Finding the Correct Sign Interchange for the Equation

The problem asks us to identify which pair of arithmetic signs, when swapped in the given equation, will make the equation mathematically correct. The given equation is:

\(17 \times 4 + 6 \div 2 - 27 = 92\)

We need to test each option by interchanging the specified signs and then evaluating the resulting expression using the BODMAS/PEMDAS rule (Brackets/Parentheses, Orders/Exponents, Division and Multiplication (from left to right), Addition and Subtraction (from left to right)).

Analyzing Option 1: Interchanging + and ×

If we interchange + and ×, the equation becomes:

\(17 + 4 \times 6 \div 2 - 27\)

Let's evaluate this expression:

• Division first: \(6 \div 2 = 3\)
• Multiplication next: \(4 \times 3 = 12\)
• Now, Addition and Subtraction from left to right: \(17 + 12 - 27 = 29 - 27 = 2\)

The result is 2, which is not equal to 92. So, interchanging + and × is not the correct solution.

Analyzing Option 2: Interchanging ÷ and ×

If we interchange ÷ and ×, the equation becomes:

\(17 \div 4 + 6 \times 2 - 27\)

Let's evaluate this expression:

• Division first: \(17 \div 4 = 4.25\)
• Multiplication next: \(6 \times 2 = 12\)
• Now, Addition and Subtraction from left to right: \(4.25 + 12 - 27 = 16.25 - 27 = -10.75\)

The result is -10.75, which is not equal to 92. So, interchanging ÷ and × is not the correct solution.

Analyzing Option 3: Interchanging + and -

If we interchange + and -, the equation becomes:

\(17 \times 4 - 6 \div 2 + 27\)

Let's evaluate this expression:

• Multiplication first: \(17 \times 4 = 68\)
• Division next: \(6 \div 2 = 3\)
• Now, Subtraction and Addition from left to right: \(68 - 3 + 27 = 65 + 27 = 92\)

The result is 92, which is equal to the right side of the given equation. So, interchanging + and - makes the equation correct.

Analyzing Option 4: Interchanging ÷ and +

If we interchange ÷ and +, the equation becomes:

\(17 \times 4 \div 6 + 2 - 27\)

Let's evaluate this expression:

• Multiplication/Division from left to right: \(17 \times 4 = 68\), then \(68 \div 6 = 11.33... (\text{approximately})\)
• Now, Addition and Subtraction from left to right: \(11.33... + 2 - 27 = 13.33... - 27 = -13.67... (\text{approximately})\)

The result is approximately -13.67, which is not equal to 92. So, interchanging ÷ and + is not the correct solution.

Conclusion

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

The corrected equation is: \(17 \times 4 - 6 \div 2 + 27 = 92\)

Revision Table: Key Concepts for Sign Interchange Problems

When solving problems involving interchanging signs in equations, remember these points:

• Always follow the order of operations (BODMAS/PEMDAS) strictly after interchanging signs.
• Test each option systematically.
• Be careful with division, especially if it results in fractions or decimals early in the calculation, as it might indicate that option is less likely unless the final answer is not a whole number.
• Double-check your calculations for each step.

Additional Information: Understanding BODMAS/PEMDAS

BODMAS (or PEMDAS) is an acronym used to remember the correct order in which mathematical operations should be performed:

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

Applying this rule consistently is crucial for correctly evaluating mathematical expressions, especially when signs are interchanged.