Solving the Equation by Sign Interchange

The problem asks us to find which interchange of two mathematical signs from the given equation will make the equation correct. The initial equation is:

\(100 + 100 \times 50 - 2 \div 3 = 51\)

Let's first evaluate the original equation using the order of operations (BODMAS/PEMDAS - Brackets, Orders, Division/Multiplication, Addition/Subtraction):

• Multiplication first: \(100 \times 50 = 5000\)
• Division next: \(2 \div 3 \approx 0.67\)

So the equation becomes approximately: \(100 + 5000 - 0.67 = 5100 - 0.67 = 5099.33\)

Clearly, \(5099.33\) is not equal to \(51\), so the original equation is incorrect. We need to test the given options by swapping the specified signs and re-evaluating the equation.

Testing Option 1: Interchanging + and ÷

If we interchange the '+' and '÷' signs, the equation becomes:

\(100 \div 100 \times 50 - 2 + 3\)

Now, let's evaluate this new equation using BODMAS:

• Division first: \(100 \div 100 = 1\)

The equation is now: \(1 \times 50 - 2 + 3\)

• Multiplication next: \(1 \times 50 = 50\)

The equation is now: \(50 - 2 + 3\)

• Now, perform addition and subtraction from left to right:
• Subtraction: \(50 - 2 = 48\)
• Addition: \(48 + 3 = 51\)

So, the equation becomes \(51\). This matches the right side of the original target equation.

\(100 \div 100 \times 50 - 2 + 3 = 51\)

Testing Other Options (for completeness)

Let's quickly look at the other options to see if they yield 51.

Option 2: Interchanging + and -

Equation becomes: \(100 - 100 \times 50 + 2 \div 3\)

Evaluation: \(100 - (100 \times 50) + (2 \div 3) = 100 - 5000 + 0.67 \approx -4899.33\). This is not 51.

Option 3: Interchanging - and ×

Equation becomes: \(100 + 100 - 50 \times 2 \div 3\)

Evaluation: \(100 + 100 - (50 \times 2) \div 3 = 200 - 100 \div 3 \approx 200 - 33.33 \approx 166.67\). This is not 51.

Option 4: Interchanging ÷ and ×

Equation becomes: \(100 + 100 \div 50 - 2 \times 3\)

Evaluation: \(100 + (100 \div 50) - (2 \times 3) = 100 + 2 - 6 = 102 - 6 = 96\). This is not 51.

Conclusion

Based on the evaluation of each option, only interchanging the '+' and '÷' signs makes the given equation correct.

Revision Table: Order of Operations (BODMAS/PEMDAS)

Additional Information: Solving Equation Puzzles

Problems involving interchanging signs or numbers in equations are common in reasoning and quantitative aptitude tests. Here are some tips for solving them:

• Understand BODMAS/PEMDAS: Always follow the correct order of operations when evaluating the equation after the interchange.
• Test Options Systematically: Go through each option one by one. Don't assume the first one you try is correct without verifying it.
• Be Careful with Calculations: Double-check your arithmetic, especially with division and subtraction.
• Estimation: Sometimes, a quick estimation after swapping signs can eliminate options that are clearly far from the target value.
• Practice: Solving more problems of this type will help you become faster and more accurate.