Solving the Mathematical Equation by Interchanging Symbols

The problem asks us to find which two mathematical symbols, when interchanged, make the given equation correct. The original equation is:

\(9 + 3 \times 2 - 8 \div 4 = 8\)

Before we start interchanging symbols, let's evaluate the original equation using the order of operations (BODMAS/PEMDAS):

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

Evaluating the original equation:

\(9 + 3 \times 2 - 8 \div 4\)

1. Perform Division: \(8 \div 4 = 2\)
2. Perform Multiplication: \(3 \times 2 = 6\)
3. Substitute back into the equation: \(9 + 6 - 2\)
4. Perform Addition: \(9 + 6 = 15\)
5. Perform Subtraction: \(15 - 2 = 13\)

So, the original equation evaluates to \(13\), which is not equal to \(8\). We need to test the given options by interchanging the specified symbols and re-evaluating the equation.

Testing the Options

Option 1: Interchange \( \div \) and \( \times \)

The equation becomes:

\(9 + 3 \div 2 - 8 \times 4\)

Evaluate:

1. Perform Division: \(3 \div 2 = 1.5\)
2. Perform Multiplication: \(8 \times 4 = 32\)
3. Substitute back: \(9 + 1.5 - 32\)
4. Perform Addition: \(9 + 1.5 = 10.5\)
5. Perform Subtraction: \(10.5 - 32 = -21.5\)

The result is \( -21.5 \), which is not equal to \( 8 \).

Option 2: Interchange \( \div \) and \( + \)

The equation becomes:

\(9 \div 3 \times 2 - 8 + 4\)

Evaluate:

1. Perform Division: \(9 \div 3 = 3\)
2. Perform Multiplication: \(3 \times 2 = 6\)
3. Substitute back: \(6 - 8 + 4\)
4. Perform Subtraction: \(6 - 8 = -2\)
5. Perform Addition: \(-2 + 4 = 2\)

The result is \( 2 \), which is not equal to \( 8 \).

Option 3: Interchange \( \times \) and \( - \)

The equation becomes:

\(9 + 3 - 2 \times 8 \div 4\)

Evaluate:

1. Perform Division: \(8 \div 4 = 2\)
2. Perform Multiplication: \(2 \times 2 = 4\)
3. Substitute back: \(9 + 3 - 4\)
4. Perform Addition: \(9 + 3 = 12\)
5. Perform Subtraction: \(12 - 4 = 8\)

The result is \( 8 \), which is equal to the right side of the original equation. This interchange corrects the equation.

Option 4: Interchange \( + \) and \( \times \)

The equation becomes:

\(9 \times 3 + 2 - 8 \div 4\)

Evaluate:

1. Perform Division: \(8 \div 4 = 2\)
2. Perform Multiplication: \(9 \times 3 = 27\)
3. Substitute back: \(27 + 2 - 2\)
4. Perform Addition: \(27 + 2 = 29\)
5. Perform Subtraction: \(29 - 2 = 27\)

The result is \( 27 \), which is not equal to \( 8 \).

Based on the evaluation of each option, interchanging the mathematical symbols \( \times \) and \( - \) correctly solves the equation.

Revision Table: Symbol Interchange Results

Additional Information: Understanding Order of Operations (BODMAS/PEMDAS)

The order of operations is a set of rules that tells us the sequence in which we should perform operations in a mathematical expression to get the correct result. It ensures consistency in calculations.

• BODMAS: Brackets, Orders, 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.

Applying BODMAS/PEMDAS is crucial when solving mathematical equations involving multiple operators, as demonstrated in this symbol interchange problem.