Understanding the Problem: Sign Interchange

The question asks us to take several mathematical equations and apply a specific rule: interchange the minus sign ($\ndash$) and the multiplication sign ($\times$). After swapping these signs, we need to evaluate each resulting equation to see if the left side equals the right side. This means checking if the equation is correctly balanced.

To evaluate the expressions, we must follow the standard order of operations, often remembered by acronyms like BODMAS or PEMDAS.

Applying the Sign Swap Rule

The rule is straightforward: everywhere you see a $\ndash$, replace it with $\times$, and everywhere you see a $\times$, replace it with $\ndash$. The division ($\div$) and addition ($+$) signs remain exactly as they are.

Evaluating Each Equation After Sign Interchange

Let's go through each given option and apply the sign interchange, then evaluate the resulting expression step-by-step using the order of operations (Division and Multiplication from left to right, then Addition and Subtraction from left to right).

Option 1: Checking Equation Balance

Original equation: $42 \div 6 \ndash 18 + 30 \times 7 = 146$

After interchanging $\ndash$ and $\times$: $42 \div 6 \times 18 + 30 \ndash 7$

Let's calculate the value of the Left Hand Side (LHS):

• First, perform division: $42 \div 6 = 7$
• Next, perform multiplication: $7 \times 18 = 126$
• Then, perform addition: $126 + 30 = 156$
• Finally, perform subtraction: $156 \ndash 7 = 149$

So, the LHS is 149. The Right Hand Side (RHS) is 146.

Since $149 \neq 146$, the equation in Option 1 is not correctly balanced after the sign interchange.

Option 2: Checking Equation Balance

Original equation: $64 \div 8 \ndash 22 \times 3 + 30 = 103$

After interchanging $\ndash$ and $\times$: $64 \div 8 \times 22 + 3 \ndash 30$

Let's calculate the value of the LHS:

• First, division: $64 \div 8 = 8$
• Next, multiplication: $8 \times 22 = 176$
• Then, addition: $176 + 3 = 179$
• Finally, subtraction: $179 \ndash 30 = 149$

So, the LHS is 149. The RHS is 103.

Since $149 \neq 103$, the equation in Option 2 is not correctly balanced after the sign interchange.

Option 3: Checking Equation Balance

Original equation: $119 \div 17 \ndash 6 + 24 \times 34 = 32$

After interchanging $\ndash$ and $\times$: $119 \div 17 \times 6 + 24 \ndash 34$

Let's calculate the value of the LHS step-by-step:

• First, perform division: $119 \div 17 = 7$
• Next, perform multiplication: $7 \times 6 = 42$
• Then, perform addition: $42 + 24 = 66$
• Finally, perform subtraction: $66 \ndash 34 = 32$

So, the LHS is 32. The RHS is 32.

Since $32 = 32$, the equation in Option 3 is correctly balanced after interchanging the signs $\ndash$ and $\times$.

Option 4: Checking Equation Balance

Original equation: $36 + 14 \times 18 \div 3 \ndash 12 = 72$

After interchanging $\ndash$ and $\times$: $36 + 14 \ndash 18 \div 3 \times 12$

Let's calculate the value of the LHS:

• First, perform division: $18 \div 3 = 6$
• Next, perform multiplication: $6 \times 12 = 72$
• Then, perform addition: $36 + 14 = 50$
• Finally, perform subtraction: $50 \ndash 72 = -22$

So, the LHS is -22. The RHS is 72.

Since $-22 \neq 72$, the equation in Option 4 is not correctly balanced after the sign interchange.

Conclusion on Balancing Equations

Based on our evaluation, only the equation in Option 3 is correctly balanced when the signs $\ndash$ and $\times$ are interchanged.

Revision Table: Order of Mathematical Operations

Remember that for operations at the same priority level (like $\div$ and $\times$, or $+$ and $\ndash$), you work from left to right across the expression.

Additional Information: Logical Reasoning with Operators

Questions like this one are common in logical reasoning and quantitative aptitude sections of many exams. They test your ability to follow rules and apply basic mathematical operations correctly, paying close attention to the order of operations. Practicing such problems helps improve analytical skills and speed in calculations.

Key aspects to remember:

• Carefully note which signs are being interchanged.
• Apply the interchange consistently throughout the equation.
• Strictly follow the BODMAS/PEMDAS rule to evaluate the expression.
• Calculate the LHS and compare it with the RHS.

This type of problem might also involve interchanging numbers or a combination of signs and numbers. The core principle remains the same: apply the given rule precisely and evaluate accurately.