Understanding Mathematical Expressions and Order of Operations

Evaluating mathematical expressions correctly requires following a specific set of rules known as the order of operations. This ensures that everyone arrives at the same answer for the same expression. The expression we need to evaluate is:

\(6 – 36 \times 3 \div 6 + 5\)

The order of operations is commonly remembered by acronyms like BODMAS or PEMDAS.

• B/P: Brackets (Parentheses) first
• O/E: Orders (Exponents, powers, square roots) next
• DM: Division and Multiplication (from left to right)
• AS: Addition and Subtraction (from left to right)

Let's apply these rules step-by-step to the given expression.

Step-by-Step Evaluation of the Expression

The expression is: \(6 – 36 \times 3 \div 6 + 5\)

There are no Brackets or Orders (Exponents) in this expression, so we move to Division and Multiplication.

Step 1: Perform Division and Multiplication from left to right.

The first operation from the left that is either Division or Multiplication is \(36 \times 3\). \(36 \times 3 = 108\)

The expression becomes: \(6 – 108 \div 6 + 5\)

Next, we perform the Division: \(108 \div 6\). \(108 \div 6 = 18\)

The expression becomes: \(6 – 18 + 5\)

Step 2: Perform Addition and Subtraction from left to right.

The first operation from the left that is either Addition or Subtraction is \(6 – 18\). \(6 – 18 = -12\)

The expression becomes: \(-12 + 5\)

Finally, perform the Addition: \(-12 + 5\). \(-12 + 5 = -7\)

So, the value of the expression \(6 – 36 \times 3 \div 6 + 5\) is \(-7\).

Let's double-check the steps: \(6 – 36 \times 3 \div 6 + 5\) Division and Multiplication from left to right:

• \(36 \times 3 = 108\)
• \(108 \div 6 = 18\)

Substitute back into the expression: \(6 – 18 + 5\)

Addition and Subtraction from left to right:

• \(6 – 18 = -12\)
• \(-12 + 5 = -7\)

The final result is \(-7\).

Revision Table: Order of Operations

Additional Information on Evaluating Expressions

The order of operations is crucial for consistency in mathematics. Without a standard order, expressions could be interpreted in multiple ways, leading to different results.

Consider the Division and Multiplication step. If an expression contains both, like \(10 \div 2 \times 5\), you must work from left to right:

• \(10 \div 2 = 5\)
• \(5 \times 5 = 25\)

Performing multiplication first (\(2 \times 5 = 10\), then \(10 \div 10 = 1\)) would give an incorrect answer. The same left-to-right rule applies to Addition and Subtraction when they appear together, like \(15 - 3 + 2\):

• \(15 - 3 = 12\)
• \(12 + 2 = 14\)

The key takeaway is to always follow the established hierarchy and the left-to-right rule for operations at the same priority level.