Understanding the Equation Interchange Problem

The question asks us to find which pair of numbers and which pair of signs, when swapped in the given equation, will make the equation correct. The original equation is: 4 + 2 ÷ 7 + 8 × 9 = 25.

To solve this, we need to test each of the given options by performing the suggested interchanges and then evaluating the resulting equation according to the order of operations (BODMAS/PEMDAS).

Evaluating Option 1: Swap 4 and 9, + and ×

If we interchange 4 and 9, and the signs + and ×, the original equation 4 + 2 ÷ 7 + 8 × 9 = 25 becomes:

9 × 2 ÷ 7 × 8 + 4 = 25

Let's evaluate the left side:

• First, multiplication/division from left to right: 9 × 2 = 18
• Equation becomes: 18 ÷ 7 × 8 + 4
• Next, 18 ÷ 7. This will result in a fraction or decimal.

Since the right side of the equation is an integer (25), a result involving fractions or decimals on the left side indicates that this interchange does not make the equation correct. So, Option 1 is incorrect.

Evaluating Option 2: Swap 4 and 7, ÷ and ×

If we interchange 4 and 7, and the signs ÷ and ×, the original equation 4 + 2 ÷ 7 + 8 × 9 = 25 becomes:

7 + 2 × 4 + 8 ÷ 9 = 25

Let's evaluate the left side:

• First, multiplication/division from left to right: 2 × 4 = 8
• Equation becomes: 7 + 8 + 8 ÷ 9
• Next, 8 ÷ 9. This will result in a fraction or decimal.

Again, since the right side is an integer (25), a fractional/decimal result on the left side means this interchange is incorrect. So, Option 2 is incorrect.

Evaluating Option 3: Swap 4 and 9, ÷ and ×

If we interchange 4 and 9, and the signs ÷ and ×, the original equation 4 + 2 ÷ 7 + 8 × 9 = 25 becomes:

9 + 2 × 7 + 8 ÷ 4 = 25

Let's evaluate the left side using BODMAS/PEMDAS:

• First, division and multiplication from left to right.
• 2 × 7 = 14
• 8 ÷ 4 = 2
• Equation becomes: 9 + 14 + 2
• Next, addition from left to right: 9 + 14 = 23
• Equation becomes: 23 + 2
• Finally: 23 + 2 = 25

The left side evaluates to 25, which is equal to the right side of the equation. Therefore, this interchange makes the equation correct.

\(9 + 2 \times 7 + 8 \div 4 = 9 + 14 + 2 = 23 + 2 = 25\)

So, Option 3 is the correct answer.

Evaluating Option 4: Swap 4 and 7, + and ×

If we interchange 4 and 7, and the signs + and ×, the original equation 4 + 2 ÷ 7 + 8 × 9 = 25 becomes:

7 × 2 ÷ 4 × 8 + 9 = 25

Let's evaluate the left side:

• First, multiplication/division from left to right: 7 × 2 = 14
• Equation becomes: 14 ÷ 4 × 8 + 9
• Next, 14 ÷ 4 = 3.5
• Equation becomes: 3.5 × 8 + 9
• Next, 3.5 × 8 = 28
• Equation becomes: 28 + 9
• Finally: 28 + 9 = 37

The left side evaluates to 37, which is not equal to 25. So, Option 4 is incorrect.

Conclusion

By testing each option and applying the correct order of operations, we found that interchanging the numbers 4 and 9, and the signs ÷ and × makes the equation correct.

Revision Table: Equation Interchange Summary

Additional Information: Order of Operations (BODMAS/PEMDAS)

When evaluating mathematical expressions, we follow a specific order to ensure consistent results. This order is often remembered using mnemonics like BODMAS or PEMDAS.

• BODMAS: Brackets, Orders (powers, square roots), Division and Multiplication (from left to right), Addition and Subtraction (from left to right).
• PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

In this problem, we used Multiplication and Division before Addition, and we performed operations from left to right when they were at the same level (like Multiplication and Division).

Applying the correct order of operations is crucial in solving these types of equation interchange problems accurately.