Understanding the Problem: Interchanging Numbers in Equations

The question asks us to identify which of the given mathematical equations becomes incorrect when the numbers 7 and 9 are interchanged. We need to take each equation, swap every instance of 7 with 9 and every instance of 9 with 7, and then evaluate the resulting equation to see if it holds true.

Applying the Order of Operations (BODMAS/PEMDAS)

To correctly evaluate each equation after interchanging the numbers, we must follow the standard order of mathematical operations. This order is commonly known as BODMAS or PEMDAS:

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

We will apply this rule to each equation after swapping 7 and 9.

Evaluating Each Equation After Interchanging 7 and 9

Option 1: Checking Equation 1 After Interchanging

The original equation is $8 \times 3 \div 6 + 9 – 7 = 2$.

After interchanging 7 and 9, the equation becomes:

$8 \times 3 \div 6 + 7 – 9$

Let's evaluate the left side:

• Division and Multiplication (left to right):
  • $8 \times 3 = 24$
  • $24 \div 6 = 4$
• Addition and Subtraction (left to right):
  • $4 + 7 = 11$
  • $11 – 9 = 2$

The left side evaluates to 2. The right side is 2. Since $2 = 2$, this equation is correct after interchanging 7 and 9.

Option 2: Checking Equation 2 After Interchanging

The original equation is $7 + 6 – 9 \times 4 \div 2 = 2$.

After interchanging 7 and 9, the equation becomes:

$9 + 6 – 7 \times 4 \div 2$

Let's evaluate the left side:

• Division and Multiplication (left to right):
  • $7 \times 4 = 28$
  • $28 \div 2 = 14$
• Addition and Subtraction (left to right):
  • $9 + 6 = 15$
  • $15 – 14 = 1$

The left side evaluates to 1. The right side is 2. Since $1 \neq 2$, this equation is not correct after interchanging 7 and 9.

Option 3: Checking Equation 3 After Interchanging

The original equation is $9 + 4 \times 3 – 7 \div 1 = 10$.

After interchanging 7 and 9, the equation becomes:

$7 + 4 \times 3 – 9 \div 1$

Let's evaluate the left side:

• Division and Multiplication (left to right):
  • $4 \times 3 = 12$
  • $9 \div 1 = 9$
• Addition and Subtraction (left to right):
  • $7 + 12 = 19$
  • $19 – 9 = 10$

The left side evaluates to 10. The right side is 10. Since $10 = 10$, this equation is correct after interchanging 7 and 9.

Option 4: Checking Equation 4 After Interchanging

The original equation is $9 \times 3 + 4 \div 2 – 7 = 14$.

After interchanging 7 and 9, the equation becomes:

$7 \times 3 + 4 \div 2 – 9$

Let's evaluate the left side:

• Division and Multiplication (left to right):
  • $7 \times 3 = 21$
  • $4 \div 2 = 2$
• Addition and Subtraction (left to right):
  • $21 + 2 = 23$
  • $23 – 9 = 14$

The left side evaluates to 14. The right side is 14. Since $14 = 14$, this equation is correct after interchanging 7 and 9.

Conclusion

After interchanging the numbers 7 and 9 in all the given equations and evaluating them using the BODMAS/PEMDAS rule, we found that only the second equation, $7 + 6 – 9 \times 4 \div 2 = 2$, results in an incorrect statement ($1 \neq 2$). Therefore, this is the equation that is not correct after the interchange.

Revision Table: Equation Evaluation Summary

Additional Information: Understanding BODMAS/PEMDAS

The BODMAS or PEMDAS rule is crucial for solving mathematical expressions involving multiple operations. Without a standard order, different people could arrive at different answers for the same expression. The rule ensures consistency.

• Parentheses/Brackets take the highest priority. Operations inside brackets are performed first.
• Exponents/Orders (like powers and square roots) are evaluated next.
• Multiplication and Division have equal priority and are performed from left to right as they appear in the expression.
• Addition and Subtraction have equal priority (lower than multiplication/division) and are performed from left to right as they appear.

Remembering this order is key to solving many quantitative aptitude problems involving equations and expressions.