Equation Balancing: Signs Interchange Problem

The main goal is to determine which pair of arithmetic operation signs, when swapped in the given equation, results in a mathematically accurate statement. The equation provided is: $$132 \div 6 - 24 \times 5 + 9 = 43$$

To solve this, we must first evaluate the original equation using the standard order of operations, commonly remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right) or BODMAS.

Original Equation Evaluation Steps

Let's apply the order of operations to the original equation:

• Division first (leftmost): $132 \div 6 = 22$
• Next, multiplication: $24 \times 5 = 120$
• Substitute these values back into the equation: $22 - 120 + 9$
• Perform subtraction: $22 - 120 = -98$
• Finally, perform addition: $-98 + 9 = -89$

The original equation evaluates to $-89$, not $43$. Therefore, the signs need to be interchanged.

Sign Interchange Testing and Verification

We will now examine each option to see which sign swap corrects the equation:

• Option 1: Interchange $ \div $ and $ \times $

  The equation becomes: $132 \times 6 - 24 \div 5 + 9 = 43$

  Evaluation: $792 - 4.8 + 9 = 796.2$. This is not equal to $43$.

• Option 2: Interchange $ + $ and $ \div $

  The equation becomes: $132 + 6 - 24 \times 5 \div 9 = 43$

  Evaluation: $132 + 6 - (24 \times 5) \div 9 = 138 - 120 \div 9 = 138 - 13.33... = 124.67...$. This is not equal to $43$.

• Option 3: Interchange $ - $ and $ + $

  The equation becomes: $132 \div 6 + 24 - 5 \times 9 = 43$

  Evaluation: $22 + 24 - 45 = 46 - 45 = 1$. This is not equal to $43$.

• Option 4: Interchange $ + $ and $ \times $

  The equation becomes: $132 \div 6 - 24 + 5 \times 9 = 43$

  Applying order of operations (PEMDAS/BODMAS):

  First, Division: $132 \div 6 = 22$

  Next, Multiplication: $5 \times 9 = 45$

  The equation transforms to: $22 - 24 + 45$

  Performing subtraction: $22 - 24 = -2$

  Performing addition: $-2 + 45 = 43$

  This result ($43$) matches the right side of the original equation ($43$). Thus, $43 = 43$. This option is correct.

Final Verification of Correct Equation

Interchanging the addition ($ + $) sign and the multiplication ($ \times $) sign successfully corrects the equation. The verified equation is:

The step-by-step calculation confirming this is:

1. Perform Division: $132 \div 6 = 22$
2. Perform Multiplication: $5 \times 9 = 45$
3. The equation is now: $22 - 24 + 45$
4. Perform Subtraction: $22 - 24 = -2$
5. Perform Addition: $-2 + 45 = 43$

Therefore, swapping the signs $ + $ and $ \times $ is the correct solution.