The following equation is incorrect: 1 ÷ 14 x 30 + 20 - 10 = 12. By interchanging exactly two operation signs among "+", "-", "x" and "÷", which pair of signs must be swapped to make the equation correct?

Introduction / Context:
This problem involves correcting an arithmetic equation by interchanging two operation signs. The left hand side expression 1 ÷ 14 x 30 + 20 - 10 does not equal 12 when evaluated with standard operator meanings and precedence. Your task is to identify which pair of signs, when swapped throughout the expression, will make the equation valid.

Given Data / Assumptions:
- Original incorrect equation: 1 ÷ 14 x 30 + 20 - 10 = 12.
- We may interchange exactly two operation signs from the set {"+", "-", "x", "÷"}.
- All other signs and numbers remain unchanged.
- Standard operator precedence holds: division and multiplication before addition and subtraction.

Concept / Approach:
We systematically test the given options, each representing a specific pair of signs to interchange wherever they occur in the expression. For each candidate pair, we swap the signs, recompute the left hand side using correct precedence, and check whether the result equals 12. The pair that produces a true equality is the required answer.

Step-by-Step Solution:
Step 1: Focus on option "+ and ÷". If we interchange "+" and "÷", then every "+" becomes "÷" and every "÷" becomes "+". Step 2: Original expression is 1 ÷ 14 x 30 + 20 - 10. Step 3: After swapping "+" and "÷", the expression becomes 1 + 14 x 30 ÷ 20 - 10. Step 4: Evaluate using precedence. First compute 14 x 30 = 420. Step 5: Next compute 420 ÷ 20 = 21. Step 6: Now compute the full expression: 1 + 21 - 10 = 22 - 10 = 12, which matches the right hand side.

Verification / Alternative check:
We can briefly check that the other options do not work. Swapping "-" and "+" changes the balance of positive and negative terms but leaves the division and multiplication untouched, leading to a value different from 12. Similarly, swapping "÷ and x" or "x and -" produces expressions whose evaluated values are not 12 under standard precedence. Only interchanging "+" and "÷" yields the correct result of 12.

Why Other Options Are Wrong:
- Swapping "- and +" changes the addition and subtraction, but the resulting expression does not simplify to 12.
- Swapping "÷ and x" distorts the relative size of the terms involving 14 and 30, making it impossible to reach 12 with the remaining additions and subtractions.
- Swapping "x and -" similarly disrupts the expression and does not lead to the target value.

Common Pitfalls:
A typical mistake is to ignore operator precedence and evaluate expressions strictly from left to right while testing sign changes, which may suggest incorrect pairs work. Another error is to change more than two signs or to alter numbers, which is not allowed. The correct strategy is to follow the rules strictly and recompute carefully for each candidate pair.

Final Answer:
The equation becomes correct if we interchange the "+" and "÷" signs. Therefore, the required pair of signs is + and ÷.