Equation Analysis

The original equation is presented as:

\( 3 \div 2 - 4 \times 5 + 5 = 1 \)

The task is to identify which pair of mathematical signs needs to be swapped to satisfy this equation. We will systematically check each option provided.

Signs Interchange Verification

Let's evaluate the equation based on the sign interchanges suggested in the options:

• Option 1: Interchanging '+' and '-'
  The equation transforms to \( 3 \div 2 - 4 \times 5 - 5 \).
  Applying the order of operations (PEMDAS/BODMAS): \( 1.5 - 20 - 5 = -23.5 \). This does not equal 1.
• Option 2: Interchanging '\(\times\)' and '-'
  The equation becomes \( 3 \div 2 + 4 - 5 \times 5 \).
  Calculation: \( 1.5 + 4 - 25 = -19.5 \). This is not equal to 1.
• Option 3: Interchanging '\(\div\)' and '\(\times\)'
  The equation changes to \( 3 \times 2 - 4 \div 5 + 5 \).
  Calculation: \( 6 - 0.8 + 5 = 10.2 \). This is not 1.
• Option 4: Interchanging '+' and '\(\div\)'
  The equation becomes \( 3 + 2 - 4 \times 5 \div 5 \).
  This option leads to the correct result. The verification follows.

Correct Equation Solution

Let's confirm the result by performing the calculations for Option 4:

Modified Equation: \( 3 + 2 - 4 \times 5 \div 5 \)

1. Division: First, calculate \( 5 \div 5 \). Result is \( 1 \).
   The equation is now \( 3 + 2 - 4 \times 1 \).
2. Multiplication: Next, calculate \( 4 \times 1 \). Result is \( 4 \).
   The equation simplifies to \( 3 + 2 - 4 \).
3. Addition: Calculate \( 3 + 2 \). Result is \( 5 \).
   The equation becomes \( 5 - 4 \).
4. Subtraction: Finally, calculate \( 5 - 4 \). Result is \( 1 \).

Since the calculation yields \( 1 \), interchanging the '+' and '\(\div\)' signs correctly resolves the equation.