The question asks us to find which two mathematical signs, when interchanged in the given equation, make the equation mathematically correct. The original equation is:

\(4 \times 5 - 24 \div 12 + 8 = 14\)

Let's evaluate the original equation using the BODMAS/PEMDAS rule (Brackets, Orders/Exponents, Division/Multiplication from left to right, Addition/Subtraction from left to right).

First, perform the multiplication and division:

\(4 \times 5 = 20\)

\(24 \div 12 = 2\)

Substitute these values back into the equation:

\(20 - 2 + 8\)

Now, perform the subtraction and addition from left to right:

\(20 - 2 = 18\)

\(18 + 8 = 26\)

So, the original equation evaluates to \(26\), which is not equal to \(14\). Therefore, the original equation is incorrect, and we need to interchange signs.

Testing Sign Interchange Options

We will now test each option by interchanging the specified signs and re-evaluating the equation using BODMAS/PEMDAS to see which interchange makes the equation equal to 14.

Option 1: Interchange + and - signs

Swap the '-' sign with the '+' sign in the original equation. The new equation becomes:

\(4 \times 5 + 24 \div 12 - 8\)

Now, evaluate this equation:

1. First, Multiplication and Division:
2. \(4 \times 5 = 20\)
3. \(24 \div 12 = 2\)
4. Substitute back: \(20 + 2 - 8\)
5. Now, Addition and Subtraction from left to right:
6. \(20 + 2 = 22\)
7. \(22 - 8 = 14\)

The result is \(14\), which matches the right side of the given equation. So, interchanging '+' and '-' signs makes the equation correct.

Option 2: Interchange ÷ and × signs

Swap the '×' sign with the '÷' sign. The new equation becomes:

\(4 \div 5 - 24 \times 12 + 8\)

Evaluate this equation:

1. First, Division and Multiplication:
2. \(4 \div 5 = 0.8\)
3. \(24 \times 12 = 288\)
4. Substitute back: \(0.8 - 288 + 8\)
5. Now, Subtraction and Addition:
6. \(0.8 - 288 = -287.2\)
7. \(-287.2 + 8 = -279.2\)

The result is \(-279.2\), which is not equal to \(14\).

Option 3: Interchange × and + signs

Swap the '×' sign with the '+' sign. The new equation becomes:

\(4 + 5 - 24 \div 12 \times 8\)

Evaluate this equation:

1. First, Division and Multiplication:
2. \(24 \div 12 = 2\)
3. \(2 \times 8 = 16\)
4. Substitute back: \(4 + 5 - 16\)
5. Now, Addition and Subtraction:
6. \(4 + 5 = 9\)
7. \(9 - 16 = -7\)

The result is \(-7\), which is not equal to \(14\).

Option 4: Interchange + and ÷ signs

Swap the '+' sign with the '÷' sign. The new equation becomes:

\(4 \times 5 - 24 + 12 \div 8\)

Evaluate this equation:

1. First, Multiplication and Division:
2. \(4 \times 5 = 20\)
3. \(12 \div 8 = 1.5\)
4. Substitute back: \(20 - 24 + 1.5\)
5. Now, Subtraction and Addition:
6. \(20 - 24 = -4\)
7. \(-4 + 1.5 = -2.5\)

The result is \(-2.5\), which is not equal to \(14\).

Based on the evaluation of all options, interchanging the '+' and '-' signs makes the equation correct.

Revision Table: Evaluating Options for Sign Interchange

Additional Information on BODMAS/PEMDAS

The BODMAS or PEMDAS rule is crucial for evaluating mathematical expressions correctly. It dictates the order in which operations should be performed:

• B / P: Brackets (Parentheses) - Operations inside brackets are performed first.
• O / E: Orders (Exponents) - Powers and square roots are evaluated next.
• D / M: Division and Multiplication - These are performed from left to right as they appear. They have equal priority.
• A / S: Addition and Subtraction - These are performed from left to right as they appear. They have equal priority.

Understanding and applying this order is essential for solving problems involving multiple operations, like the sign interchange problem we solved.