Finding the Correct Sign Interchange for the Equation

The problem asks us to determine which pair of mathematical signs, when swapped in the given equation, makes the equation true. The equation provided is: $$108 \div 2 * 10 + 70 - 10 = 480$$

To solve this type of problem, we need to test each option by interchanging the specified signs and then evaluating the resulting expression using the correct order of operations (BODMAS/PEMDAS: Brackets, Orders, Division and Multiplication (from left to right), Addition and Subtraction (from left to right)).

Testing Option 1: Interchange + and –

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

$$108 \div 2 * 10 - 70 + 10$$

Now, let's evaluate this expression step-by-step following BODMAS:

1. Division: $$108 \div 2 = 54$$ The expression becomes: $$54 * 10 - 70 + 10$$
2. Multiplication: $$54 * 10 = 540$$ The expression becomes: $$540 - 70 + 10$$
3. Subtraction and Addition (from left to right):
   • $$540 - 70 = 470$$ The expression becomes: $$470 + 10$$
   • $$470 + 10 = 480$$

The result is 480, which matches the right side of the original equation (480). Therefore, interchanging '+' and '–' makes the equation correct.

Testing Other Options

Although we found the correct interchange, let's quickly test the other options to demonstrate why they are incorrect.

Testing Option 2: Interchange ÷ and *

If we interchange '÷' and '*', the equation becomes:

$$108 * 2 \div 10 + 70 - 10$$

Evaluating using BODMAS:

1. Multiplication/Division (from left to right):
   • $$108 * 2 = 216$$ The expression becomes: $$216 \div 10 + 70 - 10$$
   • $$216 \div 10 = 21.6$$ The expression becomes: $$21.6 + 70 - 10$$
2. Addition/Subtraction (from left to right):
   • $$21.6 + 70 = 91.6$$ The expression becomes: $$91.6 - 10$$
   • $$91.6 - 10 = 81.6$$

The result is 81.6, which is not equal to 480.

Testing Option 3: Interchange + and *

If we interchange '+' and '*', the equation becomes:

$$108 \div 2 + 10 * 70 - 10$$

Evaluating using BODMAS:

1. Division: $$108 \div 2 = 54$$ The expression becomes: $$54 + 10 * 70 - 10$$
2. Multiplication: $$10 * 70 = 700$$ The expression becomes: $$54 + 700 - 10$$
3. Addition/Subtraction (from left to right):
   • $$54 + 700 = 754$$ The expression becomes: $$754 - 10$$
   • $$754 - 10 = 744$$

The result is 744, which is not equal to 480.

Testing Option 4: Interchange ÷ and –

If we interchange '÷' and '–', the equation becomes:

$$108 - 2 * 10 + 70 \div 10$$

Evaluating using BODMAS:

1. Multiplication/Division (from left to right):
   • $$2 * 10 = 20$$ The expression becomes: $$108 - 20 + 70 \div 10$$
   • $$70 \div 10 = 7$$ The expression becomes: $$108 - 20 + 7$$
2. Subtraction/Addition (from left to right):
   • $$108 - 20 = 88$$ The expression becomes: $$88 + 7$$
   • $$88 + 7 = 95$$

The result is 95, which is not equal to 480.

Based on the evaluation of each option, interchanging the '+' and '–' signs is the only swap that makes the equation correct.

Revision Table: Equation Sign Interchange

Additional Information on Order of Operations

The order of operations is a rule that tells us the sequence in which to perform mathematical operations in an expression. It is often remembered using acronyms like 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).

Following this order ensures that everyone evaluates the same expression in the same way, leading to a consistent result. In the problem above, we used Division and Multiplication before Addition and Subtraction, and handled operations of the same priority (like multiplication and division, or addition and subtraction) from left to right.