The following equation is incorrect. By interchanging two of the arithmetic signs, make the equation correct: 1 ÷ 14 x 30 + 20 - 10 = 12. Which two signs must be interchanged?

Introduction / Context:
In this problem, an equation involving several arithmetic operations is not correct as written. We are asked to choose a pair of signs and interchange them (swap their positions) so that the equation balances. This tests understanding of the order of operations and the effect of each arithmetic sign on the value of the expression.

Given Data / Assumptions:

• The original equation is 1 ÷ 14 x 30 + 20 - 10 = 12.
• We can only interchange two operation signs from among "÷", "x", "+", "-".
• After the interchange, the equation must hold true using normal arithmetic rules.
• No numbers are changed, only the positions of two signs are swapped.

Concept / Approach:
We evaluate the original left hand side and then test each proposed interchange. For each option, we swap the indicated symbols in the entire expression, then compute the new left hand side value using standard precedence (multiplication and division first, then addition and subtraction). The correct choice is the one that makes the left hand side equal to 12.

Step-by-Step Solution:
Step 1: Evaluate the original left side: 1 ÷ 14 x 30 + 20 - 10. Compute 1 ÷ 14 x 30 = (1 / 14) * 30 = 30 / 14 ≈ 2.1429, so the left side is about 2.1429 + 20 - 10 ≈ 12.1429, which is not 12. Step 2: Try swapping "÷" and "+". Then the expression becomes 1 + 14 x 30 ÷ 20 - 10. Step 3: Evaluate 14 x 30 ÷ 20. First 14 x 30 = 420, then 420 ÷ 20 = 21. Step 4: Now the left side is 1 + 21 - 10 = 22 - 10 = 12. Step 5: Since this equals the right side, the interchange "plus and divide" makes the equation correct.

Verification / Alternative check:
We can quickly confirm that the other suggested interchanges do not work. Swapping minus and plus, or divide and multiply, or multiply and minus gives different expressions whose left hand sides evaluate to values other than 12. Only the swap between "+" and "÷" produces a correct equality.

Why Other Options Are Wrong:
"Minus and plus" leads to a left side that is still not equal to 12, because the relative sizes of the terms are not adjusted correctly. "Divide and multiply" keeps the order of magnitude of the left side far away from 12. "Multiply and minus" again produces a left side that does not equal 12. None of them satisfy the equation after the interchange.

Common Pitfalls:
Students may forget to apply multiplication and division before addition and subtraction when testing each option, leading to incorrect evaluations. Another error is to assume that numbers can also be rearranged, which is not allowed. Only the positions of the two chosen signs may be swapped; all numbers and their sequence remain fixed.

Final Answer:
Therefore, the equation becomes correct when we interchange the signs for plus and divide, so the correct choice is plus and divide.