Understanding the Operator and Number Interchanges

The question asks us to apply specific rules of interchange to mathematical expressions and then determine which of the given options remains correct after these changes. The rules for interchange are:

• The operator '+' is interchanged with the operator '-'.
• The number '8' is interchanged with the number '6'.

This means wherever a '+' appears, it becomes '-', and wherever a '-' appears, it becomes '+'. Similarly, wherever '8' appears, it becomes '6', and wherever '6' appears, it becomes '8'. Other numbers and operators remain unchanged.

Applying Interchanges to Each Option

Let's go through each given option and apply the specified interchanges to the expression on the left side of the equality sign. We will then evaluate the new expression and compare it to the original result on the right side.

Option 1: Evaluating 6 - 8 - 8 + 8 = 0

• Original expression: $6 - 8 - 8 + 8$
• Applying interchanges: '6' becomes '8', '-' becomes '+', '8' becomes '6', '+' becomes '-'.
• Modified expression: $8 + 6 + 6 - 6$
• Evaluating the modified expression: $8 + 6 + 6 - 6 = 14 + 6 - 6 = 20 - 6 = 14$
• Original result: $0$
• Comparison: $14 \neq 0$. This option is not correct after the interchanges.

Option 2: Evaluating 12 + 8 - 6 = 2

• Original expression: $12 + 8 - 6$
• Applying interchanges: '+' becomes '-', '8' becomes '6', '-' becomes '+', '6' becomes '8'. The number '12' remains unchanged.
• Modified expression: $12 - 6 + 8$
• Evaluating the modified expression: $12 - 6 + 8 = 6 + 8 = 14$
• Original result: $2$
• Comparison: $14 \neq 2$. This option is not correct after the interchanges.

Option 3: Evaluating 12 + 6 - 8 = 10

• Original expression: $12 + 6 - 8$
• Applying interchanges: '+' becomes '-', '6' becomes '8', '-' becomes '+', '8' becomes '6'. The number '12' remains unchanged.
• Modified expression: $12 - 8 + 6$
• Evaluating the modified expression: $12 - 8 + 6 = 4 + 6 = 10$
• Original result: $10$
• Comparison: $10 = 10$. This option is correct after the interchanges.

Option 4: Evaluating 6 - 4 + 8 - 8 = 6

• Original expression: $6 - 4 + 8 - 8$
• Applying interchanges: '6' becomes '8', '-' becomes '+', '4' remains unchanged, '+' becomes '-', '8' becomes '6', '-' becomes '+', '8' becomes '6'.
• Modified expression: $8 + 4 - 6 + 6$
• Evaluating the modified expression: $8 + 4 - 6 + 6 = 12 - 6 + 6 = 6 + 6 = 12$
• Original result: $6$
• Comparison: $12 \neq 6$. This option is not correct after the interchanges.

Summary of Evaluation

Let's summarize the results in a table for clarity:

Based on the evaluations, only Option 3 results in the original equality holding true after applying the given interchanges.

Conclusion on Interchanging Operators and Numbers

When operators and numbers are interchanged according to the specific rules provided in the question, we must carefully substitute each instance of the specified operators and numbers in the given expressions. After performing these substitutions, we evaluate the new mathematical expression. The option for which the evaluated result matches the original result stated in the option is the correct answer.

Revision Table: Key Concepts

Additional Information: Logical Reasoning with Symbols

Problems involving interchanging symbols or numbers are common in logical reasoning or mathematical puzzle sections of exams. They test your ability to follow instructions precisely and apply transformations consistently. Here are some related concepts:

• Symbol Substitution: Replacing one symbol or character with another based on a given rule.
• Code-Breaking: Similar to substitution, where symbols represent other values or operations, and you need to decode the message or expression.
• Pattern Recognition: Sometimes the interchanges might follow a pattern rather than a simple one-to-one swap.
• Mathematical Operations: A strong understanding of basic arithmetic operations (addition, subtraction, multiplication, division) and their order is necessary to evaluate the transformed expressions correctly.

In this specific problem, the interchange is straightforward: '+' and '-' swap, and '8' and '6' swap. The key is systematic application of these rules to each part of the equation and then correctly performing the arithmetic.