Solving Equation by Interchanging Signs

The problem asks us to find which two mathematical signs in the given equation need to be swapped so that the equation becomes correct. The given equation is:

\(20 \div 10 \times 100 - 10 + 100 = 110\)

We need to check each option by interchanging the specified signs and then evaluate the resulting equation using the order of operations (BODMAS/PEMDAS) to see if it equals 110.

First, let's evaluate the original equation as it is given:

\(20 \div 10 \times 100 - 10 + 100 = 2 \times 100 - 10 + 100 = 200 - 10 + 100 = 190 + 100 = 290\)

Since 290 is not equal to 110, the original equation is incorrect, and we must interchange signs.

Testing the Options for Sign Interchange

Option 1: Swap + and ÷

Interchanging '+' and '÷', the equation becomes:

\(20 + 10 \times 100 - 10 \div 100\)

Evaluating this expression:

• Division: \(10 \div 100 = 0.1\)
• Multiplication: \(10 \times 100 = 1000\)
• Addition and Subtraction (left to right): \(20 + 1000 = 1020\)
• Subtraction: \(1020 - 0.1 = 1019.9\)

The result 1019.9 is not equal to 110.

Option 2: Swap × and -

Interchanging '×' and '-', the equation becomes:

\(20 \div 10 - 100 \times 10 + 100\)

Evaluating this expression:

• Division: \(20 \div 10 = 2\)
• Multiplication: \(100 \times 10 = 1000\)
• Subtraction and Addition (left to right): \(2 - 1000 = -998\)
• Addition: \(-998 + 100 = -898\)

The result -898 is not equal to 110.

Option 3: Swap ÷ and ×

Interchanging '÷' and '×', the equation becomes:

\(20 \times 10 \div 100 - 10 + 100\)

Evaluating this expression:

• Multiplication: \(20 \times 10 = 200\)
• Division: \(200 \div 100 = 2\)
• Subtraction and Addition (left to right): \(2 - 10 = -8\)
• Addition: \(-8 + 100 = 92\)

The result 92 is not equal to 110.

Option 4: Swap + and -

Interchanging '+' and '-', the equation becomes:

\(20 \div 10 \times 100 + 10 - 100\)

Evaluating this expression step-by-step:

• Division: \(20 \div 10 = 2\)
• The equation is now: \(2 \times 100 + 10 - 100\)
• Multiplication: \(2 \times 100 = 200\)
• The equation is now: \(200 + 10 - 100\)
• Addition (left to right): \(200 + 10 = 210\)
• The equation is now: \(210 - 100\)
• Subtraction: \(210 - 100 = 110\)

The result is 110, which matches the right-hand side of the original equation. Thus, interchanging '+' and '-' makes the equation correct.

Conclusion

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

Revision Table: Testing Equation Sign Swaps

Additional Information on Order of Operations

When solving mathematical equations or expressions with multiple operations, it is crucial to follow a specific order. The standard order is often remembered using acronyms like BODMAS or PEMDAS.

• BODMAS: Brackets, Orders (powers and square roots), Division and Multiplication (from left to right), Addition and Subtraction (from left to right).
• PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Adhering to this order ensures that everyone gets the same result when evaluating an expression. In this problem, we applied this rule by performing division and multiplication before addition and subtraction.