Understanding the Problem: Sign Interchange

The question asks us to identify which of the given equations becomes incorrect when the mathematical signs '+' (plus) and '÷' (division) are swapped. We need to go through each option, interchange these two signs, and then evaluate the resulting expression using the standard order of operations (BODMAS/PEMDAS) to see if the left side of the equation still equals the right side.

Step-by-Step Solution: Checking Each Equation

We will apply the BODMAS rule, which dictates the order of operations:

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

Let's check each option after interchanging '+' and '÷'.

Checking Option 1: \(18 + 6 – 6 ÷ 3 × 3 = 6\)

Original equation: \(18 + 6 - 6 \div 3 \times 3 = 6\)

After interchanging '+' and '÷', the equation becomes:

\(18 \div 6 - 6 + 3 \times 3\)

Let's evaluate the left side:

• First, Division and Multiplication (from left to right):
• \(18 \div 6 = 3\)
• \(3 \times 3 = 9\)
• Now the expression is: \(3 - 6 + 9\)
• Next, Addition and Subtraction (from left to right):
• \(3 - 6 = -3\)
• \(-3 + 9 = 6\)

The left side evaluates to 6. The right side is 6.

Since \(6 = 6\), this equation is correct after interchanging the signs.

Checking Option 2: \(18 + 2 × 6 ÷ 3 – 7 = 50\)

Original equation: \(18 + 2 \times 6 \div 3 - 7 = 50\)

After interchanging '+' and '÷', the equation becomes:

\(18 \div 2 \times 6 + 3 - 7\)

Let's evaluate the left side:

• First, Division and Multiplication (from left to right):
• \(18 \div 2 = 9\)
• \(9 \times 6 = 54\)
• Now the expression is: \(54 + 3 - 7\)
• Next, Addition and Subtraction (from left to right):
• \(54 + 3 = 57\)
• \(57 - 7 = 50\)

The left side evaluates to 50. The right side is 50.

Since \(50 = 50\), this equation is correct after interchanging the signs.

Checking Option 3: \(7 × 8 ÷ 4 + 2 – 6 = 49\)

Original equation: \(7 \times 8 \div 4 + 2 - 6 = 49\)

After interchanging '+' and '÷', the equation becomes:

\(7 \times 8 + 4 \div 2 - 6\)

Let's evaluate the left side:

• First, Division and Multiplication (from left to right):
• \(4 \div 2 = 2\)
• \(7 \times 8 = 56\)
• Now the expression is: \(56 + 2 - 6\)
• Next, Addition and Subtraction (from left to right):
• \(56 + 2 = 58\)
• \(58 - 6 = 52\)

The left side evaluates to 52. The right side is 49.

Since \(52 \neq 49\), this equation is not correct after interchanging the signs.

Checking Option 4: \(6 × 9 + 3 – 24 ÷ 8 = 2\)

Original equation: \(6 \times 9 + 3 - 24 \div 8 = 2\)

After interchanging '+' and '÷', the equation becomes:

\(6 \times 9 \div 3 - 24 + 8\)

Let's evaluate the left side:

• First, Division and Multiplication (from left to right):
• \(6 \times 9 = 54\)
• \(54 \div 3 = 18\)
• Now the expression is: \(18 - 24 + 8\)
• Next, Addition and Subtraction (from left to right):
• \(18 - 24 = -6\)
• \(-6 + 8 = 2\)

The left side evaluates to 2. The right side is 2.

Since \(2 = 2\), this equation is correct after interchanging the signs.

Conclusion

After interchanging the '+' and '÷' signs, the equation \(7 \times 8 \div 4 + 2 - 6 = 49\) becomes \(7 \times 8 + 4 \div 2 - 6 = 49\), which evaluates to \(52 = 49\). This is incorrect.

Therefore, the equation that will be not correct after interchanging the given two signs is \(7 × 8 ÷ 4 + 2 – 6 = 49\).

Revision Table: Sign Interchange Results

Additional Information: Order of Operations

The order of operations, like BODMAS or PEMDAS, is crucial for solving mathematical expressions consistently. Without a defined order, expressions involving multiple operations could have different results depending on which operation is performed first. BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction) and PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) are just mnemonics to remember this order. 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.