The following equation is incorrect: 40 ÷ 8 x 5 + 15 - 2 = 30. If exactly two arithmetic signs in this expression are interchanged, which pair of signs should be swapped so that the equation becomes correct?

Introduction / Context:
This question tests algebraic reasoning with operator signs. We are given a numerical expression that currently evaluates to a value different from 30. By interchanging exactly two of the operation symbols, we must transform the equation so that it becomes true. This requires systematic testing of the given options while respecting usual arithmetic precedence rules.

Given Data / Assumptions:

• Original equation: 40 ÷ 8 x 5 + 15 - 2 = 30.
• The operators present are division, multiplication, addition and subtraction.
• Exactly two signs are to be interchanged across the entire expression.
• Normal precedence applies: division and multiplication first, followed by addition and subtraction.

Concept / Approach:
First, compute the value of the left side using the original signs to confirm that it is not equal to 30. Then, for each option, interchange the specified two signs everywhere they occur in the expression, rewrite the modified expression and evaluate it. The correct option will be the one for which the modified left side equals 30 while the others do not. It is important to keep the order of numbers the same and change only the operators.

Step-by-Step Solution:
Step 1: Evaluate the original expression 40 ÷ 8 x 5 + 15 - 2. Step 2: Compute 40 ÷ 8 = 5, then 5 x 5 = 25. Now we have 25 + 15 - 2 = 38, so the equation 40 ÷ 8 x 5 + 15 - 2 = 30 is false. Step 3: Option A suggests interchanging ÷ and -. This would yield 40 - 8 x 5 + 15 ÷ 2. Evaluate: 8 x 5 = 40 and 15 ÷ 2 = 7.5, so 40 - 40 + 7.5 = 7.5, not 30. Step 4: Option B suggests interchanging + and ÷. The expression would become 40 + 8 x 5 ÷ 15 - 2. Evaluate: 8 x 5 = 40, 40 ÷ 15 is not an integer and the result is not 30, so this option fails. Step 5: Option C suggests interchanging - and +. The expression becomes 40 ÷ 8 x 5 - 15 + 2. As before, 40 ÷ 8 x 5 = 25, so 25 - 15 + 2 = 12, not 30. Step 6: Option D suggests interchanging x and -. This means every x becomes -, and every - becomes x. The expression becomes 40 ÷ 8 - 5 + 15 x 2. Step 7: Evaluate 40 ÷ 8 = 5 and 15 x 2 = 30. Now the expression is 5 - 5 + 30. Step 8: 5 - 5 = 0 and 0 + 30 = 30, so with this interchange the equation becomes true.

Verification / Alternative check:
We can verify that no other pair of sign interchanges yields 30 by checking the remaining options quickly. None of them produce the correct value when we recompute the expression with proper precedence. Only interchanging multiplication and subtraction turns the expression into 40 ÷ 8 - 5 + 15 x 2, which computes exactly to 30.

Why Other Options Are Wrong:
Swapping ÷ and - produces a result of 7.5, swapping + and ÷ gives a noninteger result far from 30 and swapping - and + yields 12. All these values contradict the required equality to 30. Therefore those options cannot correct the equation.

Common Pitfalls:
A frequent mistake is to ignore operator precedence when evaluating the modified expressions, which can lead to incorrect conclusions about which swap works. Another error is to change only one occurrence of a symbol instead of all occurrences, but the question requires all instances of the two symbols to be interchanged. Working carefully through each option step by step helps avoid these issues.

Final Answer:
The equation becomes correct when the multiplication and subtraction signs are interchanged. Therefore, the required pair of signs is x and -.