Understanding the Problem: Interchanging Signs and Numbers

The question asks us to analyze two mathematical equations after performing specific interchanges: swapping the sign of multiplication ($\times$) with addition ($+$), and swapping the number 7 with the number 4. We need to determine which of the given equations becomes incorrect after these interchanges.

Let's break down the required interchanges:

• Sign interchange: $\times \leftrightarrow +$
• Number interchange: $7 \leftrightarrow 4$

We will apply these rules to each equation one by one and evaluate the result using the order of operations (BODMAS/PEMDAS).

Analyzing Equation I after Interchange

The original Equation I is: $7 \times 8 + 4 – 6 \div 3 = 57$.

Applying the interchanges:

• Replace 7 with 4.
• Replace 4 with 7.
• Replace $\times$ with $+$.
• Replace $+$ with $\times$.

The new Equation I becomes:

$4 + 8 \times 7 – 6 \div 3$

Now, let's evaluate this expression using BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction):

First, perform Division and Multiplication from left to right:

$4 + (8 \times 7) – (6 \div 3)$

$4 + 56 – 2$

Next, perform Addition and Subtraction from left to right:

$(4 + 56) – 2$

$60 – 2$

$58$

The original equation claimed the result was 57. After the interchanges, the result is 58.

Since $58 \neq 57$, Equation I is not correct after the interchanging the given two signs and numbers.

Analyzing Equation II after Interchange

The original Equation II is: $4 \times 8 – 9 \div 3 + 7 = 3$.

Applying the interchanges:

• Replace 4 with 7.
• Replace 7 with 4.
• Replace $\times$ with $+$.
• Replace $+$ with $\times$.

The new Equation II becomes:

$7 + 8 – 9 \div 3 \times 4$

Now, let's evaluate this expression using BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). Division and Multiplication are done from left to right:

$7 + 8 – (9 \div 3) \times 4$

$7 + 8 – 3 \times 4$

$7 + 8 – (3 \times 4)$

$7 + 8 – 12$

Next, perform Addition and Subtraction from left to right:

$(7 + 8) – 12$

$15 – 12$

$3$

The original equation claimed the result was 3. After the interchanges, the result is 3.

Since $3 = 3$, Equation II is correct after the interchanging the given two signs and numbers.

Conclusion: Which Equation is Not Correct?

Based on our analysis:

• Equation I becomes $4 + 8 \times 7 – 6 \div 3 = 58$, which is not equal to the original claimed result of 57. So, Equation I is not correct.
• Equation II becomes $7 + 8 – 9 \div 3 \times 4 = 3$, which is equal to the original claimed result of 3. So, Equation II is correct.

The question asks which equation will be not correct. Only Equation I fits this description.

Therefore, only Equation I is not correct after applying the specified interchanges.

Revision Table: Interchanging Mathematical Operations

Problems involving interchanging signs and numbers require careful application of the new rules and strict adherence to the order of operations (BODMAS/PEMDAS).

• Identify the specific signs and numbers to be interchanged.
• Rewrite the expression or equation with the swapped elements.
• Evaluate the new expression step-by-step following the BODMAS/PEMDAS rule.
• Compare the calculated result with the given result to check if the equation holds true or is incorrect.

Additional Information: Order of Operations (BODMAS/PEMDAS)

The order of operations is a rule that tells us the sequence in which to perform operations in an expression. This ensures a unique and correct result.

BODMAS stands for:

• Brackets ()
• Orders (powers, roots)
• Division ($\div$) and Multiplication ($\times$) (from left to right)
• Addition ($+$) and Subtraction (–) (from left to right)

PEMDAS is another common acronym, standing for Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right).

In this problem, we used Division and Multiplication before Addition and Subtraction, following the order of operations correctly for both equations after the interchange.