Solving Mathematical Equation by Interchanging Signs

This problem asks us to identify which two mathematical signs in the given equation need to be swapped so that the equation becomes correct. We are given the equation \(18 + 24 – 6 \times 6 \div 3 = 39\).

To solve this, we will evaluate the given equation first using the standard order of operations, commonly known as BODMAS or PEMDAS. Then, we will test each given option by interchanging the specified signs and re-evaluating the modified equation to see if it results in 39.

Understanding the Order of Operations (BODMAS/PEMDAS)

The order of operations is crucial for evaluating mathematical expressions consistently:

• Brackets (Parentheses)
• Orders (Exponents, Square Roots, etc.)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

Evaluating the Original Equation

The original equation is: \(18 + 24 – 6 \times 6 \div 3\)

Let's evaluate the left side step-by-step using BODMAS:

• First, handle Division and Multiplication from left to right. Division comes first: \({6 \div 3 = 2}\)
• The expression becomes: \({18 + 24 – 6 \times 2}\)
• Next, Multiplication: \({6 \times 2 = 12}\)
• The expression becomes: \({18 + 24 – 12}\)
• Now, handle Addition and Subtraction from left to right. Addition first: \({18 + 24 = 42}\)
• The expression becomes: \({42 – 12}\)
• Finally, Subtraction: \({42 – 12 = 30}\)

So, the original equation evaluates to 30. Since \(30 \neq 39\), the original equation is incorrect.

Testing Sign Interchange Options to Correct the Equation

Now, we will examine each option and see which interchange makes the equation correct.

Option 1: Interchange ÷ and – signs

If we interchange the ÷ and – signs, the equation becomes: \({18 + 24 \div 6 \times 6 – 3}\)

Let's evaluate this new expression using BODMAS:

• Division: \({24 \div 6 = 4}\)
• The expression becomes: \({18 + 4 \times 6 – 3}\)
• Multiplication: \({4 \times 6 = 24}\)
• The expression becomes: \({18 + 24 – 3}\)
• Addition: \({18 + 24 = 42}\)
• Subtraction: \({42 – 3 = 39}\)

The result is 39, which matches the target value. Therefore, interchanging ÷ and – makes the equation correct.

Option 2: Interchange × and + signs

If we interchange the × and + signs, the equation becomes: \({18 \times 24 – 6 + 6 \div 3}\)

Let's evaluate this new expression:

• Division: \({6 \div 3 = 2}\)
• The expression becomes: \({18 \times 24 – 6 + 2}\)
• Multiplication: \({18 \times 24 = 432}\)
• The expression becomes: \({432 – 6 + 2}\)
• Subtraction: \({432 – 6 = 426}\)
• Addition: \({426 + 2 = 428}\)

The result is 428, which is not equal to 39.

Option 3: Interchange + and – signs

If we interchange the + and – signs, the equation becomes: \({18 – 24 + 6 \times 6 \div 3}\)

Let's evaluate this new expression:

• Division: \({6 \div 3 = 2}\)
• The expression becomes: \({18 – 24 + 6 \times 2}\)
• Multiplication: \({6 \times 2 = 12}\)
• The expression becomes: \({18 – 24 + 12}\)
• Subtraction: \({18 – 24 = -6}\)
• Addition: \({-6 + 12 = 6}\)

The result is 6, which is not equal to 39.

Option 4: Interchange ÷ and + signs

If we interchange the ÷ and + signs, the equation becomes: \({18 \div 24 – 6 \times 6 + 3}\)

Let's evaluate this new expression:

• Division: \({18 \div 24 = \frac{18}{24} = \frac{3}{4} = 0.75}\)
• The expression becomes: \({0.75 – 6 \times 6 + 3}\)
• Multiplication: \({6 \times 6 = 36}\)
• The expression becomes: \({0.75 – 36 + 3}\)
• Subtraction: \({0.75 – 36 = -35.25}\)
• Addition: \({-35.25 + 3 = -32.25}\)

The result is -32.25, which is not equal to 39.

Conclusion: Correct Sign Interchange

After testing all options, we found that interchanging the ÷ and – signs makes the equation evaluate to 39. Therefore, this is the correct pair of signs to interchange.

Revision Table: Order of Operations

Additional Information: Strategies for Operator Problems

When faced with problems requiring operator interchanging to balance an equation, a systematic approach is key. Testing each option methodically, while strictly following the order of operations, ensures accuracy. Pay special attention to the operators like division and multiplication, which have higher priority than addition and subtraction, and remember to work from left to right for operators of the same priority level.