Understanding the Number Interchange Problem

The question asks us to consider two given equations. We need to interchange the numbers 7 and 4 in both equations. After performing this interchange, we must evaluate each modified equation to see if it remains correct (i.e., if the left side equals the right side). The goal is to find which of the two equations will not be correct after the numbers 7 and 4 are interchanged.

When we interchange 7 and 4, every instance of the number 7 in an equation will be replaced by 4, and every instance of the number 4 will be replaced by 7. Other numbers and mathematical operators remain unchanged.

Analyzing Equation I with Number Interchange

The original Equation I is: $4 \times 3 - 7 + 8 \div 2 = 23$

Now, let's interchange the numbers 7 and 4 in this equation:

• The number 4 becomes 7.
• The number 7 becomes 4.

The modified Equation I becomes: $7 \times 3 - 4 + 8 \div 2 = 23$

Let's evaluate the left side of this modified equation following the order of operations (BODMAS/PEMDAS - Brackets, Orders, Division/Multiplication, Addition/Subtraction):

1. Perform Multiplication and Division from left to right:
2. $7 \times 3 = 21$
3. $8 \div 2 = 4$
4. The equation becomes: $21 - 4 + 4$
5. Perform Addition and Subtraction from left to right:
6. $21 - 4 = 17$
7. $17 + 4 = 21$

So, after interchanging 7 and 4, the left side of Equation I evaluates to 21. The equation becomes: $21 = 23$.

This equation is not correct because 21 is not equal to 23.

Analyzing Equation II with Number Interchange

The original Equation II is: $7 \times 5 - 9 \div 3 + 4 = 24$

Now, let's interchange the numbers 7 and 4 in this equation:

• The number 7 becomes 4.
• The number 4 becomes 7.

The modified Equation II becomes: $4 \times 5 - 9 \div 3 + 7 = 24$

Let's evaluate the left side of this modified equation following the order of operations (BODMAS/PEMDAS):

1. Perform Multiplication and Division from left to right:
2. $4 \times 5 = 20$
3. $9 \div 3 = 3$
4. The equation becomes: $20 - 3 + 7$
5. Perform Addition and Subtraction from left to right:
6. $20 - 3 = 17$
7. $17 + 7 = 24$

So, after interchanging 7 and 4, the left side of Equation II evaluates to 24. The equation becomes: $24 = 24$.

This equation is correct because 24 is equal to 24.

Conclusion on Equation Correctness After Interchange

After interchanging the numbers 7 and 4:

• Equation I ($7 \times 3 - 4 + 8 \div 2 = 23$) became $21 = 23$, which is incorrect.
• Equation II ($4 \times 5 - 9 \div 3 + 7 = 24$) became $24 = 24$, which is correct.

The question asks which of the following equations will not be correct after interchanging the given two numbers. Based on our analysis, only Equation I is not correct after the interchange.

Therefore, only Equation I will not be correct after interchanging 7 and 4.

Revision Table: Key Concepts

Additional Information: Applying Order of Operations

When evaluating expressions involving multiple operations, strictly following the order of operations is essential to get the correct result. Let's quickly review the order:

1. Brackets/Parentheses: Evaluate anything inside brackets first.
2. Orders/Exponents: Calculate powers or roots next.
3. Division and Multiplication: Perform these operations from left to right as they appear.
4. Addition and Subtraction: Perform these operations from left to right as they appear.

In this problem, both equations involved multiplication, division, addition, and subtraction. We first handled the multiplication and division parts, and then the addition and subtraction parts, moving from left to right at each stage.

For Equation I (modified): $7 \times 3 - 4 + 8 \div 2$

• $7 \times 3 = 21$ (Multiplication)
• $8 \div 2 = 4$ (Division)
• Expression becomes: $21 - 4 + 4$
• $21 - 4 = 17$ (Subtraction)
• $17 + 4 = 21$ (Addition)

For Equation II (modified): $4 \times 5 - 9 \div 3 + 7$

• $4 \times 5 = 20$ (Multiplication)
• $9 \div 3 = 3$ (Division)
• Expression becomes: $20 - 3 + 7$
• $20 - 3 = 17$ (Subtraction)
• $17 + 7 = 24$ (Addition)

This systematic approach ensures accuracy when dealing with such mathematical problems.