Understanding the Question: Number Interchange

The question asks us to identify which of the given mathematical equations will become incorrect if we swap the positions of the numbers 7 and 4 in that equation. We need to check each option by performing the interchange and then evaluating the resulting expression according to the order of operations (BODMAS/PEMDAS).

Applying the Interchange and BODMAS Rule

For each equation, we will replace every 7 with a 4 and every 4 with a 7. After the interchange, we will calculate the value of the left side of the modified equation using the BODMAS rule: Brackets, Orders (powers, roots), Division and Multiplication (from left to right), Addition and Subtraction (from left to right). We will then compare this result with the right side of the equation to see if it holds true.

Evaluating Option 1: 8 ÷ 6 × 9 + 7 - 4 = 12

Original equation: \(8 \div 6 \times 9 + 7 - 4 = 12\)

Interchanging 7 and 4, the equation becomes:

\(8 \div 6 \times 9 + 4 - 7 = 12\)

Now, let's evaluate the left side of this new equation using BODMAS:

• Perform Division: \(8 \div 6 = \frac{8}{6} = \frac{4}{3}\)
• Perform Multiplication: \(\frac{4}{3} \times 9 = 12\)
• Perform Addition and Subtraction from left to right: \(12 + 4 - 7 = 16 - 7 = 9\)

The equation simplifies to \(9 = 12\).

This resulting equation is not correct.

Evaluating Option 2: 7 × 8 + 4 - 6 ÷ 3 = 37

Original equation: \(7 \times 8 + 4 - 6 \div 3 = 37\)

Interchanging 7 and 4, the equation becomes:

\(4 \times 8 + 7 - 6 \div 3 = 37\)

Now, let's evaluate the left side:

• Perform Division: \(6 \div 3 = 2\)
• Perform Multiplication: \(4 \times 8 = 32\)
• Perform Addition and Subtraction from left to right: \(32 + 7 - 2 = 39 - 2 = 37\)

The equation simplifies to \(37 = 37\).

This resulting equation is correct.

Evaluating Option 3: 9 - 4 × 3 + 6 × 7 ÷ 1 = 12

Original equation: \(9 - 4 \times 3 + 6 \times 7 \div 1 = 12\)

Interchanging 7 and 4, the equation becomes:

\(9 - 7 \times 3 + 6 \times 4 \div 1 = 12\)

Now, let's evaluate the left side:

• Perform Multiplication and Division from left to right: \(7 \times 3 = 21\), \(6 \times 4 = 24\), \(24 \div 1 = 24\)
• Perform Addition and Subtraction from left to right: \(9 - 21 + 24 = -12 + 24 = 12\)

The equation simplifies to \(12 = 12\).

This resulting equation is correct.

Evaluating Option 4: 4 × 8 - 9 ÷ 3 + 7 = 57

Original equation: \(4 \times 8 - 9 \div 3 + 7 = 57\)

Interchanging 7 and 4, the equation becomes:

\(7 \times 8 - 9 \div 3 + 4 = 57\)

Now, let's evaluate the left side:

• Perform Division: \(9 \div 3 = 3\)
• Perform Multiplication: \(7 \times 8 = 56\)
• Perform Addition and Subtraction from left to right: \(56 - 3 + 4 = 53 + 4 = 57\)

The equation simplifies to \(57 = 57\).

This resulting equation is correct.

Conclusion: Identifying the Incorrect Equation

After interchanging the numbers 7 and 4 in each option, we found that only the first equation resulted in an incorrect mathematical statement (\(9 = 12\)). The other equations remained correct.

Revision Table: Number Interchange and Equation Verification

Additional Information: Order of Operations (BODMAS/PEMDAS)

The order of operations is crucial in mathematics to ensure consistent results. It dictates the sequence in which operations should be performed within an expression.

• BODMAS stands for Brackets, Orders, Division, Multiplication, Addition, Subtraction.
• PEMDAS stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.

Division and multiplication have the same priority; they are performed from left to right. Similarly, addition and subtraction have the same priority and are performed from left to right. Understanding and applying this order is fundamental for solving mathematical expressions correctly, especially in problems involving multiple operations like the number interchange problem discussed here.