If the signs ‘+’ and ‘÷ ' are interchanged, then which of the following equations can be correctly balanced?
- 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
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Important Questions from Logical Puzzle
Seven persons P, Q, R, S, T, U and V like different watches namely W1, W2, W3, W4, W5, W6 and W7 (not necessarily in the same order). P and R do not like odd numbered watch. T likes W5. U does not like W2 or W3 or W6 or W7. P likes prime numbered watch. Q likes W4. S likes W2 or W7. Which of the following statement(s) is/are correct ?
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II. R likes W2.
III. U likes W1.
IV. V likes W3.
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