Understanding the Sign and Number Interchange Problem

The question asks us to identify which of the given equations becomes mathematically correct after performing specific interchanges: the '+' sign is swapped with the '–' sign, and the number '2' is swapped with the number '4'. We need to apply these changes to each equation and then evaluate the resulting expression to see if it matches the right side of the original equation.

Performing Interchanges: + <-> – and 2 <-> 4

The rules for interchange are simple:

• Every '+' sign in the original equation becomes a '–' sign.
• Every '–' sign in the original equation becomes a '+' sign.
• Every '2' in the original equation becomes a '4'.
• Every '4' in the original equation becomes a '2'.

Other signs (like '×', '÷', '=') and numbers remain unchanged.

Analyzing Each Equation Option After Interchange

Let's apply these rules to each option and evaluate the result using the standard order of operations (BODMAS/PEMDAS: Brackets, Orders/Exponents, Division and Multiplication (from left to right), Addition and Subtraction (from left to right)).

Evaluating Option 1 Equation

Original Equation: \(9 – 5 ÷ 4 × 2 + 8 = 80\)

Applying Interchanges (+ <-> –, 2 <-> 4):

The equation becomes: \(9 + 5 ÷ 2 × 4 – 8\)

Now, let's evaluate the left side:

• Division: \(5 ÷ 2 = 2.5\)
• The expression is now: \(9 + 2.5 × 4 – 8\)
• Multiplication: \(2.5 × 4 = 10\)
• The expression is now: \(9 + 10 – 8\)
• Addition and Subtraction (from left to right): \(9 + 10 = 19\)
• The expression is now: \(19 – 8\)
• Subtraction: \(19 – 8 = 11\)

The result is 11. The original right side was 80. Since \(11 \neq 80\), this equation is not correct after the interchanges.

Evaluating Option 2 Equation

Original Equation: \(7 × 4 – 2 ÷ 1 + 5 = 13\)

Applying Interchanges (+ <-> –, 2 <-> 4):

The equation becomes: \(7 × 2 + 4 ÷ 1 – 5\)

Now, let's evaluate the left side:

• Multiplication: \(7 × 2 = 14\)
• Division: \(4 ÷ 1 = 4\)
• The expression is now: \(14 + 4 – 5\)
• Addition and Subtraction (from left to right): \(14 + 4 = 18\)
• The expression is now: \(18 – 5\)
• Subtraction: \(18 – 5 = 13\)

The result is 13. The original right side was 13. Since \(13 = 13\), this equation is correct after the interchanges.

Evaluating Option 3 Equation

Original Equation: \(7 – 8 ÷ 2 + 16 × 4 = 18\)

Applying Interchanges (+ <-> –, 2 <-> 4):

The equation becomes: \(7 + 8 ÷ 4 – 16 × 2\)

Now, let's evaluate the left side:

• Division: \(8 ÷ 4 = 2\)
• Multiplication: \(16 × 2 = 32\)
• The expression is now: \(7 + 2 – 32\)
• Addition and Subtraction (from left to right): \(7 + 2 = 9\)
• The expression is now: \(9 – 32\)
• Subtraction: \(9 – 32 = -23\)

The result is -23. The original right side was 18. Since \(-23 \neq 18\), this equation is not correct after the interchanges.

Evaluating Option 4 Equation

Original Equation: \(5 + 7 × 4 – 8 ÷ 2 = 5\)

Applying Interchanges (+ <-> –, 2 <-> 4):

The equation becomes: \(5 – 7 × 2 + 8 ÷ 4\)

Now, let's evaluate the left side:

• Multiplication: \(7 × 2 = 14\)
• Division: \(8 ÷ 4 = 2\)
• The expression is now: \(5 – 14 + 2\)
• Addition and Subtraction (from left to right): \(5 – 14 = -9\)
• The expression is now: \(-9 + 2\)
• Addition: \(-9 + 2 = -7\)

The result is -7. The original right side was 5. Since \(-7 \neq 5\), this equation is not correct after the interchanges.

Correct Equation After Sign and Number Swap

Based on our evaluation, only Option 2 resulted in a correct mathematical equation after interchanging '+' with '–' and '2' with '4'.

Revision Table: Key Math Operations

Additional Information: BODMAS/PEMDAS Rule

When evaluating mathematical expressions, it's crucial to follow the correct order of operations. This order is often remembered using mnemonics like BODMAS or PEMDAS.

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

Following this order ensures that everyone gets the same result for a given expression, which is essential for solving equations correctly after applying interchanges.