The equation 10 × 4 + 5 − 30 ÷ 6 = 31 is incorrect. Which two signs should be interchanged so that the equation becomes correct?

Introduction / Context:
In this problem, you are given an equation that is not correct under normal arithmetic rules. You must identify which two operator signs need to be interchanged in order to make the equation true. This tests understanding of operator precedence, the ability to manipulate symbolic expressions, and systematic trial of alternatives.

Given Data / Assumptions:

- Original equation: 10 × 4 + 5 − 30 ÷ 6 = 31.
- Signs that may be interchanged: ×, +, −, ÷.
- Each option specifies a pair of signs to swap throughout the equation.
- After swapping, we apply standard arithmetic precedence to verify the equality.

Concept / Approach:
First compute the value of the original left-hand side to confirm that the equation is indeed incorrect. Then, for each option, mentally or on paper, interchange the pair of operator symbols and evaluate the new left-hand side. The correct option is the one for which the left-hand side equals 31. While it is possible to reason cleverly, a careful trial of the few given options is usually quickest and safe.

Step-by-Step Solution:
Original left-hand side: 10 × 4 + 5 − 30 ÷ 6. Compute with normal precedence: 10 × 4 = 40 and 30 ÷ 6 = 5. So LHS = 40 + 5 − 5 = 40, not 31. The equation is incorrect. Now test option A: interchange "× and −". Replace every "×" with "−" and every "−" with "×". The equation becomes: 10 − 4 + 5 × 30 ÷ 6 = 31. Compute step by step: First multiplication and division: 5 × 30 = 150, then 150 ÷ 6 = 25. Now the expression is: 10 − 4 + 25. 10 − 4 = 6, and 6 + 25 = 31. Thus the equation 10 − 4 + 25 = 31 is correct. Therefore, interchanging "× and −" yields a true equation.

Verification / Alternative check:
We can quickly check other options to ensure uniqueness. Interchanging "÷ and −" or "+ and ÷" produces left-hand sides that clearly do not simplify to 31 when evaluated with correct precedence. Interchanging "− and +" keeps the total effectively unchanged and still yields 40. Hence, only option A results in a correct equation, confirming our answer.

Why Other Options Are Wrong:

- Option B: swapping "÷ and −" leads to a peculiar arrangement where the division is moved, and the resulting value does not approach 31.
- Option C: swapping "+ and ÷" distorts the balance between multiplication, division, and addition, again leading to a different total.
- Option D: swapping "− and +" still produces an incorrect left-hand side that evaluates to 40, not 31.

Common Pitfalls:
A common error is forgetting operator precedence after the interchange and evaluating from left to right. Another mistake is misapplying the swaps, changing only one symbol or missing an occurrence. Writing the transformed equation clearly and then calculate in stages helps avoid these errors.

Final Answer:
The equation becomes correct when the signs "× and −" are interchanged. Therefore, the correct choice is × and −.