Solving Mathematical Equations by Interchanging Signs

The problem asks us to find which pair of mathematical signs, when interchanged in the given equation, makes the equation correct. The equation is:

756 ÷ 252 + 89 × 180 - 63 = 384

First, let's evaluate the original equation using the standard order of operations (BODMAS/PEMDAS: Brackets, Orders (powers, roots), Division and Multiplication (left-to-right), Addition and Subtraction (left-to-right)).

Original Equation Evaluation:

• Perform Division: $\frac{756}{252} = 3$
• The equation becomes: $3 + 89 \times 180 - 63$
• Perform Multiplication: $89 \times 180 = 16020$
• The equation becomes: $3 + 16020 - 63$
• Perform Addition: $3 + 16020 = 16023$
• The equation becomes: $16023 - 69$
• Perform Subtraction: $16023 - 63 = 15960$

So, $15960 \neq 384$. The original equation is incorrect. We need to test the given options by interchanging the specified signs and re-evaluating the equation.

Testing Sign Interchange Options

Let's examine the effect of interchanging signs as per the options.

Option 1: Interchanging '+' and '×'

If we interchange the '+' and '×' signs, the equation becomes:

756 ÷ 252 × 89 + 180 - 63

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

• Perform Division (leftmost of division/multiplication): $\frac{756}{252} = 3$
• The equation becomes: $3 \times 89 + 180 - 63$
• Perform Multiplication (next division/multiplication): $3 \times 89 = 267$
• The equation becomes: $267 + 180 - 63$
• Perform Addition (leftmost of addition/subtraction): $267 + 180 = 447$
• The equation becomes: $447 - 63$
• Perform Subtraction: $447 - 63 = 384$

The result of this evaluation is 384, which matches the right-hand side of the given equation. Therefore, interchanging the '+' and '×' signs makes the equation correct.

Let's briefly look at why other options would likely not work (though a full calculation confirms):

• Option 2: ÷ and - would change the equation to $756 - 252 + 89 \times 180 \div 63$. This introduces a division $180 \div 63$ which is not a whole number, making it unlikely to result in a whole number like 384.
• Option 3: and + This option seems incomplete as written. Assuming it means '-' and '+', the equation becomes $756 \div 252 - 89 \times 180 + 63$. This is $3 - 16020 + 63 = -15954$, which is incorrect.
• Option 4: ÷ and + would change the equation to $756 + 252 \div 89 \times 180 - 63$. This involves the division $252 \div 89$, which is not a whole number, again making it unlikely to result in 384.

Confirming the Correct Sign Interchange

Based on our step-by-step evaluation, interchanging the '+' and '×' signs in the equation 756 ÷ 252 + 89 × 180 - 63 = 384 results in the correct equation 756 ÷ 252 × 89 + 180 - 63 = 384, as both sides evaluate to 384.

The correct signs to be interchanged are + and ×.

Revision Table: Order of Operations (BODMAS/PEMDAS)

Additional Information: Strategies for Equation Problems

When tackling problems that require interchanging mathematical signs to balance an equation, follow these steps:

1. Write down the original equation clearly.
2. Evaluate the original equation to confirm it is incorrect.
3. For each given option, rewrite the equation with the specified signs swapped.
4. Carefully re-evaluate the modified equation using the correct order of operations (BODMAS/PEMDAS).
5. If the result matches the value on the right-hand side of the equation, that option is correct.
6. Pay close attention to division, especially if it results in fractions, as the target number is usually an integer.
7. Eliminating options that involve non-integer divisions when the target is an integer can save time.