Understanding the Mathematical Operation Interchange Problem

This question asks us to determine which mathematical equation becomes correct when two specific mathematical signs and two specific numbers are interchanged according to a given rule. The interchange rule is that the '+' sign is swapped with the '×' sign, and the number '1' is swapped with the number '2'. We need to apply this rule to each of the given options and then evaluate the resulting expression using the standard order of operations (BODMAS/PEMDAS) to see if it equals the value on the right side of the original equation.

Applying the Interchange Rule: + <--> ×, 1 <--> 2

The rule requires us to make the following substitutions in each equation:

• Replace every '+' with '×'.
• Replace every '×' with '+'.
• Replace every '1' with '2'.
• Replace every '2' with '1'.

Other signs (–, ÷) and numbers remain unchanged.

Evaluating Each Option After Interchange

Let's apply the interchange rule to each given option and then calculate the result. Remember to follow the BODMAS/PEMDAS rule: Brackets, Orders (powers/roots), Division and Multiplication (from left to right), Addition and Subtraction (from left to right).

Option 1 Analysis

Original Equation: \(8 \times 3 \ndash 4 \div 2 + 1 = 7\)

Applying the interchange rule (+ <--> ×, 1 <--> 2):

\(8 + 3 \ndash 4 \div 1 \times 2\)

Now, evaluate the expression:

\(8 + 3 \ndash 4 \div 1 \times 2\)

Perform Division:

\(8 + 3 \ndash 4 \times 2\)

Perform Multiplication:

\(8 + 3 \ndash 8\)

Perform Addition:

\(11 \ndash 8\)

Perform Subtraction:

\(3\)

The result is 3. The original equation stated the result should be 7. Since \(3 \neq 7\), this equation is not correct after the interchange.

Option 2 Analysis

Original Equation: \(2 \times 3 \ndash 8 \div 1 + 9 = 18\)

Applying the interchange rule (+ <--> ×, 1 <--> 2):

\(1 + 3 \ndash 8 \div 2 \times 9\)

Now, evaluate the expression:

\(1 + 3 \ndash 8 \div 2 \times 9\)

Perform Division:

\(1 + 3 \ndash 4 \times 9\)

Perform Multiplication:

\(1 + 3 \ndash 36\)

Perform Addition:

\(4 \ndash 36\)

Perform Subtraction:

\(\ndash 32\)

The result is -32. The original equation stated the result should be 18. Since \(\ndash 32 \neq 18\), this equation is not correct after the interchange.

Option 3 Analysis

Original Equation: \(8 \times 3 \ndash 4 \div 2 + 1 = 3\)

Applying the interchange rule (+ <--> ×, 1 <--> 2):

\(8 + 3 \ndash 4 \div 1 \times 2\)

Now, evaluate the expression:

\(8 + 3 \ndash 4 \div 1 \times 2\)

Perform Division:

\(8 + 3 \ndash 4 \times 2\)

Perform Multiplication:

\(8 + 3 \ndash 8\)

Perform Addition:

\(11 \ndash 8\)

Perform Subtraction:

\(3\)

The result is 3. The original equation stated the result should be 3. Since \(3 = 3\), this equation is correct after the interchange.

Option 4 Analysis

Original Equation: \(2 \times 3 \ndash 8 \div 1 + 2 = 5\)

Applying the interchange rule (+ <--> ×, 1 <--> 2):

\(1 + 3 \ndash 8 \div 2 \times 1\)

Now, evaluate the expression:

\(1 + 3 \ndash 8 \div 2 \times 1\)

Perform Division:

\(1 + 3 \ndash 4 \times 1\)

Perform Multiplication:

\(1 + 3 \ndash 4\)

Perform Addition:

\(4 \ndash 4\)

Perform Subtraction:

\(0\)

The result is 0. The original equation stated the result should be 5. Since \(0 \neq 5\), this equation is not correct after the interchange.

Conclusion

After applying the given interchange rule (+ <--> ×, 1 <--> 2) to each equation, only the equation in Option 3 results in a correct mathematical statement.

Revision Table: Understanding BODMAS/PEMDAS

The order of operations is crucial for evaluating mathematical expressions correctly. The acronyms BODMAS or PEMDAS help remember the sequence.

Additional Information: Types of Mathematical Reasoning Questions

Mathematical reasoning questions often involve applying specific rules or patterns to numbers, signs, or equations. Common types include:

• Sign Interchange: Swapping two arithmetic signs (+, –, ×, ÷) to make an equation correct.
• Number Interchange: Swapping two numbers to make an equation correct.
• Sign and Number Interchange: Swapping both signs and numbers simultaneously, as seen in this problem.
• Equation Correction: Identifying which sign(s) or number(s) need to be changed to correct a given incorrect equation.
• Finding the Missing Term: Determining the correct sign or number that should be placed in a blank to satisfy a mathematical relationship.

Practicing these different types helps build proficiency in applying rules and evaluating expressions accurately.