Understanding the Mathematical Problem

The question asks us to find which pair of mathematical signs, when interchanged in the given equation, will make the equation correct. The initial equation is: $3 \times 12 - 6 \div 2 + 12 = 16$. Our goal is to perform sign interchanges as suggested by the options and evaluate the resulting equation to see if it equals 16.

Evaluating the Original Equation

Let's first evaluate the given equation as it is, following the order of operations (BODMAS/PEMDAS).

The order of operations is:

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

Original expression: $3 \times 12 - 6 \div 2 + 12$

Perform multiplication and division from left to right:

• $3 \times 12 = 36$
• $6 \div 2 = 3$

The expression becomes: $36 - 3 + 12$

Perform addition and subtraction from left to right:

• $36 - 3 = 33$
• $33 + 12 = 45$

So, $3 \times 12 - 6 \div 2 + 12 = 45$. Since $45 \neq 16$, the original equation is incorrect.

Testing Options for Sign Interchange

Now, let's test each given option by interchanging the specified signs and evaluating the new expression.

Testing Option 1: Interchange $\div$ and $+$

If we interchange $\div$ and $+$ signs, the equation becomes:

$3 \times 12 - 6 + 2 \div 12 = 16$

Let's evaluate the left side using BODMAS:

$3 \times 12 - 6 + 2 \div 12$

Perform multiplication and division:

• $3 \times 12 = 36$
• $2 \div 12 = \frac{2}{12} = \frac{1}{6}$

The expression becomes: $36 - 6 + \frac{1}{6}$

Perform subtraction and addition:

• $36 - 6 = 30$
• $30 + \frac{1}{6} = 30\frac{1}{6}$

So, $3 \times 12 - 6 + 2 \div 12 = 30\frac{1}{6}$. Since $30\frac{1}{6} \neq 16$, this option is incorrect.

Testing Option 2: Interchange $\times$ and $\div$

If we interchange $\times$ and $\div$ signs, the equation becomes:

$3 \div 12 - 6 \times 2 + 12 = 16$

Let's evaluate the left side using BODMAS:

$3 \div 12 - 6 \times 2 + 12$

Perform multiplication and division:

• $3 \div 12 = \frac{3}{12} = \frac{1}{4}$
• $6 \times 2 = 12$

The expression becomes: $\frac{1}{4} - 12 + 12$

Perform subtraction and addition:

• $\frac{1}{4} - 12 = -11\frac{3}{4}$
• $-11\frac{3}{4} + 12 = \frac{1}{4}$

So, $3 \div 12 - 6 \times 2 + 12 = \frac{1}{4}$. Since $\frac{1}{4} \neq 16$, this option is incorrect.

Testing Option 3: Interchange $-$ and $\div$

If we interchange $-$ and $\div$ signs, the equation becomes:

$3 \times 12 \div 6 - 2 + 12 = 16$

Let's evaluate the left side using BODMAS:

$3 \times 12 \div 6 - 2 + 12$

Perform multiplication and division from left to right:

• $3 \times 12 = 36$
• $36 \div 6 = 6$

The expression becomes: $6 - 2 + 12$

Perform subtraction and addition from left to right:

• $6 - 2 = 4$
• $4 + 12 = 16$

So, $3 \times 12 \div 6 - 2 + 12 = 16$. This matches the value on the right side of the original equation. Thus, interchanging the $-$ and $\div$ signs makes the equation correct.

Testing Option 4: Interchange $-$ and $+$

If we interchange $-$ and $+$ signs, the equation becomes:

$3 \times 12 + 6 \div 2 - 12 = 16$

Let's evaluate the left side using BODMAS:

$3 \times 12 + 6 \div 2 - 12$

Perform multiplication and division:

• $3 \times 12 = 36$
• $6 \div 2 = 3$

The expression becomes: $36 + 3 - 12$

Perform addition and subtraction from left to right:

• $36 + 3 = 39$
• $39 - 12 = 27$

So, $3 \times 12 + 6 \div 2 - 12 = 27$. Since $27 \neq 16$, this option is incorrect.

Conclusion on Sign Interchange

After testing all options, we found that interchanging the $-$ and $\div$ signs makes the equation $3 \times 12 - 6 \div 2 + 12 = 16$ correct.

Revision Table: Testing Sign Swaps

Additional Information: Order of Operations (BODMAS/PEMDAS)

The order of operations is a rule used to clarify which operations should be performed first in a mathematical expression. This ensures that everyone gets the same answer when evaluating an expression.

• BODMAS stands for:
  • Brackets
  • Orders (powers, square roots, etc.)
  • Division and Multiplication (from left to right)
  • Addition and Subtraction (from left to right)
• PEMDAS is another common acronym and stands for:
  • Parentheses
  • Exponents
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)

Both acronyms represent the same order of operations. The key is to remember that division/multiplication are at the same level and should be done from left to right as they appear, and similarly, addition/subtraction are at the same level and done from left to right.

Understanding and correctly applying the order of operations is crucial for solving mathematical problems involving multiple operations, like the sign interchange problem we just solved.