Consider the following incorrect equation involving the standard arithmetic symbols:\n\n15 x 12 + 40 ÷ 40 - 6 = 21\n\nBy interchanging exactly two of the operation signs, the equation can be made correct.\nWhich pair of signs should be interchanged so that the equality holds true?

Introduction / Context:
This question checks your ability to manipulate arithmetic equations by interchanging operation signs. The original equation is numerically false, but if we swap the correct pair of signs everywhere they occur, we can restore the equality. This type of puzzle evaluates logical reasoning with basic arithmetic skills.

Given Data / Assumptions:

• Original equation: 15 x 12 + 40 ÷ 40 - 6 = 21.
• We may interchange only the two signs mentioned in an option.
• After the swap, all other signs keep their original meanings.
• We use normal operator precedence for multiplication, division, addition and subtraction.

Concept / Approach:
The systematic method is to test each option one by one. For each suggested pair of signs, we swap all their occurrences in the expression, rewrite the equation with ordinary symbols, and then simplify the left-hand side. If the result equals 21, that option is the required answer.

Step-by-Step Solution:
Step 1: Try option (a): interchange + and x. Step 2: Every x becomes + and every + becomes x in the left-hand side. Step 3: The new equation becomes 15 + 12 x 40 ÷ 40 - 6 = 21. Step 4: Evaluate the left-hand side with standard precedence. Step 5: Compute 12 x 40 = 480. Step 6: Then 480 ÷ 40 = 12. Step 7: Now compute 15 + 12 - 6 = 27 - 6 = 21. Step 8: The left-hand side equals 21, matching the right-hand side. So option (a) works.

Verification / Alternative check:
We can quickly inspect another option to ensure it fails. If we swapped + and ÷ instead, the equation would become 15 x 12 ÷ 40 + 40 - 6, which simplifies to a number different from 21. This confirms that the successful repair is unique among the given choices.

Why Other Options Are Wrong:
Swapping + and ÷, or - and +, or ÷ and x leads to left-hand sides that are clearly not equal to 21 after correct evaluation. Either the result is a fraction or a much larger integer, so these options do not fix the equation and must be discarded.

Common Pitfalls:
A typical mistake is to interchange signs only at one place instead of every occurrence, or to recompute without respecting operator precedence. Another pitfall is performing mental arithmetic too quickly and miscalculating intermediate products or quotients. Writing each intermediate step clearly helps avoid such errors.

Final Answer:
The equation becomes correct only when we interchange the plus and multiplication signs, so the correct pair is + and x, corresponding to option (a).