The equation 18 + 12 ÷ 9 - 30 x 16 = 10 is incorrect. If we interchange exactly two different arithmetic signs everywhere they occur, which pair of signs must be swapped to make the equation true?

Introduction / Context:
This is another equation correction question using sign interchange. We are given a numerical equation that does not balance, and we have to identify which two arithmetic operators, when swapped globally, will make the equation valid. This type of reasoning question tests careful evaluation and systematic trial of different possibilities rather than quick guessing.

Given Data / Assumptions:

• Original incorrect equation: 18 + 12 ÷ 9 - 30 x 16 = 10.
• We must interchange exactly two different signs throughout the equation.
• Candidate pairs are plus and multiplication, plus and division, minus and plus, and division and multiplication, with one extra distractor pair.
• Standard precedence of multiplication and division over addition and subtraction applies after the swap.

Concept / Approach:
The approach is to treat each option as a possible pair of signs to swap and then compute the result of the new left hand side. A swap is global, so every occurrence of the first sign becomes the second and vice versa. After swapping, we must evaluate the resulting expression carefully with the correct order of operations. Only one pair should lead to a left hand side equal to 10. This systematic checking keeps the reasoning transparent and avoids confusion from mental juggling of operations.

Step-by-Step Solution:
Step 1: Represent the equation in compact symbolic form as 18 + 12 / 9 - 30 * 16 = 10.Step 2: Consider swapping plus and multiplication. This means every plus becomes multiplication and every multiplication becomes plus.Step 3: Apply this swap to obtain 18 * 12 / 9 - 30 + 16.Step 4: Evaluate using precedence rules. First compute 18 * 12 = 216.Step 5: Then compute 216 / 9 = 24. The expression now is 24 - 30 + 16.Step 6: Perform subtraction and addition from left to right: 24 - 30 = -6, and then -6 + 16 = 10.Step 7: The left hand side equals the right hand side 10, so swapping plus and multiplication corrects the equation.

Verification / Alternative check:
To be certain, we can briefly check other options. Swapping plus and division produces expressions with non integer values that do not equal 10. Swapping minus and plus or swapping division and multiplication also yields totals far from 10 after evaluation. Therefore the only pair that works is plus and multiplication, matching option plus and x in the original coded form where x denotes multiplication. This confirms our solution.

Why Other Options Are Wrong:
Swapping plus and division would move division into the position of plus and vice versa, causing the expression to involve divisions where additions should be, resulting in a value that does not match 10. Swapping minus and plus effectively rearranges the low priority operations and leaves the 30 times 16 term intact, producing a large negative number. Swapping division and multiplication changes the structure of the high priority operations but still does not produce the required value. The extra distractor pair minus and x also fails when evaluated.

Common Pitfalls:
One common mistake is to interchange signs in only one place instead of across the entire equation. Another is to misapply order of operations after the swap, treating all operations with equal precedence. Some learners also forget that multiplication and division remain higher priority even after signs are swapped. Writing out the modified equation clearly and evaluating step by step helps avoid these issues.

Final Answer:
The equation becomes correct only when the plus and multiplication signs are interchanged, so the required pair of signs to swap is + and x.