An incorrect arithmetic equation is written as 5 + 3 × 8 - 12 ÷ 4 = 3. Which one of the following interchanges of signs (swapping every occurrence of the two chosen signs) will make this equation mathematically correct?

Introduction / Context:
This question is about operation symbol interchange in an arithmetic equation. The given equation 5 + 3 × 8 - 12 ÷ 4 = 3 is not correct under normal arithmetic rules. You are asked which pair of signs, when interchanged everywhere they appear in the equation, will make the equation true. This type of problem tests your understanding of how changing operators affects the value of an expression and your ability to systematically try each possibility without making precedence mistakes.

Given Data / Assumptions:

• Original equation (with usual meanings): 5 + 3 × 8 - 12 ÷ 4 = 3.
• Available sign pairs to interchange: (- and ÷), (+ and x), (+ and ÷), (+ and -), and (x and ÷).
• Interchange means swap meanings wherever the two chosen signs appear in the equation.
• Normal precedence of operations (multiplication and division before addition and subtraction) continues to apply after the interchange.

Concept / Approach:
The safe method is to consider each suggested interchange one by one, rewrite the equation with swapped signs, and then evaluate it carefully using correct precedence. The equation must hold exactly as written after the swap. Instead of guessing, you should organise the work by writing down the new equation explicitly. When you check each option, always recompute multiplication and division first, followed by addition and subtraction from left to right, to avoid accidental arithmetic mistakes.

Step-by-Step Solution:
Step 1: Start from 5 + 3 × 8 - 12 ÷ 4. Step 2: Try interchanging - and ÷ (option A). This means every minus becomes division and every division becomes minus. Step 3: After swapping - and ÷, the expression becomes 5 + 3 × 8 ÷ 12 - 4. Step 4: Evaluate using precedence. First compute multiplication and division. 3 × 8 = 24. 24 ÷ 12 = 2. Step 5: Substitute back: 5 + 2 - 4. Step 6: Now evaluate from left to right for addition and subtraction. 5 + 2 = 7. 7 - 4 = 3. Step 7: The resulting value is 3, which matches the right side, so the equation becomes correct with this interchange.

Verification / Alternative check:
To be thorough, you can quickly test another option to see that it fails. For example, if you swap + and ×, the equation becomes 5 × 3 + 8 - 12 ÷ 4, which evaluates to 15 + 8 - 3 = 20, not 3. Similarly, other proposed swaps produce values different from 3. Because only the swap of minus and division converts the left side exactly to 3, we can be confident that option A is the unique correct interchange.

Why Other Options Are Wrong:
Option B (+ and x) changes all additions into multiplications and vice versa, giving a much larger result. Option C (+ and ÷) mixes addition and division, but the resulting value does not equal 3. Option D (+ and -) only toggles between plus and minus, leaving the multiplication and division parts untouched, and again does not produce 3. Option E (x and ÷) swaps multiplication and division but keeps addition and subtraction the same, and the modified expression still does not evaluate to 3. Only the specific swap of minus and division leads to a correct equation.

Common Pitfalls:
A frequent mistake is to apply the interchange inconsistently, changing some occurrences of a sign but not all. Another is to ignore operator precedence and compute strictly from left to right, which can give incorrect values. To avoid these issues, always rewrite the entire expression after the swap and underline the multiplication and division operations to evaluate them first. This disciplined approach is crucial in time-pressured exams.

Final Answer:
The equation becomes correct when minus and division are interchanged, so the required pair is - and ÷.