Analyzing the Equation to Find Correct Sign Interchange

The problem asks us to identify which two mathematical signs in the given equation must be swapped to make the equation mathematically correct. The given equation is:

\(55 \div 5 + 15 \times 45 – 30 = 180\)

First, let's evaluate the original equation following the order of operations (BODMAS/PEMDAS):

• Division: \(55 \div 5 = 11\)
• Multiplication: \(15 \times 45 = 675\)
• Addition and Subtraction: \(11 + 675 – 30 = 686 – 30 = 656\)

So, the original equation evaluates to \(656 = 180\), which is incorrect. We need to test each given option by interchanging the specified signs and re-evaluating the equation.

Testing Sign Interchange Options

Option 1: Interchange \( \div \) and \( \times \)

If we swap the division and multiplication signs, the equation becomes:

\(55 \times 5 + 15 \div 45 – 30\)

Let's evaluate this new expression:

• Multiplication: \(55 \times 5 = 275\)
• Division: \(15 \div 45 = \frac{15}{45} = \frac{1}{3}\)
• Addition and Subtraction: \(275 + \frac{1}{3} – 30 = (275 – 30) + \frac{1}{3} = 245 + \frac{1}{3} = \frac{735+1}{3} = \frac{736}{3}\)

The equation becomes \(\frac{736}{3} = 180\), which is not correct.

Option 2: Interchange \( \times \) and \( + \)

If we swap the multiplication and addition signs, the equation becomes:

\(55 \div 5 \times 15 + 45 – 30\)

Let's evaluate this new expression:

• Division: \(55 \div 5 = 11\)
• Multiplication: \(11 \times 15 = 165\)
• Addition and Subtraction: \(165 + 45 – 30 = 210 – 30 = 180\)

The equation becomes \(180 = 180\), which is correct.

Option 3: Interchange \( \div \) and \( + \)

If we swap the division and addition signs, the equation becomes:

\(55 + 5 \div 15 \times 45 – 30\)

Let's evaluate this new expression:

• Division: \(5 \div 15 = \frac{5}{15} = \frac{1}{3}\)
• Multiplication: \(\frac{1}{3} \times 45 = 15\)
• Addition and Subtraction: \(55 + 15 – 30 = 70 – 30 = 40\)

The equation becomes \(40 = 180\), which is not correct.

Option 4: Interchange \( \times \) and \( - \)

If we swap the multiplication and subtraction signs, the equation becomes:

\(55 \div 5 + 15 – 45 \times 30\)

Let's evaluate this new expression:

• Division: \(55 \div 5 = 11\)
• Multiplication: \(45 \times 30 = 1350\)
• Addition and Subtraction: \(11 + 15 – 1350 = 26 – 1350 = -1324\)

The equation becomes \(-1324 = 180\), which is not correct.

Conclusion on Sign Interchange

By testing each option, we found that interchanging the multiplication (\( \times \)) and addition (\( + \)) signs makes the given equation correct. The resulting correct equation is \(55 \div 5 \times 15 + 45 – 30 = 180\).

Revision Table: Testing Each Interchange

Additional Information on Order of Operations

When solving mathematical expressions, it is crucial to follow the correct order of operations. This is commonly remembered using acronyms like BODMAS or PEMDAS.

• BODMAS: Brackets, Orders (powers/roots), Division, Multiplication, Addition, Subtraction.
• PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.

Multiplication and Division have the same priority and are performed from left to right. Similarly, Addition and Subtraction have the same priority and are performed from left to right after multiplication and division. In this problem, applying the correct order of operations is essential to evaluate the equation after interchanging signs.