Problem 51 - 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

The simplified expression is -5.

Step 1: Apply Division in Parentheses

First, we need to simplify the expression inside the parentheses. Here, we have \(-100 \div 10\). To get the result, divide \(-100\) by \(10\). The result is \(-10\). Replace the expression in parentheses with this result, so the expression becomes \(-10 \div 2\).

Step 2: Simplify the Remaining Division

Now, simplify the expression \(-10 \div 2\). Divide \(-10\) by \(2\) to get the final result of the expression. The result is \(-5\).

Key Concepts

Division

Division is one of the four fundamental arithmetic operations. It involves splitting a number (the dividend) into equal parts. In the expression \(-100 \div 10\), \(-100\) is the dividend, and \(10\) is the divisor. The operation \(-100 \div 10\) asks how many times \(10\) fits into \(-100\). Dividing \(-100\) by \(10\) results in \(-10\). Key points to remember about division include:

• The result of dividing two numbers is called the quotient.
• When dividing by a positive number, the sign of the result will be the same as the dividend.
• If both numbers (dividend and divisor) have the same sign, the result is positive. If they have different signs, the result is negative.

Parentheses

Parentheses are used in mathematics to clarify which operations should be performed first in an expression. When evaluating expressions, always start with operations inside parentheses to follow the order of operations correctly. In our example, we handle the parentheses first with the division \((-100 \div 10)\), as denoted in the original exercise. This ensures the expression is simplified step-by-step, avoiding mistakes. Parentheses ensure that expressions are evaluated properly and help us easily identify which part of an expression to solve first.

• They have a high priority in the order of operations, often represented by the acronym PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction).
• Always solve expressions within parentheses before moving on to other operations outside them.
• Using parentheses effectively can change the outcome of a calculation by altering the sequence in which operations are performed.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form by following the rules of arithmetic operations. In this context, simplification makes an expression easier to understand and work with. In our example \((-100 \div 10) \div 2\), we completed the simplification by dealing with each division operation step-by-step.

• First, perform any operations within parentheses, as shown by solving \(-100 \div 10\) first.
• Once the expression inside the parentheses is simplified, proceed with simplifying the remaining operations. Here, that last step is to simplify \(-10 \div 2\).
• The goal is to transform the expression into a more straightforward form that reveals its value or meaning more clearly.

Simplifying expressions is crucial for solving problems accurately and efficiently, making complex problems more manageable.