Solving Mathematical Expressions by Interchanging Signs

The problem requires us to evaluate two mathematical expressions after interchanging the signs '×' (multiplication) and '+' (addition).

We need to apply the rule of BODMAS/PEDMAS (Brackets, Orders/Exponents, Division and Multiplication (from left to right), Addition and Subtraction (from left to right)) when evaluating the expressions.

Evaluating Expression I

The original Expression I is:
\(5 \times 12 - 15 + 20 \div 25\)

After interchanging '×' and '+', the expression becomes:
\(5 + 12 - 15 \times 20 \div 25\)

Now, let's evaluate this modified expression following the BODMAS/PEDMAS rule:

1. Division: Evaluate \(20 \div 25\).
   \(20 \div 25 = \frac{20}{25} = \frac{4}{5} = 0.8\)
2. The expression is now: \(5 + 12 - 15 \times 0.8\)
3. Multiplication: Evaluate \(15 \times 0.8\).
   \(15 \times 0.8 = 12\)
4. The expression is now: \(5 + 12 - 12\)
5. Addition and Subtraction: Perform addition and subtraction from left to right.
   \(5 + 12 = 17\)
   \(17 - 12 = 5\)

The value of Expression I after interchanging signs is 5.

Evaluating Expression II

The original Expression II is:
\(36 + 3 \times 2 - 15 \div 5\)

After interchanging '+' and '×', the expression becomes:
\(36 \times 3 + 2 - 15 \div 5\)

Now, let's evaluate this modified expression following the BODMAS/PEDMAS rule:

1. Division: Evaluate \(15 \div 5\).
   \(15 \div 5 = 3\)
2. The expression is now: \(36 \times 3 + 2 - 3\)
3. Multiplication: Evaluate \(36 \times 3\).
   \(36 \times 3 = 108\)
4. The expression is now: \(108 + 2 - 3\)
5. Addition and Subtraction: Perform addition and subtraction from left to right.
   \(108 + 2 = 110\)
   \(110 - 3 = 107\)

The value of Expression II after interchanging signs is 107.

So, after interchanging the signs '×' and '+', the values of expression (I) and (II) are 5 and 107 respectively.

The values are 5 and 107.

Revision Table: Order of Operations (BODMAS/PEDMAS)

Additional Information: Significance of Order of Operations

The order of operations is crucial in mathematics to ensure that everyone gets the same result when evaluating an expression. Without a standard order, an expression could have multiple possible values, leading to confusion and errors. BODMAS or PEDMAS provides a consistent method for solving mathematical problems involving multiple operations. When operations of the same priority level appear (like multiplication and division, or addition and subtraction), they should be performed from left to right.