Finding the Correct Sign Interchange in an Equation

The problem asks us to determine which interchange of two mathematical signs in the given equation will make it correct. The given equation is:

\(288 \times 72 + 36 \div 12 - 6 = 20770\)

We need to test each option by swapping the specified signs and then evaluating the new expression using the order of operations (BODMAS/PEMDAS: Brackets, Orders, Division/Multiplication, Addition/Subtraction) to see if the result equals 20770.

Testing Sign Interchange Options

Option 1: Interchange × and −

Original equation: \(288 \times 72 + 36 \div 12 - 6\)

Interchange × and −: \(288 - 72 + 36 \div 12 \times 6\)

Evaluate the expression:

• First, Division: \(36 \div 12 = 3\)
• Then, Multiplication: \(3 \times 6 = 18\)
• Equation becomes: \(288 - 72 + 18\)
• Perform Subtraction and Addition from left to right:
• \(288 - 72 = 216\)
• \(216 + 18 = 234\)

Result: \(234\). This does not equal \(20770\).

Option 2: Interchange − and ÷

Original equation: \(288 \times 72 + 36 \div 12 - 6\)

Interchange − and ÷: \(288 \times 72 + 36 - 12 \div 6\)

Evaluate the expression using the order of operations:

• First, Multiplication: \(288 \times 72 = 20736\)
• Then, Division: \(12 \div 6 = 2\)
• Equation becomes: \(20736 + 36 - 2\)
• Perform Addition and Subtraction from left to right:
• \(20736 + 36 = 20772\)
• \(20772 - 2 = 20770\)

Result: \(20770\). This equals the right side of the original equation.

Since Option 2 results in the correct value, this is the correct sign interchange.

We can briefly check the other options to confirm they do not yield the correct result.

Option 3: Interchange + and ×

Original equation: \(288 \times 72 + 36 \div 12 - 6\)

Interchange + and ×: \(288 + 72 \times 36 \div 12 - 6\)

Evaluate:

• Multiplication: \(72 \times 36 = 2592\)
• Division: \(2592 \div 12 = 216\)
• Equation becomes: \(288 + 216 - 6\)
• \(288 + 216 = 504\)
• \(504 - 6 = 498\)

Result: \(498\). Does not equal \(20770\).

Option 4: Interchange ÷ and +

Original equation: \(288 \times 72 + 36 \div 12 - 6\)

Interchange ÷ and +: \(288 \times 72 \div 36 + 12 - 6\)

Evaluate:

• Multiplication/Division from left to right:
• \(288 \times 72 = 20736\)
• \(20736 \div 36 = 576\)
• Equation becomes: \(576 + 12 - 6\)
• \(576 + 12 = 588\)
• \(588 - 6 = 582\)

Result: \(582\). Does not equal \(20770\).

Therefore, only interchanging the − and ÷ signs makes the equation correct.

Revision Table: Sign Interchange Results

Additional Information: Order of Operations (BODMAS/PEMDAS)

The order of operations is a rule used to clarify the sequence of operations in a mathematical expression. It ensures that everyone gets the same result when evaluating an expression. The acronyms BODMAS and PEMDAS are commonly used:

• BODMAS:
  • B - Brackets
  • O - Orders (powers, square roots, etc.)
  • D - Division
  • M - Multiplication
  • A - Addition
  • S - Subtraction
• PEMDAS:
  • P - Parentheses
  • E - Exponents
  • M - Multiplication
  • D - Division
  • A - Addition
  • S - Subtraction

Division and Multiplication have the same priority and are performed from left to right. Similarly, Addition and Subtraction have the same priority and are performed from left to right.

In this problem, we strictly followed this order after interchanging the signs to correctly evaluate each resulting expression.