Solving Equation by Interchanging Signs

The problem asks us to find which pair of mathematical signs in the equation $48 – 8 ÷ 4 + 5 × 6 = 32$ needs to be interchanged to make the equation correct. Currently, the equation is incorrect.

Evaluating the Original Equation

Let's first calculate the value of the left side of the original equation using the standard order of operations (BODMAS/PEMDAS):

The equation is: $48 – 8 ÷ 4 + 5 × 6$

• First, perform division and multiplication from left to right:
• $8 ÷ 4 = 2$
• $5 × 6 = 30$
• Now substitute these values back into the equation:
• $48 – 2 + 30$
• Next, perform addition and subtraction from left to right:
• $48 – 2 = 46$
• $46 + 30 = 76$

So, the original equation $48 – 8 ÷ 4 + 5 × 6$ evaluates to $76$. Since $76 \neq 32$, the equation is indeed incorrect.

Testing Sign Interchanges to Correct the Equation

We need to test the given options by interchanging the specified signs and then evaluating the new equation to see if it equals 32.

Interchanging ÷ and – Signs

Let's test the option where the ÷ and – signs are interchanged. The original equation is $48 – 8 ÷ 4 + 5 × 6 = 32$.

Interchanging – and ÷, the new equation becomes:

$48 ÷ 8 – 4 + 5 × 6$

Now, let's evaluate this new equation using the order of operations:

• First, perform division and multiplication from left to right:
• $48 ÷ 8 = 6$
• $5 × 6 = 30$
• Substitute these values into the equation:
• $6 – 4 + 30$
• Next, perform addition and subtraction from left to right:
• $6 – 4 = 2$
• $2 + 30 = 32$

The new equation evaluates to $32$. This matches the right side of the original equation ($32$). Therefore, interchanging the ÷ and – signs makes the equation $48 \div 8 – 4 + 5 \times 6 = 32$ correct.

Let's briefly look at why other options would not work (though we have found the correct one):

• Interchanging × and +: $48 – 8 ÷ 4 × 5 + 6$. This evaluates to $48 - 2 \times 5 + 6 = 48 - 10 + 6 = 38 + 6 = 44$. $44 \neq 32$.
• Interchanging × and ÷: $48 – 8 × 4 + 5 ÷ 6$. $5 \div 6$ is not a whole number, likely not intended for this type of problem, but would be $48 - 32 + 0.833... \approx 16.833... \neq 32$.
• Interchanging ÷ and +: $48 – 8 + 4 ÷ 5 × 6$. This involves $4 \div 5$, similar to the previous case, resulting in $48 - 8 + 0.8 \times 6 = 40 + 4.8 = 44.8 \neq 32$.

Based on our evaluation, only interchanging ÷ and – makes the equation correct.

The correct option is interchanging ÷ and –.

Revision Table: Understanding Operators

Additional Information: Order of Operations (BODMAS/PEMDAS)

To correctly evaluate mathematical expressions involving multiple operations, we follow a specific order. This is commonly known as BODMAS or PEMDAS.

• B/P: Brackets (Parentheses) - Solve expressions inside brackets first.
• O/E: Orders (Exponents) - Solve powers and roots next.
• DM: Division and Multiplication - Perform these operations from left to right.
• AS: Addition and Subtraction - Perform these operations from left to right.

Understanding the order of operations is crucial for solving equations correctly, especially when interchanging signs changes the sequence of calculations.