Problem 50 - 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] $$-500 \div(50 \div 10)$$

Step-by-Step Solution

Answer

The simplified expression is \(-100\).

Step 1: Understanding Order of Operations

To solve this expression, we must follow the order of operations, usually abbreviated as PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

Step 2: Simplify Inside the Parentheses

The first operation to perform is inside the parentheses. The expression is: \(50 \div 10\). Divide 50 by 10, which simplifies to 5.

Step 3: Substitute Simplified Value

Replace the simplified expression in the parentheses with the result. The expression now becomes: \(-500 \div 5\).

Step 4: Perform Division Operation

Now, perform the division: Divide \(-500\) by 5. The result is \(-100\).

Key Concepts

PEMDAS

When tackling mathematical expressions, clarity in the sequence of operations is essential. That's where the order of operations comes into play, famously known as PEMDAS. This acronym stands for:

• Parentheses – Always solve expressions inside parentheses first.
• Exponents – Next, handle powers or square roots.
• Multiplication and Division – Address these operations from left to right, as they appear in the expression.
• Addition and Subtraction – Finally, execute these operations from left to right.

This sequence ensures consistent results in mathematical problem-solving. A common mistake is to think that multiplication precedes division or addition precedes subtraction, but the correct approach is to work left to right when dealing with these pairs. For example, in the expression \( 8 \div 4 \times 2 \), PEMDAS tells us to divide 8 by 4 first, and then multiply the result by 2.

simplifying expressions

Simplifying expressions is all about making them easier to understand or solve. When you simplify, you're breaking down the structure of the expression as much as possible. The goal is to reduce clutter and make any formula less complex. In our given example with \(-500 \div (50 \div 10)\), we start by resolving operations inside parentheses. This means simplifying \( 50 \div 10 \) to get 5. Now, the problem appears less daunting as it becomes \(-500 \div 5\). Why is simplification vital? It helps identify the primary operations needed to solve the problem. You get rid of unnecessary complexity, which minimizes chances for errors. Always look for grouping symbols like parentheses or brackets to find starting points for simplifying.

division operations

Division operations are fundamental in arithmetic and involve splitting a number into equal parts. In the expression we're focused on, after handling the parentheses, the key operation left was division: \(-500 \div 5\). Here's how division is approached in math:

• Identify the dividend and divisor. In \(-500 \div 5\), -500 is the dividend, and 5 is the divisor.
• Divide the dividend by the divisor. This means seeing how many times the divisor fits into the dividend.
• In our example, 5 fits into 500 exactly 100 times. Considering the negative sign, \(-500 \div 5\) results in \(-100\).

Understanding division operations can simplify both everyday math tasks and complex calculations. Just remember that when dividing with negative numbers, a negative divided by a positive yields a negative result.