Finding the Correct Sign Interchange

We are given the equation \(162 \div 3 + 5 \times 6 - 2 = 274\) and asked to find which interchange of signs will make the equation correct. To solve this, we will test each option by swapping the specified signs in the original equation and then evaluating the resulting expression using the standard order of operations (BODMAS/PEMDAS).

Understanding the Order of Operations (BODMAS/PEMDAS)

The correct order for performing mathematical operations is essential. We follow these steps:

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

Testing Each Sign Interchange Option

Let's apply the BODMAS/PEMDAS rule after performing the sign interchanges suggested by the options.

Option 1: Interchange + and -

Original Equation: \(162 \div 3 + 5 \times 6 - 2\)

After interchanging '+' and '-': \(162 \div 3 - 5 \times 6 + 2\)

Now, evaluate this expression:

• Division: \(162 \div 3 = 54\)
• Expression becomes: \(54 - 5 \times 6 + 2\)
• Multiplication: \(5 \times 6 = 30\)
• Expression becomes: \(54 - 30 + 2\)
• Subtraction and Addition (left to right): \(54 - 30 = 24\), then \(24 + 2 = 26\)

The result is \(26\). Since \(26 \neq 274\), this interchange does not make the equation correct.

Option 2: Interchange ÷ and ×

Original Equation: \(162 \div 3 + 5 \times 6 - 2\)

After interchanging '÷' and '×': \(162 \times 3 + 5 \div 6 - 2\)

Now, evaluate this expression:

• Multiplication: \(162 \times 3 = 486\)
• Expression becomes: \(486 + 5 \div 6 - 2\)
• Division: \(5 \div 6 = \frac{5}{6}\)
• Expression becomes: \(486 + \frac{5}{6} - 2\)
• Addition and Subtraction: \(486 + \frac{5}{6} - 2 = 484 + \frac{5}{6} \approx 484.833\)

The result is approximately \(484.833\). Since \(484.833 \neq 274\), this interchange does not make the equation correct.

Option 4: Interchange + and ×

Original Equation: \(162 \div 3 + 5 \times 6 - 2\)

After interchanging '+' and '×': \(162 \div 3 \times 5 + 6 - 2\)

Now, evaluate this expression using BODMAS/PEMDAS:

• Division: \(162 \div 3 = 54\)
• Expression becomes: \(54 \times 5 + 6 - 2\)
• Multiplication: \(54 \times 5 = 270\)
• Expression becomes: \(270 + 6 - 2\)
• Addition: \(270 + 6 = 276\)
• Expression becomes: \(276 - 2\)
• Subtraction: \(276 - 2 = 274\)

The result is \(274\). Since \(274 = 274\), this interchange makes the equation correct.

Conclusion on Sign Interchange

By interchanging the '+' and '×' signs in the original equation \(162 \div 3 + 5 \times 6 - 2 = 274\), we get the new equation \(162 \div 3 \times 5 + 6 - 2\). Evaluating this new expression using the BODMAS/PEMDAS rule yields a result of \(274\), which matches the right side of the given equation. Therefore, the interchange of '+' and '×' signs is the correct option.

Revision Table: Testing Sign Interchanges

Additional Information: Importance of Operation Order

Mathematical expressions must be evaluated in a specific order to ensure a single correct answer. The BODMAS/PEMDAS rule provides this standard order. Without it, solving an expression like \(10 + 2 \times 5\) could lead to \(10 + 10 = 20\) (correct) or \(12 \times 5 = 60\) (incorrect). In problems involving sign interchanges, correctly applying this order after swapping signs is the key to finding the correct option.