Understanding the Equation and the Problem

The given equation is: \(95 \times 5 + 19 \div 2 - 21 = 36\).

Our goal is to find which two mathematical signs, when interchanged, will make this equation correct, meaning the left side evaluates to 36.

Analyzing the Original Equation

Let's first evaluate the given equation with the original signs following the order of operations (Division and Multiplication before Addition and Subtraction):

• First, calculate multiplication: \(95 \times 5 = 475\)
• Next, calculate division: \(19 \div 2 = 9.5\)
• Now, perform addition and subtraction from left to right: \(475 + 9.5 - 21\)
• \(475 + 9.5 = 484.5\)
• \(484.5 - 21 = 463.5\)

So, the original equation evaluates to \(463.5 = 36\), which is false.

Testing Sign Interchange Options

We need to test each option provided by swapping the indicated signs in the original equation and re-evaluating it.

Evaluating Option 1: Interchange ÷ and *

If we interchange the division (÷) and multiplication (*) signs, the equation becomes:

\(95 \div 5 + 19 \times 2 - 21 = 36\)

Let's evaluate this new equation:

• First, perform division and multiplication from left to right:
• Division: \(95 \div 5 = 19\)
• Multiplication: \(19 \times 2 = 38\)
• Now, perform addition and subtraction from left to right: \(19 + 38 - 21\)
• Addition: \(19 + 38 = 57\)
• Subtraction: \(57 - 21 = 36\)

This evaluation gives us \(36 = 36\), which is true.

Evaluating Other Options

Let's quickly consider what happens with the other options:

• Option 2: Interchange + and -
  Equation becomes: \(95 \times 5 - 19 \div 2 + 21\). Evaluating this gives \(475 - 9.5 + 21 = 465.5 + 21 = 486.5\). Not 36.
• Option 3: Interchange ÷ and -
  Equation becomes: \(95 \times 5 + 19 - 2 \div 21\). Evaluating this gives \(475 + 19 - (2 \div 21)\). This results in a value close to \(494\), not 36.
• Option 4: Interchange + and *
  Equation becomes: \(95 + 5 \times 19 \div 2 - 21\). Evaluating this gives \(95 + (5 \times 19 \div 2) - 21 = 95 + (95 \div 2) - 21 = 95 + 47.5 - 21 = 142.5 - 21 = 121.5\). Not 36.

Conclusion: Correct Sign Interchange

Based on the evaluation of each option, interchanging the division (÷) and multiplication (*) signs makes the equation correct.

Revision Table: Mathematical Operations

Additional Information: Order of Operations

When evaluating mathematical expressions involving multiple operations, we follow a specific order to ensure a consistent result. A common acronym for this order is BODMAS or PEMDAS.

• BODMAS: Brackets, Orders (powers/roots), Division, Multiplication, Addition, Subtraction.
• PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.

Division and Multiplication have the same priority, and are performed from left to right. Similarly, Addition and Subtraction have the same priority and are performed from left to right after multiplication and division.

In this problem, we applied this order when evaluating the original and modified equations.