Math Equation Correction by Swapping Numbers

This problem challenges us to find two specific numbers within a given mathematical equation. When these two numbers are interchanged (swapped), the equation should become correct, evaluating to the specified result.

Original Equation Calculation and Result

First, we must evaluate the provided equation in its current state to determine its outcome. The equation given is:

$21 \times 5 - 35 + (42 \div 7) + 28 = 96$

We apply the standard order of operations (PEMDAS/BODMAS) to solve this:

1. Division: We start with the expression inside the parentheses: $\qquad (42 \div 7)$.
   $\qquad 42 \div 7 = 6$
   The equation simplifies to: $21 \times 5 - 35 + 6 + 28 = 96$
2. Multiplication: Next, we perform the multiplication: $21 \times 5$.
   $\qquad 21 \times 5 = 105$
   The equation now looks like: $105 - 35 + 6 + 28 = 96$
3. Subtraction: Following the order, we perform subtraction: $105 - 35$.
   $\qquad 105 - 35 = 70$
   The equation simplifies further to: $70 + 6 + 28 = 96$
4. Addition: We proceed with addition from left to right: $70 + 6$.
   $\qquad 70 + 6 = 76$
   The equation becomes: $76 + 28 = 96$
5. Final Addition: The last step is adding the remaining numbers: $76 + 28$.
   $\qquad 76 + 28 = 104$

The original equation evaluates to $104$, which is not equal to the target value of $96$. This means we need to swap two numbers.

Testing Swaps to Make Equation Correct

We will now examine the given options to see which pair of numbers, when interchanged, rectifies the equation.

Analyzing Swap: 35 and 42

Let's consider the option suggesting we interchange the numbers $35$ and $42$. We replace every occurrence of $35$ with $42$ and every occurrence of $42$ with $35$ in the original equation:

The modified equation becomes:
$21 \times 5 - 42 + (35 \div 7) + 28 = 96$

Now, let's evaluate this new equation using the order of operations:

1. Division: Calculate $(35 \div 7)$.
   $\qquad 35 \div 7 = 5$
   The equation transforms to: $21 \times 5 - 42 + 5 + 28 = 96$
2. Multiplication: Perform the multiplication: $21 \times 5$.
   $\qquad 21 \times 5 = 105$
   The equation simplifies to: $105 - 42 + 5 + 28 = 96$
3. Subtraction: Perform the subtraction: $105 - 42$.
   $\qquad 105 - 42 = 63$
   The equation becomes: $63 + 5 + 28 = 96$
4. Addition: Perform the addition from left to right: $63 + 5$.
   $\qquad 63 + 5 = 68$
   The equation becomes: $68 + 28 = 96$
5. Final Addition: The last step is adding the remaining numbers: $68 + 28$.
   $\qquad 68 + 28 = 96$

With this swap, the equation correctly evaluates to $96$.

Conclusion

Interchanging the numbers $35$ and $42$ is the correct action needed to make the given mathematical equation true.