The following coded arithmetic equation is incorrect when the ordinary meanings of the operators are used: 13 - 14 ÷ 30 x 7 + 15 = 13. By interchanging exactly two of the operator signs among plus (+), minus (-), division (÷), and multiplication (x), the equation can be made correct. Which two operator signs must be interchanged so that the equation holds true?

Introduction / Context:
This is a sign interchange puzzle where an equation appears incorrect under normal arithmetic rules. The task is to identify which two operator signs have been misplaced so that after swapping them, the equation becomes numerically correct. These questions test careful substitution, understanding of operator precedence, and a systematic trial of possible interchanges instead of random guessing.

Given Data / Assumptions:

• Original equation: 13 - 14 ÷ 30 x 7 + 15 = 13. • Available operators: plus (+), minus (-), division (÷), multiplication (x). • We must interchange exactly two operator signs. • After interchange, we evaluate the equation with usual precedence rules.

Concept / Approach:
The safest approach is to keep the numbers fixed and try interchanging pairs of operators one pair at a time. For each pair, rewrite the equation, apply normal precedence (division and multiplication before addition and subtraction), and then check whether the result equals 13. Rather than doing this completely at random, it is useful to suspect that the division or multiplication sign may be in the wrong place, because these operators have a larger impact on the result.

Step-by-Step Solution:
Step 1: Evaluate the original equation to confirm it is incorrect. 13 - 14 ÷ 30 x 7 + 15 is not equal to 13. Step 2: Now consider swapping + and ÷ in the expression. The minus and multiplication signs remain as they are. Step 3: After swapping, the operators between the numbers become: 13 minus 14 plus 30 x 7 ÷ 15. Step 4: In symbolic form, the new expression is 13 - 14 + 30 x 7 ÷ 15. Step 5: Apply operator precedence. First handle multiplication and division: 30 x 7 = 210, then 210 ÷ 15 = 14. Step 6: Now evaluate the remaining addition and subtraction from left to right: 13 - 14 + 14. Step 7: Compute 13 - 14 = -1, then -1 + 14 = 13. Step 8: The right hand side is 13, which matches the given value, so this interchange makes the equation correct.

Verification / Alternative check:
You can verify that no other pair of sign interchanges yields 13. For example, swapping - and +, or ÷ and x, or x and - leads to values that are clearly not equal to 13. With + and ÷ swapped, everything fits neatly with standard arithmetic, so this pair is uniquely correct. Once one valid pair is found and checked, there is no need to keep testing further if the options insist on a single correct choice.

Why Other Options Are Wrong:
Option "- and +" changes the balance of positive and negative contributions but does not fix the large effect of the division and multiplication, so the result does not become 13. Option "÷ and x" only swaps the two high precedence operators and results in a substantially different but still incorrect value. Option "x and -" alters both a high and a low precedence operator, producing an incorrect final total. Option "÷ and -" also fails to yield 13 and therefore cannot be the required interchange.

Common Pitfalls:
A common mistake is to forget to apply operator precedence after interchanging signs and instead evaluate strictly from left to right, which can give misleading results. Another pitfall is to change more than two operators at a time mentally, which does not correspond to the requirement of a single interchange of two signs. Writing each trial equation clearly helps avoid confusion.

Final Answer:
The equation becomes correct when the plus and division signs are interchanged. Therefore, the required interchange is + and ÷.