The following equation with coded operators is incorrect: 18 x 12 - 3 ÷ 10 + 2 = 64. If exactly two mathematical signs are interchanged to make the equation correct, which two signs must be swapped?

Introduction / Context:
This question checks understanding of arithmetic equations with incorrect operator placements. We are given an equation that does not hold true and asked to correct it by interchanging exactly two types of operators everywhere they appear. Such problems test attention to detail, operator precedence, and the ability to reason systematically through multiple possibilities, rather than relying on guesswork.

Given Data / Assumptions:

• The intended full equation is 18 x 12 - 3 ÷ 10 + 2 = 64, with the usual multiplication and division symbols.
• Exactly two different signs among plus, minus, multiplication, and division must be interchanged.
• Interchanging means that wherever one of these two signs appears, it is replaced with the other, and this swap is global across the equation.
• The final corrected equation must be numerically true when evaluated with standard operator precedence.

Concept / Approach:
We should treat each option as a candidate pair of operators to swap and then check whether the resulting equation evaluates to 64. Because the equation contains multiplication, subtraction, division, and addition, changing any pair may significantly alter the result. The systematic way is to translate the expression into a compact form, test each swap option step by step, and compute the value using normal precedence: multiplication and division first, then addition and subtraction from left to right.

Step-by-Step Solution:
Step 1: Rewrite the original equation in symbolic form as 18 * 12 - 3 / 10 + 2.Step 2: Consider the option of interchanging the division and minus signs. This means every division sign becomes minus and every minus sign becomes division.Step 3: Apply this swap: 18 * 12 / 3 - 10 + 2 becomes 18 * 12 / 3 - 10 + 2 after replacement, which simplifies to 18 * 12 / 3 - 10 + 2 using standard symbols.Step 4: Evaluate with precedence. First compute 18 * 12 = 216. Then 216 / 3 = 72. The equation now reads 72 - 10 + 2.Step 5: Perform the remaining operations from left to right: 72 - 10 = 62 and 62 + 2 = 64.Step 6: The final left hand side equals 64, which matches the right hand side, so swapping division and minus corrects the equation.

Verification / Alternative check:
To ensure this is the only correct swap, quickly test the other pairs. Swapping plus with multiplication or plus with division gives values far from 64 when evaluated. Similarly, swapping minus with plus or multiplication with division does not balance the equation. Only the interchange of division and minus yields exactly 64. This confirms that the pair of signs that must be interchanged is division and minus.

Why Other Options Are Wrong:
If plus and multiplication are swapped, the expression becomes 18 + 12 * 3 / 10 - 2, which evaluates to a non integer or at least not 64 when calculated correctly. Swapping plus and division or minus and plus also leads to values that are clearly not 64, either very large or significantly smaller. The distractor involving multiplication and division simply rearranges the high priority operations without bringing the value anywhere near 64. Therefore none of those swaps restore equality.

Common Pitfalls:
Many learners try to mentally juggle sign changes without carefully writing out each transformed expression, which can lead to confusion. Others inadvertently change only a single occurrence of a sign instead of applying the swap consistently to every instance. Another frequent issue is ignoring multiplication and division precedence and evaluating strictly left to right, leading to incorrect conclusions about which swap works. Writing down each candidate equation and calculating step by step avoids these errors.

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