Solving the Equation Correction Problem

The question asks us to correct a given mathematical equation by interchanging two arithmetic signs. The original equation is:

\( 4 \times 2 + 6 \div 2 - 12 = 2 \)

We need to find which pair of signs, when swapped, makes this equation true. We will test each option provided by performing the calculation using the standard order of operations (BODMAS/PEMDAS).

Understanding BODMAS/PEMDAS

The order of operations is crucial for evaluating mathematical expressions correctly. It stands for:

• Brackets (Parentheses)
• Orders (Exponents, Roots)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

Evaluating the Original Equation

Let's first evaluate the left side of the original equation \( 4 \times 2 + 6 \div 2 - 12 \) using BODMAS:

1. Division and Multiplication (from left to right):
   • \( 4 \times 2 = 8 \)
   • \( 6 \div 2 = 3 \)
2. Substitute back into the equation: \( 8 + 3 - 12 \)
3. Addition and Subtraction (from left to right):
   • \( 8 + 3 = 11 \)
   • \( 11 - 12 = -1 \)

So, the original equation evaluates to \( -1 = 2 \), which is false.

Testing the Options

Analyzing Option 1: Swapping ÷ and -

If we interchange the \(\div\) and \(-\) signs, the equation becomes:

\( 4 \times 2 + 6 - 2 \div 12 \)

Now, let's evaluate this using BODMAS:

1. Division and Multiplication (from left to right):
   • \( 4 \times 2 = 8 \)
   • \( 2 \div 12 = \frac{2}{12} = \frac{1}{6} \)
2. Substitute back into the equation: \( 8 + 6 - \frac{1}{6} \)
3. Addition and Subtraction (from left to right):
   • \( 8 + 6 = 14 \)
   • \( 14 - \frac{1}{6} = \frac{84}{6} - \frac{1}{6} = \frac{83}{6} \)

The result is \( \frac{83}{6} \), which is not equal to 2. So, Option 1 is incorrect.

Analyzing Option 2: Swapping + and -

If we interchange the \(+\) and \(-\) signs, the equation becomes:

\( 4 \times 2 - 6 \div 2 + 12 \)

Now, let's evaluate this using BODMAS:

1. Division and Multiplication (from left to right):
   • \( 4 \times 2 = 8 \)
   • \( 6 \div 2 = 3 \)
2. Substitute back into the equation: \( 8 - 3 + 12 \)
3. Addition and Subtraction (from left to right):
   • \( 8 - 3 = 5 \)
   • \( 5 + 12 = 17 \)

The result is 17, which is not equal to 2. So, Option 2 is incorrect.

Analyzing Option 3: Swapping × and -

If we interchange the \(\times\) and \(-\) signs, the equation becomes:

\( 4 - 2 + 6 \div 2 \times 12 \)

Now, let's evaluate this using BODMAS:

1. Division and Multiplication (from left to right):
   • \( 6 \div 2 = 3 \)
   • \( 3 \times 12 = 36 \)
2. Substitute back into the equation: \( 4 - 2 + 36 \)
3. Addition and Subtraction (from left to right):
   • \( 4 - 2 = 2 \)
   • \( 2 + 36 = 38 \)

The result is 38, which is not equal to 2. So, Option 3 is incorrect.

Analyzing Option 4: Swapping ÷ and ×

If we interchange the \(\div\) and \(\times\) signs, the equation becomes:

\( 4 \div 2 + 6 \times 2 - 12 \)

Now, let's evaluate this using BODMAS:

1. Division and Multiplication (from left to right):
   • \( 4 \div 2 = 2 \)
   • \( 6 \times 2 = 12 \)
2. Substitute back into the equation: \( 2 + 12 - 12 \)
3. Addition and Subtraction (from left to right):
   • \( 2 + 12 = 14 \)
   • \( 14 - 12 = 2 \)

The result is 2, which is equal to the right side of the original equation. So, Option 4 is correct.

Interchanging the \(\div\) and \(\times\) signs makes the equation \( 4 \div 2 + 6 \times 2 - 12 = 2 \) true.

Summary of Calculations

Revision Table: Key Concepts

Additional Information: Mathematical Reasoning Problems

Problems involving correcting equations by swapping signs or numbers are common in logical reasoning and quantitative aptitude sections of various tests. These problems assess your ability to understand and apply mathematical rules, such as the order of operations, and your systematic approach to problem-solving.

• Always start by evaluating the original expression to understand where it goes wrong.
• Systematically test each option provided. Do not guess.
• Carefully apply the order of operations at each step of the calculation after swapping signs.
• Pay close attention to positive and negative signs.

Practicing these types of problems helps improve your speed and accuracy in applying basic arithmetic rules and problem-solving strategies.