Solving the Equation by Interchanging Signs

The question asks us to find which pair of mathematical signs, when swapped in the given equation, makes the equation correct. The original equation is:

\(35 + 7 \times 12 - 6 \div 2 = 6\)

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

• Division: \(6 \div 2 = 3\)
• Multiplication: \(7 \times 12 = 84\)
• Addition/Subtraction (from left to right): \(35 + 84 - 3 = 119 - 3 = 116\)

So, \(35 + 7 \times 12 - 6 \div 2 = 116\). The equation \(116 = 6\) is false. We need to interchange signs to make it true.

Testing the Options for Sign Interchange

We will test each option by swapping the indicated signs and re-evaluating the equation.

Option 1: Interchange - and ×

Swap the minus sign (-) and the multiplication sign (×) in the original equation. The new equation becomes:

\(35 + 7 - 12 \times 6 \div 2\)

Now, let's evaluate this new equation using BODMAS/PEMDAS:

• Multiplication: \(12 \times 6 = 72\)
• Division: \(72 \div 2 = 36\)
• Addition/Subtraction (from left to right): \(35 + 7 - 36 = 42 - 36 = 6\)

So, \(35 + 7 - 12 \times 6 \div 2 = 6\). The equation \(6 = 6\) is true.

This option makes the equation correct.

Option 2: Interchange - and ÷

Swap the minus sign (-) and the division sign (÷) in the original equation. The new equation becomes:

\(35 + 7 \times 12 \div 6 - 2\)

Evaluate using BODMAS/PEMDAS:

• Multiplication: \(7 \times 12 = 84\)
• Division: \(84 \div 6 = 14\)
• Addition/Subtraction (from left to right): \(35 + 14 - 2 = 49 - 2 = 47\)

The equation becomes \(47 = 6\), which is false.

Option 3: Interchange + and ×

Swap the addition sign (+) and the multiplication sign (×) in the original equation. The new equation becomes:

\(35 \times 7 + 12 - 6 \div 2\)

Evaluate using BODMAS/PEMDAS:

• Division: \(6 \div 2 = 3\)
• Multiplication: \(35 \times 7 = 245\)
• Addition/Subtraction (from left to right): \(245 + 12 - 3 = 257 - 3 = 254\)

The equation becomes \(254 = 6\), which is false.

Option 4: Interchange × and ÷

Swap the multiplication sign (×) and the division sign (÷) in the original equation. The new equation becomes:

\(35 + 7 \div 12 - 6 \times 2\)

Evaluate using BODMAS/PEMDAS:

• Division: \(7 \div 12 = \frac{7}{12}\)
• Multiplication: \(6 \times 2 = 12\)
• Addition/Subtraction (from left to right): \(35 + \frac{7}{12} - 12 = 23 + \frac{7}{12}\)

The equation becomes \(23 + \frac{7}{12} = 6\), which is false.

Based on the testing, interchanging the minus (-) and multiplication (×) signs makes the equation correct.

Step-by-Step Solution for the Correct Sign Interchange

The correct option is to interchange the signs - and ×.

Original Equation: \(35 + 7 \times 12 - 6 \div 2 = 6\)

Step 1: Identify the signs to be interchanged, which are - and ×.

Step 2: Rewrite the equation with the signs interchanged.

The original equation has × between 7 and 12, and - between 12 and 6.

After interchanging, - will be between 7 and 12, and × will be between 12 and 6.

New equation: \(35 + 7 - 12 \times 6 \div 2\)

Step 3: Apply the order of operations (BODMAS/PEMDAS) to evaluate the new equation.

• First, evaluate Multiplication and Division from left to right.
• Multiplication: \(12 \times 6 = 72\). The equation becomes \(35 + 7 - 72 \div 2\).
• Division: \(72 \div 2 = 36\). The equation becomes \(35 + 7 - 36\).
• Next, evaluate Addition and Subtraction from left to right.
• Addition: \(35 + 7 = 42\). The equation becomes \(42 - 36\).
• Subtraction: \(42 - 36 = 6\).

Step 4: Compare the result with the right side of the original equation.

The result is 6, which matches the right side of the original equation \(... = 6\).

Thus, interchanging the signs - and × makes the equation \(35 + 7 - 12 \times 6 \div 2 = 6\) correct.

Revision Table: Understanding Operations Order

Understanding the correct order of mathematical operations is crucial for solving these types of problems correctly. The widely used acronyms are BODMAS or PEMDAS.

Additional Information: Types of Mathematical Operations

Mathematical operations are fundamental actions performed on numbers. The basic arithmetic operations are:

• Addition (+): Combining two or more numbers to find their sum. Example: \(5 + 3 = 8\).
• Subtraction (-): Finding the difference between two numbers. Example: \(8 - 3 = 5\).
• Multiplication (× or *): Repeated addition or scaling numbers. Example: \(5 \times 3 = 15\).
• Division (÷ or /): Splitting a number into equal parts. Example: \(15 \div 3 = 5\).

Problems involving interchanging signs test your understanding of how these operations interact and the critical importance of following the correct order of operations to arrive at the accurate result.