Which of the following interchanges of signs and numbers will make the equation 8 x 6 + 4 = 52 correct?

Introduction / Context:
This question involves an incorrect arithmetic equation and asks you to correct it by interchanging certain operation signs and numbers. The original equation is 8 x 6 + 4 = 52. Each option specifies a pair of signs and a pair of numbers to exchange positions. After performing the specified swaps, the equation should become true under standard operator precedence. This problem tests careful manipulation and evaluation of algebraic expressions.

Given Data / Assumptions:

• The incorrect starting equation is 8 x 6 + 4 = 52.
• We may interchange one pair of operation signs and one pair of numbers as specified in the chosen option.
• The set of operations involved is multiplication and addition.
• We evaluate expressions using normal precedence, where multiplication is done before addition.
• Our goal is to find the option that produces a valid equality.

Concept / Approach:
For each option, we apply the indicated interchanges step by step. First, swap the signs in all positions where they occur, then swap the indicated numbers. After performing both swaps, we evaluate the new left hand side and compare it with the right hand side. Only those changes that make the equality true are acceptable. Because the expression is short, we can test all options reliably.

Step-by-Step Solution:
Step 1: Consider option c, which suggests interchanging the signs x and +, and the numbers 8 and 4.Step 2: Start from 8 x 6 + 4 = 52. First swap x and +. This gives 8 + 6 x 4 = 52.Step 3: Next, interchange the numbers 8 and 4. In the expression 8 + 6 x 4 = 52, the 8 on the left becomes 4, and the 4 in the product becomes 8, giving 4 + 6 x 8 = 52.Step 4: Evaluate the left hand side using BODMAS. Compute 6 x 8 = 48 first, then add 4 to 48 to get 52.Step 5: Thus the new equation 4 + 6 x 8 = 52 is true, since both sides equal 52.Step 6: For comparison, check option a briefly. Swapping x and + and then 8 and 52 leads to 52 + 6 x 4 = 8, which evaluates to 52 + 24 = 76 on the left, not equal to 8 on the right, so it is incorrect.Step 7: Options b and d similarly fail when tested, leaving option c as the only correct choice.

Verification / Alternative check:
We can verify the final corrected equation independently. Starting from 4 + 6 x 8, perform the multiplication first: 6 x 8 = 48. Then add 4 to get 52. This exactly matches the right hand side of 52, confirming that the equation is valid after the described interchanges. No other sequence of swaps from the options yields such a clean equality.

Why Other Options Are Wrong:
Option a gives 52 + 6 x 4 = 8, which simplifies to 76 = 8, clearly false. Option b, after swapping + and x and then 6 and 4, results in 8 + 4 x 6 = 52, which simplifies to 8 + 24 = 32, not 52. Option d gives 6 + 8 x 4 = 52, which simplifies to 6 + 32 = 38, also incorrect. Only option c correctly transforms the equation into a true statement.

Common Pitfalls:
One frequent mistake is to perform the swaps in the wrong order or to forget that both the sign pair and number pair must be swapped in all their occurrences. Another pitfall is evaluating the resulting expression from left to right without respecting multiplication precedence. Carefully applying each change and then using BODMAS avoids these issues.

Final Answer:
The equation becomes correct when we interchange x and +, and 8 and 4, corresponding to option c.