Analyzing the Equation for Correctness

The problem asks us to identify two numbers and two signs in the given mathematical expression that need to be swapped to make the equation valid. The original equation presented is:

$$840 \div 38 - 7 + 10 \times 12 = 462$$

To find the correct combination, we need to test each option by applying the suggested interchanges to the original equation and then evaluating the result using the standard order of operations (BODMAS/PEMDAS: Brackets, Orders, Division/Multiplication from left to right, Addition/Subtraction from left to right).

Testing Interchange Option 2

Option 2 proposes interchanging the numbers 12 and 38, along with the signs × and -.

Let's first identify these elements in the original equation:

840 ÷ 38 - 7 + 10 × 12

Now, we apply the swaps indicated by Option 2:

• The number 38 is replaced by 12.
• The number 12 is replaced by 38.
• The sign - is replaced by ×.
• The sign × is replaced by -.

After performing these interchanges, the equation becomes:

$$840 \div 12 \times 7 + 10 - 38$$

Step-by-Step Calculation of the Modified Equation

We will now evaluate this new equation step-by-step, adhering strictly to the BODMAS order:

1. Division: Perform the division first ($840 \div 12$). $$840 \div 12 = 70$$ The equation simplifies to: $70 \times 7 + 10 - 38$.
2. Multiplication: Next, perform the multiplication ($70 \times 7$). $$70 \times 7 = 490$$ The equation now is: $490 + 10 - 38$.
3. Addition: Carry out the addition ($490 + 10$). $$490 + 10 = 500$$ The equation becomes: $500 - 38$.
4. Subtraction: Finally, perform the subtraction ($500 - 38$). $$500 - 38 = 462$$

The result of evaluating the modified equation is 462.

Conclusion: Verifying Option 2

The calculated value of 462 perfectly matches the value given on the right side of the original equation. This confirms that swapping the numbers 12 and 38 and the signs × and - is the correct operation needed to make the equation true.

Therefore, Option 2 is the correct answer.