• The core task is to determine which mathematical equation becomes correctly balanced after swapping the '+' (addition) and '÷' (division) signs.
• We need to examine each option provided, apply the sign interchange rule, and verify the resulting equation using the order of operations (BODMAS/PEMDAS).

Analyzing Sign Interchange: '+' and '÷'

The rule requires us to replace every '+' sign with a '÷' sign and every '÷' sign with a '+' sign within the given equations.

Option 1: `22 + 11 × 8 ÷ 3 = 19`

• Original Equation: 22 + 11 × 8 ÷ 3 = 19
• Applying the Interchange Rule: Replace '+' with '÷' and '÷' with '+'. The equation becomes: 22 ÷ 11 × 8 + 3 = 19
• Calculation Steps (BODMAS/PEMDAS):
  • Division first: 22 ÷ 11 = 2
  • Multiplication next: 2 × 8 = 16
  • Addition last: 16 + 3 = 19
• Result: The calculation yields 19, which matches the right side of the equation (19 = 19).
• Conclusion: This equation is correctly balanced after the sign interchange.

Option 2: `24 – 12 ÷ 6 + 3 = 20`

• Original Equation: 24 – 12 ÷ 6 + 3 = 20
• Applying the Interchange Rule: The equation becomes: 24 – 12 + 6 ÷ 3 = 20
• Calculation Steps (BODMAS/PEMDAS):
  • Division first: 6 ÷ 3 = 2
  • Subtraction next: 24 – 12 = 12
  • Addition last: 12 + 2 = 14
• Result: The calculation yields 14, which does not match the right side of the equation (14 &neq 20).
• Conclusion: This equation is not correctly balanced.

Option 3: `12 + 4 – 8 ÷ 3 = 11`

• Original Equation: 12 + 4 – 8 ÷ 3 = 11
• Applying the Interchange Rule: The equation becomes: 12 ÷ 4 – 8 + 3 = 11
• Calculation Steps (BODMAS/PEMDAS):
  • Division first: 12 ÷ 4 = 3
  • Subtraction next: 3 – 8 = -5
  • Addition last: -5 + 3 = -2
• Result: The calculation yields -2, which does not match the right side of the equation (-2 &neq 11).
• Conclusion: This equation is not correctly balanced.

Option 4: `16 + 4 × 8 ÷ 3 = 45`

• Original Equation: 16 + 4 × 8 ÷ 3 = 45
• Applying the Interchange Rule: The equation becomes: 16 ÷ 4 × 8 + 3 = 45
• Calculation Steps (BODMAS/PEMDAS):
  • Division first: 16 ÷ 4 = 4
  • Multiplication next: 4 × 8 = 32
  • Addition last: 32 + 3 = 35
• Result: The calculation yields 35, which does not match the right side of the equation (35 &neq 45).
• Conclusion: This equation is not correctly balanced.

Final Conclusion on Balanced Equations

After interchanging the '+' and '÷' signs, only the first equation results in a true statement. The other equations do not balance when the signs are swapped according to the specified rule.

Therefore, the correctly balanced equation after interchanging '+' and '÷' is:

22 ÷ 11 × 8 + 3 = 19