Understanding the Equation Interchange Problem

The main goal is to identify which pair of numbers must be swapped within the given mathematical equation to make it true. The original equation presented is:

\( 108 \div 3 - 126 \div ( 5 \times 3 + 12) + 4 = 11 \)

We need to systematically check the provided options, which suggest pairs of numbers to interchange.

Evaluating the Original Equation's Validity

Before testing the swaps, let's calculate the value of the equation as it is written, strictly following the order of operations (PEMDAS/BODMAS).

1. Calculate inside the Parentheses: First, evaluate \( ( 5 \times 3 + 12) \).
   • Multiplication first: \( 5 \times 3 = 15 \).
   • Then, addition: \( 15 + 12 = 27 \).
   So, the expression inside the parentheses is \( 27 \).
2. Rewrite the equation: The equation now looks like \( 108 \div 3 - 126 \div 27 + 4 \).
3. Perform Divisions: Next, we do the divisions from left to right.
   • \( 108 \div 3 = 36 \).
   • \( 126 \div 27 \). This simplifies to \( \frac{14}{3} \) (by dividing both numbers by 9).
   The equation becomes \( 36 - \frac{14}{3} + 4 \).
4. Perform Addition/Subtraction: Finally, carry out the addition and subtraction from left to right.
   • \( 36 - \frac{14}{3} = \frac{108}{3} - \frac{14}{3} = \frac{94}{3} \).
   • \( \frac{94}{3} + 4 = \frac{94}{3} + \frac{12}{3} = \frac{106}{3} \).

The calculation shows the left side equals \( \frac{106}{3} \), which is approximately \( 35.33 \). Since this does not equal 11, the original equation is indeed incorrect.

Testing the Number Interchange Options

Let's see which pair swap corrects the equation. We focus on the numbers present within the equation structure itself.

Checking the Interchange of 12 and 3 (Option 2)

This option suggests swapping the numbers 12 and 3. Let's replace every '3' with '12' and every '12' with '3' in the original equation:

Original: \( 108 \div 3 - 126 \div ( 5 \times 3 + 12) + 4 = 11 \)

After Swap: \( 108 \div 12 - 126 \div ( 5 \times 12 + 3) + 4 = 11 \)

Now, let's evaluate this new equation:

1. Parentheses: Calculate \( ( 5 \times 12 + 3) \).
   • Multiplication: \( 5 \times 12 = 60 \).
   • Addition: \( 60 + 3 = 63 \).
   The expression is now \( 108 \div 12 - 126 \div 63 + 4 \).
2. Divisions: Perform the divisions.
   • \( 108 \div 12 = 9 \).
   • \( 126 \div 63 = 2 \).
   The equation becomes \( 9 - 2 + 4 \).
3. Addition/Subtraction: Perform from left to right.
   • \( 9 - 2 = 7 \).
   • \( 7 + 4 = 11 \).

The result is exactly 11. This confirms that swapping 12 and 3 makes the equation correct.

Checking Other Swaps Briefly

To be thorough, let's quickly verify another option, for instance, swapping 126 and 108 (Option 4).

Original: \( 108 \div 3 - 126 \div ( 5 \times 3 + 12) + 4 = 11 \)

Swap 108 and 126: \( 126 \div 3 - 108 \div ( 5 \times 3 + 12) + 4 = 11 \)

1. Parentheses: \( ( 5 \times 3 + 12) = 15 + 12 = 27 \).
2. Equation: \( 126 \div 3 - 108 \div 27 + 4 \).
3. Divisions: \( 126 \div 3 = 42 \) and \( 108 \div 27 = 4 \).
4. Calculation: \( 42 - 4 + 4 = 42 \).

Since \( 42 \neq 11 \), this swap is incorrect.

Final Conclusion

Based on the detailed step-by-step evaluation, interchanging the numbers 12 and 3 is the correct action required to satisfy the equation, making the final result equal to 11.