Solving Operator Replacement Equations

This question asks us to evaluate a mathematical expression where letters represent arithmetic operators. We are given the mapping for each letter and the expression string.

The given expression string is: 27B29A45C9D4

The operator replacements are defined as:

• 'A' is replaced by '+' (addition)
• 'B' is replaced by '-' (subtraction)
• 'C' is replaced by '÷' (division)
• 'D' is replaced by 'x' (multiplication)

Step-by-Step Evaluation

First, we substitute the letters in the expression with the corresponding operators:

Original expression string: 27B29A45C9D4

Substituting operators: 27 - 29 + 45 ÷ 9 x 4

Now, we need to evaluate this mathematical expression following the order of operations, commonly known as BODMAS or PEMDAS.

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

Let's evaluate the expression \(27 - 29 + 45 \div 9 \times 4\) step-by-step:

Step 1: Perform Division. There is one division operation: \(45 \div 9\).

\(45 \div 9 = 5\)

The expression becomes: \(27 - 29 + 5 \times 4\)

Step 2: Perform Multiplication. There is one multiplication operation: \(5 \times 4\).

\(5 \times 4 = 20\)

The expression becomes: \(27 - 29 + 20\)

Step 3: Perform Addition and Subtraction (from left to right). We have subtraction and addition remaining.

First, perform the subtraction: \(27 - 29\).

\(27 - 29 = -2\)

The expression becomes: \(-2 + 20\)

Next, perform the addition: \(-2 + 20\).

\(-2 + 20 = 18\)

The final value of the expression is 18.

Let's summarize the steps in a table:

Thus, the value of the given equation after substituting the operators is 18.

The calculation sequence is crucial for arriving at the correct answer. Following the BODMAS/PEMDAS rule ensures that operations are performed in the correct order.

Revision Table: Operator Precedence (BODMAS/PEMDAS)

Additional Information: Algebraic Expressions and Substitution

This problem involves a form of algebraic substitution, where letters stand for specific mathematical operations rather than variables representing unknown numbers. The core concept is to replace symbols with their assigned values or meanings before performing the required operations.

Understanding the order of operations (BODMAS/PEMDAS) is fundamental in mathematics to ensure consistent and correct evaluation of expressions, especially when they involve multiple operations.

In more complex scenarios, expressions might include brackets, exponents, and a mix of positive and negative numbers, requiring careful application of these rules. Practice with different combinations helps in mastering these concepts.