The following equation is not correct when the usual arithmetic symbols are used:\n\n6 ÷ 17 x 51 + 6 – 12 = – 4\n\nBy interchanging exactly two of the operation signs, the equation can be made correct.\nWhich one of the following pairs of signs must be interchanged to make the equation true?

Introduction / Context:
This problem belongs to the mathematical operations segment of verbal reasoning, where you must repair an incorrect equation by interchanging two operation signs. The key skill is to systematically test possible sign swaps and check which one makes the numerical equality valid.

Given Data / Assumptions:

• Original equation: 6 ÷ 17 x 51 + 6 – 12 = -4.
• We are allowed to interchange only two operation signs from among ÷, x, + and –.
• After the swap, we evaluate the left-hand side using normal operator precedence.

Concept / Approach:
The basic idea is to pick each suggested pair of signs, swap all their occurrences in the equation, and then evaluate the new left-hand side. If the resulting numerical value is -4, that choice is correct. Because there are only four options, a structured trial approach is reasonable and efficient.

Step-by-Step Solution:
Step 1: Consider option (a), swapping x and ÷ everywhere. Step 2: The equation becomes 6 x 17 ÷ 51 + 6 - 12. Step 3: Replace x with multiplication and ÷ with division: 6 × 17 ÷ 51 + 6 - 12. Step 4: Evaluate: 6 × 17 = 102. Step 5: Then 102 ÷ 51 = 2. Step 6: Now compute 2 + 6 - 12 = 8 - 12 = -4. Step 7: The left-hand side equals -4, matching the right-hand side, so option (a) works.

Verification / Alternative check:
We can briefly check another option to see that it fails. For instance, if we swap + and ÷, the expression becomes 6 + 17 x 51 ÷ 6 - 12, which evaluates to a large positive number, not -4. This confirms that not all swaps work and that the successful one is special.

Why Other Options Are Wrong:
Swapping + and ÷, or + and –, or – and ÷ leads to left-hand sides that are far from -4 when calculated with correct precedence. They either yield positive numbers much larger than -4 or other unrelated values, so those options cannot repair the equation.

Common Pitfalls:
Many learners try to mentally evaluate several swaps at once and lose track of the replaced symbols. Another pitfall is ignoring operator precedence and performing operations strictly from left to right, which produces incorrect results. Writing the modified expression clearly and simplifying carefully avoids these mistakes.

Final Answer:
The equation becomes correct only when we interchange the division and multiplication signs, that is, swap x and ÷, so the correct choice is x and ÷ (option (a)).