Problem 49 - Prealgebra

Question

Use any of the rules developed in this chapter and the rule for order of operations to simplify each of the following expressions as much as possible. [Examples 6–9] $$-100 \div(10 \div 2)$$

Step-by-Step Solution

Answer

Simplified expression is -20.

Step 1: Solve the Inner Parenthesis

First, identify the expression inside the parenthesis. Here, it is \(10 \div 2\). Perform the division inside the parenthesis. \(10 \div 2 = 5\). Now, the expression becomes \(-100 \div 5\).

Step 2: Divide

Now, carry out the remaining division. Divide \(-100\) by \(5\). The operation is straightforward: \(-100 \div 5 = -20\).

Key Concepts

Understanding Division

Division is one of the fundamental operations in mathematics. It involves dividing a number, known as the dividend, by another number, called the divisor. The result is known as the quotient. In the expression \(-100 \/ \(10 \/ 2\)\), division plays a central role. Here's how:

• The dividend is \-100\.
• The divisor is the result of the inner division \(10 \/ 2\).
• The quotient is the final result after all operations are applied.

When performing division, ensure that the divisor is not zero, as division by zero is undefined. Practice dividing both positive and negative numbers to build confidence in handling different scenarios. Breaking down complex expressions into smaller divisions can simplify problems and lead to accurate results.
Remember, division is often sequential in mathematical operations, and its position can change results if order of operations isn't followed correctly.

Working with Parentheses

Parentheses are used in mathematical expressions to dictate the order in which operations are carried out. They override the standard order of operations (PEMDAS/BODMAS rules), making them a critical concept to grasp.
In the given expression \(-100 \/ \(10 \/ 2\)\), parentheses indicate that the operation inside \(10 \/ 2\) must be performed first.

• This ensures clarity, leaving no room for ambiguity about which operations are prioritized.
• When simplifying expressions, always start by calculating expressions within parentheses.

After evaluating the contents of the parentheses, the expression becomes simpler, often reducing complexity significantly. This method is essential for solving algebraic expressions efficiently and accurately.

Simplification Process

Simplification is the process of reducing an expression to its simplest form. In mathematical problems, this often involves performing operations in a strategic order. For \(-100 \/ \(10 \/ 2\)\), simplification enhances clarity and removes unnecessary complexity from the calculation.

Steps in Simplification:

• Solve Within Parentheses: Initial simplification starts by solving expressions inside parentheses. Here, \(10 \/ 2 = 5\).
• Remove Parentheses: Once solved, the parentheses are no longer needed, resulting in \-100 \/ 5\.
• Perform Remaining Operations: Finally, conduct any remaining operations following the order of operations to reach the simplest form, \-20\.

Simplification helps present a clear solution and is especially useful in environments that require exact computations, like exam settings. It also aids in verifying the accuracy of results.

Handling Negative Numbers

Negative numbers can initially seem tricky, but understanding them is key to effectively solving expressions. The expression \(-100 \/ \(10 \/ 2\)\) involves negative dividends, demonstrating how to operate with negatives.

Key Points on Negative Numbers:

• Maintain attentiveness when working with negative signs, as they can alter the outcome of operations significantly.
• Dividing a negative number by a positive number results in a negative quotient. For example, \-100 \/ 5 = -20\.
• When dividing two negative numbers, the quotient becomes positive, e.g., \-10 \/ -2 = 5\.

Understanding negative numbers is crucial as they frequently appear in calculations across various mathematical contexts. Staying mindful of their influence ensures accuracy and avoids common pitfalls like misplaced signs.