Understanding the Problem: Interchanging Numbers in Equations

The question asks us to examine two mathematical equations and determine which one(s) become correct when the numbers 5 and 9 are interchanged. We are specifically told to interchange the numbers themselves, not just the digits within a number.

To solve this, we need to apply the standard order of mathematical operations, often remembered by the acronyms BODMAS or PEMDAS.

• B/P: Brackets/Parentheses
• O/E: Orders/Exponents
• D/M: Division/Multiplication (from left to right)
• A/S: Addition/Subtraction (from left to right)

We will analyze each equation separately after performing the number interchange.

Analyzing Equation I by Interchanging 5 and 9

The original Equation I is: \(7 \times 9 + 5 - 8 \div 4 = 42\)

Now, let's interchange the numbers 5 and 9. This means every instance of 9 becomes 5, and every instance of 5 becomes 9.

The new Equation I becomes: \(7 \times 5 + 9 - 8 \div 4\)

Let's evaluate the left side of this new equation using the BODMAS/PEMDAS rule:

1. Division: \(8 \div 4 = 2\)
2. Multiplication: \(7 \times 5 = 35\)
3. Addition: \(35 + 9 = 44\)
4. Subtraction: \(44 - 2 = 42\)

So, the left side of the new Equation I evaluates to 42.

The right side of the original equation is 42. Since the new left side (42) equals the right side (42), Equation I is correct after interchanging 5 and 9.

Analyzing Equation II by Interchanging 5 and 9

The original Equation II is: \(9 \times 3 + 5 \div 1 - 4 = 20\)

Now, let's interchange the numbers 5 and 9. Replace 9 with 5 and 5 with 9.

The new Equation II becomes: \(5 \times 3 + 9 \div 1 - 4\)

Let's evaluate the left side of this new equation using the BODMAS/PEMDAS rule:

1. Division: \(9 \div 1 = 9\)
2. Multiplication: \(5 \times 3 = 15\)
3. Addition: \(15 + 9 = 24\)
4. Subtraction: \(24 - 4 = 20\)

So, the left side of the new Equation II evaluates to 20.

The right side of the original equation is 20. Since the new left side (20) equals the right side (20), Equation II is correct after interchanging 5 and 9.

Conclusion: Which Equations are Correct?

Based on our analysis:

• Equation I becomes correct after interchanging 5 and 9.
• Equation II becomes correct after interchanging 5 and 9.

Therefore, both Equation I and Equation II are correct when the numbers 5 and 9 are interchanged.

Revision Table: Interchanging Numbers 5 and 9

Additional Information: Order of Operations (BODMAS/PEMDAS)

The order of operations is crucial in evaluating mathematical expressions to ensure a consistent result. Without a defined order, expressions could have multiple interpretations. The acronyms BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction) and PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) represent this standard order.

• Operations within brackets or parentheses are always performed first.
• Next, evaluate any exponents or orders (like square roots).
• Then, perform multiplication and division. These have equal priority and are done from left to right as they appear in the expression.
• Finally, perform addition and subtraction. These also have equal priority and are done from left to right.

Understanding and correctly applying the order of operations is fundamental to solving mathematical equations accurately.