In the expression 5 + 3 × 8 − 12 ÷ 4 = 3, two of the arithmetic signs have been interchanged by mistake. Which pair of signs should be swapped with each other so that, using the usual order of operations, the equation becomes numerically correct?

Introduction / Context:
This problem checks your understanding of the order of operations and your ability to repair an incorrect numerical equation by interchanging two operation signs. The goal is to identify which pair of symbols, when swapped, turns the expression into a true arithmetic statement while still following the standard BODMAS or PEMDAS rules for evaluating expressions.

Given Data / Assumptions:

• Original expression: 5 + 3 × 8 − 12 ÷ 4 = 3.
• We may interchange exactly one pair of operation signs from the set {+, ×, −, ÷}.
• Standard precedence applies: multiplication and division are performed before addition and subtraction.
• All numbers are real and calculations are exact.

Concept / Approach:
The direct approach is to systematically test each suggested pair of signs. For each option, swap the two indicated operators everywhere they appear in the expression, then evaluate the new expression using the correct order of operations. The correct option will be the one that yields a left-hand side equal to 3, which matches the right-hand side of the equation.

Step-by-Step Solution:
First evaluate the original left-hand side: 5 + 3 × 8 − 12 ÷ 4 = 5 + 24 − 3 = 26, which is not equal to 3.Try option A: swap + and −. The expression becomes 5 − 3 × 8 + 12 ÷ 4 = 5 − 24 + 3 = −16, not 3.Try option B: swap − and ÷. The expression becomes 5 + 3 × 8 ÷ 12 − 4. Evaluate: 3 × 8 ÷ 12 = 24 ÷ 12 = 2, so we get 5 + 2 − 4 = 3, which matches the right-hand side.Since we have already found a working pair, we can still briefly check that the other suggested interchanges do not work.

Verification / Alternative check:
For option C (swap + and ×), we get 5 × 3 + 8 − 12 ÷ 4 = 15 + 8 − 3 = 20, not 3.For option D (swap + and ÷), we get 5 ÷ 3 × 8 − 12 + 4, which simplifies to a noninteger value and clearly does not equal 3.Therefore, only swapping the minus and division signs gives exactly 3 on the left-hand side.

Why Other Options Are Wrong:
Swapping + and − makes the subtraction too large in magnitude, giving a negative result rather than 3.Swapping + and × increases the multiplication effect and pushes the result well above 3.Swapping + and ÷ introduces division where addition was, producing a fractional value that does not match the integer 3.

Common Pitfalls:
A frequent mistake is to ignore operator precedence and simply evaluate from left to right, which leads to incorrect checks for each option.Another mistake is to change more than the specified two signs mentally, which effectively tests a different equation than the one described.

Final Answer:
The equation becomes correct only when we interchange the minus and division signs, so the correct choice is - and /.