In the equation 7 x 9 + 5 = 68 given in an arithmetic reasoning puzzle, which of the following interchanges of signs and digits will make the equation mathematically correct?

Introduction / Context:
This question involves correcting a wrong arithmetic equation by interchanging signs and digits. Such problems test logical thinking about how arithmetic operations and numbers interact. You are given a false equation, 7 x 9 + 5 = 68, and several options specifying which pair of signs and which pair of digits should be interchanged to make the equation true. Correct interpretation of the option statements and careful evaluation of the resulting expressions are essential.

Given Data / Assumptions:
- Original equation: 7 x 9 + 5 = 68. - Standard arithmetic precedence applies, with multiplication performed before addition. - Each option specifies two signs and two digits to interchange. - Interchanging digits means swapping their positions wherever they appear in the equation. - We must find the option that leads to a correct numerical equality.

Concept / Approach:
The strategy is to interpret each option as a set of swaps, apply those swaps to the equation, and then check if the resulting equation becomes numerically correct. When swapping signs, multiplication and addition symbols change places. When swapping digits, the indicated digits exchange positions wherever they occur. By evaluating each transformed equation, we can identify which one satisfies the equality. Because the numbers are small, direct calculation is convenient and accurate.

Step-by-Step Solution:
Step 1: Consider option c: interchange x and +, and interchange digits 5 and 7. Step 2: First interchange x and + in the original equation 7 x 9 + 5 = 68. This gives 7 + 9 x 5 = 68. Step 3: Now interchange digits 5 and 7 in this new expression. Wherever 7 appears, write 5, and wherever 5 appears, write 7. The expression becomes 5 + 9 x 7 = 68. Step 4: Evaluate 5 + 9 x 7 using precedence (multiplication first). We have 9 x 7 = 63, then 5 + 63 = 68. Step 5: Therefore, after these interchanges, the equation 5 + 9 x 7 = 68 is true, so option c achieves a correct equality. Step 6: Briefly test other options: they do not yield a true equation when their prescribed swaps are applied, so they can be rejected.

Verification / Alternative check:
We can verify option c directly by rechecking operations. Starting with 5 + 9 x 7, applying multiplication first gives 63, and adding 5 results in 68, exactly matching the right hand side. For comparison, if we try option d, which interchanges x and +, and 7 and 9, we get 9 + 7 x 5 = 68, which evaluates to 9 + 35 = 44, not 68. Similarly, options a and b lead to equations that are clearly false. This confirms that option c is uniquely correct.

Why Other Options Are Wrong:
- Option a (+ and =, 5 and 6) produces an equation that results in unequal sides when evaluated. - Option b (+ and =, 7 and 9) also fails to balance the two sides of the equation. - Option d (x and +, 7 and 9) leads to 9 + 7 x 5 = 68, which is numerically 44, not 68. - Option e is a distractor and likewise does not produce a valid equality. None of these options transforms the original equation into a true statement.

Common Pitfalls:
Candidates sometimes ignore the order of operations and treat addition and multiplication as if they occur left to right without precedence. Others misinterpret what interchanging digits means and may replace a digit in only one place instead of consistently. Another frequent error is to assume that swapping only signs will be sufficient, while this problem clearly needs both sign and digit changes. Careful reading and methodical checking of each option prevent these errors.

Final Answer:
The interchange that makes the equation correct is interchanging x and +, and interchanging digits 5 and 7, which corresponds to option c.