Understanding the Sign Interchange Problem

The problem asks us to identify which two mathematical signs in the given equation need to be swapped to make the equation mathematically correct. The original equation is: \( 171 \div 3 – 16 + 72 \times 412 = 572 \). Currently, the left side of the equation does not equal the right side (572).

Method for Testing Sign Interchanges

To solve this, we need to test each option provided. For each option, we will interchange the specified signs in the original equation and then evaluate the left side of the new equation using the BODMAS or PEMDAS rule to see if it equals 572.

Testing Option 1: Interchanging × and –

Let's swap the multiplication sign (\( \times \)) and the subtraction sign (\( – \)) in the original equation.

Original Equation: \( 171 \div 3 – 16 + 72 \times 412 = 572 \)

After interchanging \( \times \) and \( – \), the equation becomes:

\( 171 \div 3 \times 16 + 72 – 412 \)

Now, let's evaluate the left side step-by-step following the BODMAS/PEMDAS order (Division and Multiplication first, then Addition and Subtraction):

• First, perform the division: \( 171 \div 3 = 57 \)
• Next, perform the multiplication: \( 57 \times 16 = 912 \)
• Then, perform the addition: \( 912 + 72 = 984 \)
• Finally, perform the subtraction: \( 984 – 412 = 572 \)

So, the left side evaluates to 572. The equation becomes \( 572 = 572 \). This is a correct equality.

Testing Other Options (for completeness)

Although we found the correct interchange in Option 1, let's quickly consider why other options would not yield the correct result.

Testing Option 2: Interchanging ÷ and –

Swap \( \div \) and \( – \): \( 171 – 3 \div 16 + 72 \times 412 \). Evaluating this would involve \( 3 \div 16 \) which is a fraction/decimal, and a large multiplication \( 72 \times 412 \). The result would not be the integer 572.

Testing Option 3: Interchanging ÷ and +

Swap \( \div \) and \( +\): \( 171 + 3 – 16 \div 72 \times 412 \). This involves \( 16 \div 72 \) which is a fraction/decimal, making the final result unlikely to be 572.

Testing Option 4: Interchanging – and +

Swap \( – \) and \( +\): \( 171 \div 3 + 16 – 72 \times 412 \). Evaluating this gives \( 57 + 16 – 29664 \), which is a large negative number, not 572.

Conclusion

Based on the step-by-step evaluation of the equation after interchanging signs, we find that only interchanging the \( \times \) and \( – \) signs results in a correct mathematical equation.

Revision Table: Understanding Arithmetic Operators

Additional Information: Importance of BODMAS/PEMDAS

The order of operations is a fundamental concept in arithmetic. It ensures that any mathematical expression is evaluated consistently, leading to a single correct answer. The acronyms BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) or PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) help remember this order.

• Operations within brackets or parentheses are always performed first.
• Next come powers or exponents, and roots.
• Then, multiplication and division are performed from left to right as they appear.
• Finally, addition and subtraction are performed from left to right as they appear.

Understanding and applying this rule is critical for solving equations and expressions correctly, especially in problems that involve rearranging parts of the equation like sign interchanging.