Understanding Sign Interchange in Mathematical Equations

The question asks us to find which of the given equations becomes correct if we interchange the '+' and '–' signs. This means wherever there is a '+' sign in the original equation, it will be replaced by a '–' sign, and wherever there is a '–' sign, it will be replaced by a '+' sign. Other signs like '×' (multiplication) and '÷' (division) remain unchanged.

Applying the Order of Operations (BODMAS/PEMDAS)

To evaluate each equation after interchanging the signs, we must follow the standard order of operations. The acronyms BODMAS or PEMDAS help remember the order:

• B/P: Brackets first / Parentheses first
• O/E: Orders (powers, square roots) / Exponents
• D/M: Division and Multiplication (from left to right)
• A/S: Addition and Subtraction (from left to right)

We will evaluate each option by first interchanging the '+' and '–' signs and then applying the BODMAS/PEMDAS rule.

Evaluating Option 1

Original Equation: \(55 \div 11 – 15 + 3 \times 2 = 15\)

After interchanging '+' and '–' signs:

\(55 \div 11 + 15 – 3 \times 2\)

Now, let's evaluate using BODMAS/PEMDAS:

• First, Division and Multiplication (from left to right):
  • \(55 \div 11 = 5\)
  • \(3 \times 2 = 6\)
• Substitute back into the expression:
  • \(5 + 15 – 6\)
• Next, Addition and Subtraction (from left to right):
  • \(5 + 15 = 20\)
  • \(20 – 6 = 14\)

The result is 14. The original equation states the result should be 15. Since \(14 \neq 15\), Option 1 is incorrect after the sign interchange.

Evaluating Option 2

Original Equation: \(4 \times 3 – 6 \div 2 + 7 = 7\)

After interchanging '+' and '–' signs:

\(4 \times 3 + 6 \div 2 – 7\)

Now, let's evaluate using BODMAS/PEMDAS:

• First, Multiplication and Division (from left to right):
  • \(4 \times 3 = 12\)
  • \(6 \div 2 = 3\)
• Substitute back into the expression:
  • \(12 + 3 – 7\)
• Next, Addition and Subtraction (from left to right):
  • \(12 + 3 = 15\)
  • \(15 – 7 = 8\)

The result is 8. The original equation states the result should be 7. Since \(8 \neq 7\), Option 2 is incorrect after the sign interchange.

Evaluating Option 3

Original Equation: \(15 - 7 \times 8 + 18 \div 3 = 65\)

After interchanging '+' and '–' signs:

\(15 + 7 \times 8 - 18 \div 3\)

Now, let's evaluate using BODMAS/PEMDAS:

• First, Multiplication and Division (from left to right):
  • \(7 \times 8 = 56\)
  • \(18 \div 3 = 6\)
• Substitute back into the expression:
  • \(15 + 56 - 6\)
• Next, Addition and Subtraction (from left to right):
  • \(15 + 56 = 71\)
  • \(71 - 6 = 65\)

The result is 65. The original equation states the result should be 65. Since \(65 = 65\), Option 3 is correct after the sign interchange.

Evaluating Option 4

Original Equation: \(529 \div 23 + 10 - 5 \times 4 = 30\)

After interchanging '+' and '–' signs:

\(529 \div 23 - 10 + 5 \times 4\)

Now, let's evaluate using BODMAS/PEMDAS:

• First, Division and Multiplication (from left to right):
  • \(529 \div 23 = 23\)
  • \(5 \times 4 = 20\)
• Substitute back into the expression:
  • \(23 - 10 + 20\)
• Next, Addition and Subtraction (from left to right):
  • \(23 - 10 = 13\)
  • \(13 + 20 = 33\)

The result is 33. The original equation states the result should be 30. Since \(33 \neq 30\), Option 4 is incorrect after the sign interchange.

Conclusion

After evaluating all the options by interchanging the '+' and '–' signs and applying the order of operations, only Option 3 results in a correct mathematical equation.

Revision Table: Key Concepts

Additional Information: Mathematical Operations and Symbols

Mathematical operations are fundamental actions performed on numbers. The basic operations are addition, subtraction, multiplication, and division. Understanding how these operations work and their symbols is crucial for solving mathematical problems.

• Addition (\(+\)): Combining quantities.
• Subtraction (\(-\)): Finding the difference between quantities.
• Multiplication (\(\times\)): Repeated addition or scaling.
• Division (\(\div\)): Sharing equally or splitting into groups.

In problems involving interchanging signs, it's important to clearly identify the signs to be swapped and then carefully rewrite the equation before performing the calculations according to the order of operations. Errors often occur if the order of operations is not strictly followed or if the sign interchange is done incorrectly.