Analyzing Mathematical Operations with Sign Interchange

The question asks us to determine which of the given equations will not be correct after interchanging the signs '+' and '×'. We need to evaluate each equation after the sign change and compare the result with the expected value.

The proposed interchange is between the addition sign (\(+\)) and the multiplication sign (\(\times\)).

Evaluating Equation I after Sign Interchange

The original equation I is:

\(180 - 32 + 3 \times 148 \div 37 = 88\)

After interchanging '+' and '×', the equation becomes:

\(180 - 32 \times 3 + 148 \div 37\)

Now, let's evaluate this modified equation using the BODMAS (or PEMDAS) rule, which defines the order of operations (Brackets, Orders/Exponents, Division/Multiplication, Addition/Subtraction):

• Step 1: Evaluate Division (\(\div\))
• \(\quad 148 \div 37 = 4\)
• The equation becomes: \(180 - 32 \times 3 + 4\)
• Step 2: Evaluate Multiplication (\(\times\))
• \(\quad 32 \times 3 = 96\)
• The equation becomes: \(180 - 96 + 4\)
• Step 3: Evaluate Subtraction (-) and Addition (+) from left to right
• \(\quad 180 - 96 = 84\)
• The equation becomes: \(84 + 4\)
• \(\quad 84 + 4 = 88\)

The evaluated value of the modified equation I is 88. The expected value in the question is also 88. Therefore, equation I is correct after interchanging the signs.

Evaluating Equation II after Sign Interchange

The original equation II is:

\(204 \div 17 \times 23 + 5 - 97 = 50\)

After interchanging '+' and '×', the equation becomes:

\(204 \div 17 + 23 \times 5 - 97\)

Now, let's evaluate this modified equation using the BODMAS (or PEMDAS) rule:

• Step 1: Evaluate Division (\(\div\))
• \(\quad 204 \div 17 = 12\)
• The equation becomes: \(12 + 23 \times 5 - 97\)
• Step 2: Evaluate Multiplication (\(\times\))
• \(\quad 23 \times 5 = 115\)
• The equation becomes: \(12 + 115 - 97\)
• Step 3: Evaluate Addition (+) and Subtraction (-) from left to right
• \(\quad 12 + 115 = 127\)
• The equation becomes: \(127 - 97\)
• \(\quad 127 - 97 = 30\)

The evaluated value of the modified equation II is 30. The expected value in the question is 50. Therefore, equation II is NOT correct after interchanging the signs.

Conclusion on Correctness after Sign Interchange

Based on our calculations:

• Equation I is correct after interchanging '+' and '×'.
• Equation II is NOT correct after interchanging '+' and '×'.

The question asks which equation will NOT be correct. This is only Equation II.

Revision Table: Summary of Sign Interchange Results

Additional Information: Importance of Order of Operations

When evaluating mathematical expressions with multiple operations, the order in which you perform the operations is critical. Using a standard order ensures everyone gets the same result for the same expression. BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) or PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) are standard acronyms that help remember this order.

Division and Multiplication have the same priority and are performed from left to right as they appear in the expression. Similarly, Addition and Subtraction have the same priority and are also performed from left to right.

In problems involving sign interchanging, applying the correct order of operations after swapping the signs is essential to verify the correctness of the modified equation.