: \(b \div c\) inside \(a \div (b \div c)\).
• Then proceed to the division \(a \div \text{(result from previous step)}\).
• Compare this with performing the divisions from left to right, as in \((a \div b) \div c\).

Failing to adhere to this order can lead to incorrect solutions. The exercise highlights how different results arise when failing to correctly follow the order of operations with division inside and outside parentheses. By understanding and applying these steps, you ensure that calculations are performed accurately and consistently.

Algebraic Expressions

Algebraic expressions consist of variables, numbers, and operations like addition, subtraction, multiplication, and division. They represent real-world phenomena in terms where unknowns and arithmetic operations are involved. Mastery of algebraic expressions is crucial for solving equations and higher-level mathematical problems. In this exercise, the expression \(a \div (b \div c)\) vs. \((a \div b) \div c\) represents two different algebraic expressions due to varying use of parentheses:

• Variables and Constants: Here, \(a\), \(b\), and \(c\) are used as variables that can take any numerical value. They are placeholders for numbers, which allow us to make general arithmetic checks.
• Structural Differences: The difference in the structure of these expressions bred by parentheses leads to differing operations and hence diverging results. Understanding how structural differences in algebraic expressions affect their outcomes is essential.

Recognizing and manipulating algebraic expressions accurately allows you to transition smoothly from numerical problems to broader algebraic problem-solving.