Understanding the Operator Substitution Problem

This question asks us to find the missing operator in a mathematical equation where standard arithmetic operators (+, ×, -, ÷) are represented by letters (A, B, C, D). We need to substitute the letters with their corresponding operators and then solve the equation following the order of operations (BODMAS/PEMDAS) to find which operator makes the equation true.

Operator Mapping Details

The question provides the following mapping for the operators:

• A denotes ‘+’ (Addition)
• B denotes ‘ב (Multiplication)
• C denotes ‘−’ (Subtraction)
• D denotes ‘÷‘ (Division)

Analyzing the Given Equation

The equation is given as: 77 A 23 B 2 A 25 D 5 = 8 B 32 D 4 ? 64

We need to substitute the given operators into this equation. Let's break it down into the Left-Hand Side (LHS) and the Right-Hand Side (RHS).

Solving the Left-Hand Side (LHS)

LHS: 77 A 23 B 2 A 25 D 5

Substitute the letters with the actual operators:

\( 77 + 23 \times 2 + 25 \div 5 \)

Now, we apply the BODMAS (or PEMDAS) rule, which dictates the order of operations: Brackets/Parentheses, Orders/Exponents, Division and Multiplication (from left to right), Addition and Subtraction (from left to right).

Step 1: Perform Division

\( 25 \div 5 = 5 \)

The equation becomes:

\( 77 + 23 \times 2 + 5 \)

Step 2: Perform Multiplication

\( 23 \times 2 = 46 \)

The equation becomes:

\( 77 + 46 + 5 \)

Step 3: Perform Addition

\( 77 + 46 = 123 \)

\( 123 + 5 = 128 \)

So, the value of the Left-Hand Side is 128.

\( \text{LHS} = 128 \)

Solving the Right-Hand Side (RHS) with Unknown Operator

RHS: 8 B 32 D 4 ? 64

Substitute the known letters with operators:

\( 8 \times 32 \div 4 ? 64 \)

Again, apply the BODMAS rule.

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

\( 32 \div 4 = 8 \)

The equation becomes:

\( 8 \times 8 ? 64 \)

Step 2: Perform Multiplication

\( 8 \times 8 = 64 \)

The equation becomes:

\( 64 ? 64 \)

Now we need to find which operator ('?') makes this expression equal to the LHS value, which is 128.

\( 64 ? 64 = 128 \)

Testing the Options for the Missing Operator

Let's test each operator option (D, A, C, B) for the place of '?'.

• Option 1: ? is D (÷) Substitute ÷ for '?': \( 64 \div 64 = 1 \). Does \( 1 = 128 \)? No.
• Option 2: ? is A (+) Substitute + for '?': \( 64 + 64 = 128 \). Does \( 128 = 128 \)? Yes.
• Option 3: ? is C (-) Substitute - for '?': \( 64 - 64 = 0 \). Does \( 0 = 128 \)? No.
• Option 4: ? is B (×) Substitute × for '?': \( 64 \times 64 = 4096 \). Does \( 4096 = 128 \)? No.

The only operator that satisfies the equation is 'A', which represents '+'.

Final Answer Determination

Based on the analysis, the operator 'A' when placed in the position of '?' makes the equation true (128 = 128). Therefore, the correct answer is A.

Revision Table: Operator Substitutions

Additional Information: Order of Operations (BODMAS/PEMDAS)

When solving mathematical expressions involving multiple operations, we follow a specific order to ensure a consistent result. This order is commonly known by acronyms like BODMAS or PEMDAS.

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

In the problem, we used this rule to correctly evaluate both sides of the equation after substituting the operators.