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

\(18 \ndash 60 \div 45 + 6 \times 128 = 138\)

We need to check each option by interchanging the specified signs and then evaluating the resulting expression using the BODMAS (or PEMDAS) rule to see if it equals 138.

Analyzing Sign Interchange Options

Option 1: Interchange + and ×

Let's swap the addition (+) and multiplication (×) signs in the original equation.

Original: \(18 \ndash 60 \div 45 + 6 \times 128\)

After swapping + and ×:

\(18 \ndash 60 \div 45 \times 6 + 128\)

Now, let's evaluate this expression following the BODMAS rule:

• Brackets (None)
• Orders (None)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

Evaluation Steps:

1. Perform Division: \(60 \div 45 = \frac{60}{45} = \frac{4}{3}\)
2. The expression becomes: \(18 \ndash \frac{4}{3} \times 6 + 128\)
3. Perform Multiplication: \(\frac{4}{3} \times 6 = \frac{24}{3} = 8\)
4. The expression becomes: \(18 \ndash 8 + 128\)
5. Perform Subtraction: \(18 \ndash 8 = 10\)
6. The expression becomes: \(10 + 128\)
7. Perform Addition: \(10 + 128 = 138\)

The result of the expression after swapping + and × is 138, which matches the right-hand side of the original equation. Thus, interchanging + and × makes the equation correct.

Option 2: Interchange + and ÷

Let's swap the addition (+) and division (÷) signs.

Original: \(18 \ndash 60 \div 45 + 6 \times 128\)

After swapping + and ÷:

\(18 \ndash 60 + 45 \div 6 \times 128\)

Evaluation Steps (BODMAS):

1. Perform Division: \(45 \div 6 = \frac{45}{6} = \frac{15}{2} = 7.5\)
2. The expression becomes: \(18 \ndash 60 + 7.5 \times 128\)
3. Perform Multiplication: \(7.5 \times 128 = 960\)
4. The expression becomes: \(18 \ndash 60 + 960\)
5. Perform Subtraction: \(18 \ndash 60 = -42\)
6. The expression becomes: \(-42 + 960\)
7. Perform Addition: \(-42 + 960 = 918\)

The result 918 does not equal 138. So, this swap is incorrect.

Option 3: Interchange × and -

Let's swap the multiplication (×) and subtraction (-) signs.

Original: \(18 \ndash 60 \div 45 + 6 \times 128\)

After swapping × and -:

\(18 \times 60 \div 45 + 6 \ndash 128\)

Evaluation Steps (BODMAS):

1. Perform Division: \(60 \div 45 = \frac{4}{3}\)
2. The expression becomes: \(18 \times \frac{4}{3} + 6 \ndash 128\)
3. Perform Multiplication: \(18 \times \frac{4}{3} = 6 \times 4 = 24\)
4. The expression becomes: \(24 + 6 \ndash 128\)
5. Perform Addition: \(24 + 6 = 30\)
6. The expression becomes: \(30 \ndash 128\)
7. Perform Subtraction: \(30 \ndash 128 = -98\)

The result -98 does not equal 138. So, this swap is incorrect.

Option 4: Interchange - and =

Interchanging an operation sign with the equality sign (=) does not fit the structure of a standard equation where one side is evaluated to equal the other side. This option is not relevant to making the equation correct by changing operations.

Summary of Results

Based on the evaluations, only interchanging the + and × signs results in the equation being correct, yielding 138.

Revision Table: Key Concepts in Solving Equation Problems

Additional Information: Understanding BODMAS/PEMDAS for Accurate Calculations

The order of operations is fundamental in arithmetic and algebra. It ensures that everyone gets the same result when evaluating a given expression. The acronym BODMAS or PEMDAS helps remember the order:

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

In our problem, we strictly followed this order. First, division \(60 \div 45\), then multiplication of that result by 6, and finally the subtraction of this product from 18, followed by the addition of 128.

It's important to perform multiplication and division, and addition and subtraction, from left to right if they appear consecutively in the expression.