Understanding Coded Operator Expressions

This question asks us to evaluate a mathematical expression where the standard arithmetic operators (+, -, *, /) are represented by letters (A, B, C, D). We first need to decode the expression using the given mapping and then solve it following the correct order of operations.

Decoding the Operators

The question provides the following mapping:

• 'A' denotes 'addition' ($+$)
• 'B' denotes 'multiplication' ($\times$)
• 'C' denotes 'subtraction' ($-$)
• 'D' denotes 'division' ($\div$)

Converting the Expression

The original expression given is:

55 A 5 B (7 A 4) C 75 D (25 D 5)

Now, let's replace the letters with their corresponding mathematical operators:

55 $+$ 5 $\times$ (7 $+$ 4) $-$ 75 $\div$ (25 $\div$ 5)

Solving the Expression Step-by-Step

To solve this mathematical expression, we must follow the order of operations, often remembered by the acronym BODMAS or PEMDAS:

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

Let's evaluate the expression step-by-step:

1. Solve the operations inside the parentheses:
   • First parenthesis: (7 $+$ 4)
   • $7 + 4 = 11$
   • Second parenthesis: (25 $\div$ 5)
   • $25 \div 5 = 5$
   The expression becomes: 55 $+$ 5 $\times$ 11 $-$ 75 $\div$ 5
2. Perform Multiplication and Division from left to right:
   • Multiplication: 5 $\times$ 11
   • $5 \times 11 = 55$
   • Division: 75 $\div$ 5
   • $75 \div 5 = 15$
   The expression becomes: 55 $+$ 55 $-$ 15
3. Perform Addition and Subtraction from left to right:
   • Addition: 55 $+$ 55
   • $55 + 55 = 110$
   • Subtraction: 110 $-$ 15
   • $110 - 15 = 95$

Final Result

The value of the expression 55 A 5 B (7 A 4) C 75 D (25 D 5) is 95.

Additional Information: Order of Operations

The order of operations is a set of rules that dictates the sequence in which operations should be performed in a mathematical expression. This ensures that everyone gets the same result when evaluating an expression.

Commonly used acronyms include:

• BODMAS: Brackets, Orders (powers, square roots), Division and Multiplication, Addition and Subtraction.
• PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction.

Division and Multiplication have the same priority and are performed from left to right. Similarly, Addition and Subtraction have the same priority and are performed from left to right.