Problem 85 - Prealgebra

Question

Use the rule for order of operations to simplify each expression. $$30 \div 5 \cdot 2$$

Step-by-Step Solution

Answer

The simplified expression is 12.

Step 1: Identify Operations

The expression given is \(30 \div 5 \cdot 2\). To solve this expression, we'll need to apply 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)). The operations involved here are division and multiplication.

Step 2: Perform Division First

According to the order of operations, we perform multiplication and division from left to right. Thus, before multiplying, we need to perform the division \(30 \div 5\). Compute \(30 \div 5\) which equals 6.

Step 3: Perform Multiplication

Now that we've handled the division, the expression simplifies to \(6 \cdot 2\). Next, perform the multiplication: \(6 \times 2 = 12\).

Key Concepts

Understanding PEMDAS

PEMDAS is a mnemonic device that helps us remember the order in which we should evaluate different operations in a mathematical expression. It stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This order is crucial because mathematical expressions can yield different results if operations are performed in a different sequence.

When you encounter a mathematical expression, start by looking for any parentheses (grouping symbols) and solve what's inside first. If there are exponents, handle them next. After that, focus on multiplication and division, which should be processed from left to right as they appear in the expression. Finally, address addition and subtraction, also from left to right.

Multiplication and Division in Order

In mathematical expressions, multiplication and division are of equal priority. They are performed based on their position from left to right. This means that you do not necessarily perform multiplication before division or vice versa.

In the expression 30 divided by 5 multiplied by 2, according to PEMDAS, we first perform the division because it appears to the left of multiplication.

• Calculate \(30 \div 5\) which simplifies to 6.
• Then, multiply the result by 2: \(6 \times 2 = 12\).

Understanding this left-to-right rule eliminates confusion and ensures consistency in solving expressions.

Working with Mathematical Expressions

A mathematical expression is a combination of numbers, operators, and sometimes variables that represent a mathematical relationship. Simplifying expressions involves following the established rules for the order of operations.

To simplify the expression \(30 \div 5 \cdot 2\), we applied the order of operations principles.

• The original expression features two operations: division and multiplication.
• By handling the division first, you reduce the chance of making errors.
• Finally, any remaining operations are completed, resulting in a coherent and simplified expression.

Using these rules helps demystify mathematical expressions, making them easier and more straightforward to solve.