Understanding the Symbol Substitutions

The problem asks us to substitute specific mathematical operators for letters given in the equation. We are told the following:

• A denotes ‘+’ (addition)
• B denotes ‘×’ (multiplication)
• C denotes ‘−’ (subtraction)
• D denotes ‘÷’ (division)

We need to apply these substitutions to the given equation and then solve for the unknown value represented by ‘?’.

Rewriting the Equation with Operators

The original equation is: 146 A 4 B 2 = 73 B ? A 8

Replacing the letters with their corresponding operators based on the definitions:

• 'A' becomes '+'
• 'B' becomes '×'

The equation transforms into:

146 + 4 × 2 = 73 × ? + 8

Solving the Equation for the Unknown Value

Now we need to solve the transformed equation. We follow the order of operations (BODMAS/PEMDAS), which dictates that multiplication should be performed before addition or subtraction.

Step 1: Solve the left side of the equation

The left side is \(146 + 4 \times 2\).

• First, perform the multiplication: \(4 \times 2 = 8\)
• Then, perform the addition: \(146 + 8 = 154\)

So, the left side of the equation simplifies to 154.

Step 2: The equation now becomes

\(154 = 73 \times ? + 8\)

Step 3: Isolate the term containing '?'

To find the value of \(73 \times ?\), we need to subtract 8 from both sides of the equation:

\(154 - 8 = 73 \times ?\)

\(146 = 73 \times ?\)

Step 4: Solve for '?'

To find '?', we need to divide 146 by 73:

\(? = \frac{146}{73}\)

\(? = 2\)

Therefore, the value that comes in place of '?' is 2.

Verification

Let's check if substituting ? = 2 makes the equation true.

Original equation with ? = 2: \(146 A 4 B 2 = 73 B 2 A 8\)

Substitute operators: \(146 + 4 \times 2 = 73 \times 2 + 8\)

Left side: \(146 + (4 \times 2) = 146 + 8 = 154\)

Right side: \((73 \times 2) + 8 = 146 + 8 = 154\)

Since the left side equals the right side \(154 = 154\), our solution ? = 2 is correct.

Revision Table: Operator Substitution and Equation Solving

Additional Information: Order of Operations

When solving mathematical expressions that involve multiple operations, it's crucial to follow a specific order to ensure a unique and correct answer. This order is often remembered using mnemonics like BODMAS or PEMDAS.

• BODMAS:
  • B: Brackets (Parentheses)
  • O: Orders (Exponents, Roots)
  • D: Division
  • M: Multiplication
  • A: Addition
  • S: Subtraction
• PEMDAS:
  • P: Parentheses
  • E: Exponents
  • M: Multiplication
  • D: Division
  • A: Addition
  • S: 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 this problem, we first performed multiplication (\(4 \times 2\)) before addition (146 + 8) on the left side, following this rule.