Solving Equations by Interchanging Signs

This problem requires us to find which pair of arithmetic signs, when interchanged in the given equation, will make the equation mathematically correct. The given equation is:

\(12 - 6 \div 12 \times 6 + 6 = 9\)

Let's first evaluate the original equation using the order of operations (BODMAS/PEMDAS - Brackets, Orders, Division and Multiplication (from left to right), Addition and Subtraction (from left to right)):

\(12 - (6 \div 12) \times 6 + 6\)
\(12 - 0.5 \times 6 + 6\)
\(12 - 3 + 6\)
\(9 + 6\)
\(15\)

The original equation evaluates to 15, which is not equal to 9. We need to test each option by interchanging the specified signs and re-evaluating the equation.

Testing Options for Sign Interchange

We will check each provided option one by one.

Option 1: Interchange + and ÷

If we interchange '+' and '÷', the equation becomes:

\(12 - 6 + 12 \times 6 \div 6\)

Now, let's evaluate this new equation:

\(12 - 6 + 12 \times (6 \div 6)\)
\(12 - 6 + 12 \times 1\)
\(12 - 6 + 12\)
\(6 + 12\)
\(18\)

This result (18) is not equal to 9. So, Option 1 is incorrect.

Option 2: Interchange × and +

If we interchange '×' and '+', the equation becomes:

\(12 - 6 \div 12 + 6 \times 6\)

Now, let's evaluate this new equation:

\(12 - (6 \div 12) + (6 \times 6)\)
\(12 - 0.5 + 36\)
\(11.5 + 36\)
\(47.5\)

This result (47.5) is not equal to 9. So, Option 2 is incorrect.

Option 3: Interchange - and +

If we interchange '-' and '+', the equation becomes:

\(12 + 6 \div 12 \times 6 - 6\)

Now, let's evaluate this new equation:

\(12 + (6 \div 12) \times 6 - 6\)
\(12 + 0.5 \times 6 - 6\)
\(12 + 3 - 6\)
\(15 - 6\)
\(9\)

This result (9) is equal to the target value in the original equation. So, interchanging '-' and '+' makes the equation correct.

Option 4: Interchange ÷ and ×

If we interchange '÷' and '×', the equation becomes:

\(12 - 6 \times 12 \div 6 + 6\)

Now, let's evaluate this new equation:

\(12 - (6 \times 12 \div 6) + 6\)
\(12 - (72 \div 6) + 6\)
\(12 - 12 + 6\)
\(0 + 6\)
\(6\)

This result (6) is not equal to 9. So, Option 4 is incorrect.

Conclusion on Sign Interchange

Based on our evaluation of each option, interchanging the '-' and '+' signs in the original equation \(12 - 6 \div 12 \times 6 + 6 = 9\) makes the equation correct, as it results in 9.

Revision Table: Understanding Sign Interchange Problems

Solving problems involving sign interchange requires careful application of the order of operations. Here’s a quick summary:

• Identify the original equation and the target result.
• Understand the pairs of signs to be interchanged for each option.
• For each option, rewrite the equation with the signs swapped.
• Evaluate the new equation strictly following the order of operations (BODMAS/PEMDAS).
• Compare the result with the target result to determine if the interchange makes the equation correct.

Additional Information: Order of Operations (BODMAS/PEMDAS)

The order of operations is a rule used to clarify the sequence of operations in an expression. It ensures that everyone gets the same answer when evaluating an expression.

• BODMAS: Brackets, Orders (powers, roots), Division and Multiplication (left-to-right), Addition and Subtraction (left-to-right).
• PEMDAS: Parentheses, Exponents, Multiplication and Division (left-to-right), Addition and Subtraction (left-to-right).

In our calculations, we performed division and multiplication before addition and subtraction, working from left to right for operations at the same level.