Problem 25 - Prealgebra

Question

Find the value of each expression. $$56 \div(7 \cdot 2) \times 6$$

Step-by-Step Solution

Answer

The value of the expression is 24.

Step 1: Identify the Order of Operations

According to the order of operations, follow the sequence of Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (PEMDAS/BODMAS). In this expression, resolve any operations within the parentheses first.

Step 2: Compute the Expression Inside the Parentheses

Inside the parentheses, we have the multiplication \(7  2\). Compute this as: \(7  2 = 14\). Now the expression becomes \(56 14  6\).

Step 3: Perform the Division

Next, according to the order of operations, perform the division operation. Divide 56 by the result from the parentheses: \(56 14 = 4\). Now the expression becomes \(4  6\).

Step 4: Perform the Multiplication

Finally, multiply the result by 6: \(4  6 = 24\).

Key Concepts

Arithmetic Expressions

Arithmetic expressions are combinations of numbers, operations, and sometimes, parentheses. They are solved following a defined order to ensure consistency in results. In math, an arithmetic expression could have operations like addition, subtraction, multiplication, and division. Any arithmetic expression will ultimately result in a final numeric value once evaluated completely.

Understanding arithmetic expressions is crucial because it helps us organize and simplify complex problems into manageable steps. With arithmetic expressions, we can represent operations compactly and solve them systematically. For example, in the given problem, the expression is \(56 \div(7 \cdot 2) \times 6\). Each part of this expression has to be evaluated in a proper order to get the correct solution.

PEMDAS/BODMAS

PEMDAS or BODMAS is a rule used to clarify which operations should be done first in an expression. The acronym PEMDAS stands for:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

Similarly, BODMAS is the same principle but in a different notation:

• Brackets
• Orders (i.e., exponents)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

These rules are essential because they ensure that everyone gets the same result when interpreting an expression.

In the exercise \(56 \div(7 \cdot 2) \times 6\), we follow the order of operations to first solve the operation inside the parentheses \((7 \cdot 2)\). Once the parentheses are resolved, we proceed with division and then multiplication.

Division and Multiplication

Division and multiplication are both part of the set of basic arithmetic operations. In expressions where both appear, they have the same priority and should be performed from left to right. This order ensures that arithmetic expressions are evaluated in a consistent way.

In our expression example, we first perform the division after resolving the parentheses, turning the expression \(56 \div(7 \cdot 2) \times 6\) into \(56 \div 14 \times 6\). The division \(56 \div 14\) results in 4. After completing this step, we proceed to perform the multiplication \(4 \times 6\), which results in 24.

Keeping track of the order, especially when both division and multiplication are present, is crucial. It ensures we follow the correct steps and achieve the right outcome.