Finding the Correct Sign Interchange

The question asks us to find which two mathematical signs in the given equation need to be swapped to make the equation result in 50. The original equation is: $27 + 13 \times 12 - 6 \div 3 = 50$.

First, let's evaluate the original equation using the BODMAS/PEMDAS rule (Brackets, Orders, Division and Multiplication, Addition and Subtraction). Operations are performed from left to right for Division/Multiplication and Addition/Subtraction.

Evaluating the Original Equation

• Original Equation: $27 + 13 \times 12 - 6 \div 3$
• Step 1: Perform Division: $6 \div 3 = 2$. Equation becomes: $27 + 13 \times 12 - 2$.
• Step 2: Perform Multiplication: $13 \times 12 = 156$. Equation becomes: $27 + 156 - 2$.
• Step 3: Perform Addition: $27 + 156 = 183$. Equation becomes: $183 - 2$.
• Step 4: Perform Subtraction: $183 - 2 = 181$.

The original equation evaluates to 181, which is not 50. So, we need to interchange signs.

Testing Sign Interchanges

Let's test each given option by swapping the signs and evaluating the new equation.

Option 1: Interchange + and ×

Swap the '+' and '$\times$' signs in the original equation.

• New Equation: $27 \times 13 + 12 - 6 \div 3$
• Step 1: Perform Division: $6 \div 3 = 2$. Equation becomes: $27 \times 13 + 12 - 2$.
• Step 2: Perform Multiplication: $27 \times 13 = 351$. Equation becomes: $351 + 12 - 2$.
• Step 3: Perform Addition: $351 + 12 = 363$. Equation becomes: $363 - 2$.
• Step 4: Perform Subtraction: $363 - 2 = 361$.

The result is 361. This is not equal to 50. So, this option is incorrect.

Option 2: Interchange + and ÷

Swap the '+' and '$\div$' signs in the original equation.

• New Equation: $27 \div 13 \times 12 - 6 + 3$
• Step 1: Perform Division: $27 \div 13 \approx 2.077$. Equation becomes: $2.077 \times 12 - 6 + 3$ (Using approximate value).
• Step 2: Perform Multiplication: $2.077 \times 12 \approx 24.924$. Equation becomes: $24.924 - 6 + 3$.
• Step 3: Perform Subtraction/Addition from left to right: $24.924 - 6 = 18.924$. Equation becomes: $18.924 + 3$.
• Step 4: Perform Addition: $18.924 + 3 = 21.924$.

The result is approximately 21.924. This is not equal to 50. So, this option is incorrect.

Note: When dealing with integer targets, decimal results from division usually indicate the option is incorrect unless the subsequent operations cancel out the decimal part, which is rare in such problems.

Option 3: Interchange - and ÷

Swap the '-' and '$\div$' signs in the original equation.

• New Equation: $27 + 13 \times 12 \div 6 - 3$
• Step 1: Perform Division: $12 \div 6 = 2$. Equation becomes: $27 + 13 \times 2 - 3$.
• Step 2: Perform Multiplication: $13 \times 2 = 26$. Equation becomes: $27 + 26 - 3$.
• Step 3: Perform Addition: $27 + 26 = 53$. Equation becomes: $53 - 3$.
• Step 4: Perform Subtraction: $53 - 3 = 50$.

The result is 50. This matches the target value. So, this option represents the correct sign interchange.

Option 4: Interchange + and -

Swap the '+' and '-' signs in the original equation.

• New Equation: $27 - 13 \times 12 + 6 \div 3$
• Step 1: Perform Division: $6 \div 3 = 2$. Equation becomes: $27 - 13 \times 12 + 2$.
• Step 2: Perform Multiplication: $13 \times 12 = 156$. Equation becomes: $27 - 156 + 2$.
• Step 3: Perform Subtraction/Addition from left to right: $27 - 156 = -129$. Equation becomes: $-129 + 2$.
• Step 4: Perform Addition: $-129 + 2 = -127$.

The result is -127. This is not equal to 50. So, this option is incorrect.

Conclusion on Sign Interchange

By testing all the options and following the order of operations, we found that interchanging the - and ÷ signs makes the equation correct, resulting in 50.

Revision Table: Key Concepts in Equation Solving

Reviewing the key concepts involved in solving sign interchange problems.

Additional Information: Mathematical Operations and Logic

Understanding the properties of mathematical operations is fundamental to solving these types of logical reasoning problems based on equations.

• Arithmetic Operators: The basic operators are addition (+), subtraction (-), multiplication ($\times$), and division ($\div$).
• Order of Operations: Following a standard order like BODMAS or PEMDAS is essential to get a unique and correct result for any expression. Without a defined order, the same expression could yield different results depending on which operation is performed first.
• Logic and Reasoning: These problems test your ability to apply mathematical rules logically and systematically test different conditions (the sign interchanges) to achieve a specific outcome. They are common in reasoning sections of competitive exams.
• Checking Your Work: Always re-evaluate the equation with the proposed correct interchange to double-check your calculation.