Understanding the Equation and the Task

The problem asks us to find a specific set of changes (interchanges of signs and numbers) in the given mathematical expression $6 \times 4 + 2 = 16$ so that the Left Hand Side (LHS) becomes equal to the Right Hand Side (RHS). Currently, the LHS of the expression is $6 \times 4 + 2$. Following the order of operations (multiplication before addition), we calculate: $6 \times 4 + 2 = 24 + 2 = 26$. Since $26 \neq 16$, the original expression is false. We need to test the given options by interchanging the specified signs and numbers in the original equation to see which one makes the equation true.

Applying the Correct Interchange

We are given an option that suggests interchanging the multiplication sign ($\times$) with the addition sign ($+$), and the number 4 with the number 6. Let's apply these two changes simultaneously to the original equation $6 \times 4 + 2 = 16$.

Original Equation: $6 \times 4 + 2 = 16$

Interchange $\times$ with $+$:

The expression becomes $6 + 4 \times 2 = 16$. (Note: We only changed the sign here, not the numbers yet)

Now, interchange 4 with 6 in this new expression:

Anywhere there is a 6, replace it with 4. Anywhere there is a 4, replace it with 6.

The expression $6 + 4 \times 2 = 16$ becomes $4 + 6 \times 2 = 16$.

So, after the suggested interchanges, the new expression is $4 + 6 \times 2 = 16$.

Evaluating the New Expression and Verifying Equality

Now, let's evaluate the LHS of the new expression, $4 + 6 \times 2$, using the order of operations (BODMAS/PEMDAS), which states that multiplication should be performed before addition.

First, calculate the multiplication: $6 \times 2 = 12$.

Next, perform the addition: $4 + 12 = 16$.

So, the evaluated LHS is 16.

The RHS of the equation is also 16.

Comparing the LHS and RHS:

LHS $= 16$

RHS $= 16$

Since LHS $=$ RHS ($16 = 16$), the equation is now true after performing the specified interchanges.

This confirms that interchanging $\times$ with $+$ and 4 with 6 makes the original equation $6 \times 4 + 2 = 16$ a correct statement.

Summary of Interchange and Result

Original Equation: $6 \times 4 + 2 = 16$
After Interchanges: $4 + 6 \times 2 = 16$
Evaluation: $4 + 12 = 16$
Result: $16 = 16$ (True)

Revision Table: Key Concepts in Equation Solving

Additional Information: Order of Operations (BODMAS/PEMDAS)

The order of operations is crucial when evaluating mathematical expressions to ensure everyone gets the same result. A commonly used acronym is BODMAS or PEMDAS.

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

In the expression $4 + 6 \times 2$, according to BODMAS/PEMDAS, multiplication ($6 \times 2$) is performed before addition ($4 + \text{result}$). This is why $4 + 6 \times 2$ evaluates to $4 + 12 = 16$, not $(4+6) \times 2 = 10 \times 2 = 20$. Understanding and correctly applying the order of operations is vital for solving mathematical problems involving multiple operations.