Understanding the Equation Interchange Problem

The problem asks us to find which pair of numbers and which pair of mathematical signs, when swapped in the given equation, will make the equation mathematically correct. The original equation is:

\(264 + 24 \times 9 \div 60 - 4 = 630\)

We need to test each option provided by performing the suggested interchanges and then evaluating the resulting equation using the standard order of operations (BODMAS/PEDMAS).

Evaluating Each Interchange Option

Let's carefully examine each option and see if the interchange makes the equation true.

Option 1: Interchange 264 and 9, $\div$ and $-$

If we interchange 264 and 9, and the signs $\div$ and $-$, the new equation becomes:

\(9 + 24 \times 264 - 60 \div 4 = 630\)

Let's evaluate this using BODMAS (Brackets, Orders, Division/Multiplication, Addition/Subtraction):

• Division: \(60 \div 4 = 15\)
• Multiplication: \(24 \times 264 = 6336\)
• Addition: \(9 + 6336 = 6345\)
• Subtraction: \(6345 - 15 = 6330\)

So, \(9 + 24 \times 264 - 60 \div 4 = 6330\). Since \(6330 \neq 630\), this option is incorrect.

Option 2: Interchange 264 and 9, + and $-$

If we interchange 264 and 9, and the signs + and $-$, the new equation becomes:

\(9 - 24 \times 60 \div 264 + 4 = 630\)

Let's evaluate this:

• Division: \(60 \div 264 \approx 0.227\)
• Multiplication: \(24 \times 0.227 \approx 5.45\)
• Subtraction: \(9 - 5.45 \approx 3.55\)
• Addition: \(3.55 + 4 = 7.55\)

So, the left side is approximately 7.55. Since \(7.55 \neq 630\), this option is incorrect.

(Note: Even with exact fractions, the result will not be 630)

• Division: \(60 \div 264 = 60/264 = 5/22\)
• Multiplication: \(24 \times (5/22) = 120/22 = 60/11\)
• Subtraction: \(9 - 60/11 = (99 - 60)/11 = 39/11\)
• Addition: \(39/11 + 4 = (39 + 44)/11 = 83/11 \approx 7.54\)

Option 3: Interchange 4 and 24, $\div$ and +

If we interchange the numbers 4 and 24, and the signs $\div$ and +, the new equation becomes:

\(264 \div 4 \times 9 + 60 - 24 = 630\)

Let's evaluate this using the BODMAS rule:

• Division: Perform division first. \(264 \div 4 = 66\). The equation is now \(66 \times 9 + 60 - 24 = 630\).
• Multiplication: Next, perform multiplication. \(66 \times 9 = 594\). The equation is now \(594 + 60 - 24 = 630\).
• Addition: Perform addition from left to right. \(594 + 60 = 654\). The equation is now \(654 - 24 = 630\).
• Subtraction: Finally, perform subtraction. \(654 - 24 = 630\).

The left side of the equation is 630, which is equal to the right side (630). Thus, the equation becomes correct after this interchange.

\(630 = 630\)

This option makes the equation correct.

Option 4: Interchange 4 and 9, $-$ and $\times$

If we interchange the numbers 4 and 9, and the signs $-$ and $\times$, the new equation becomes:

\(264 + 24 - 4 \div 60 \times 9 = 630\)

Let's evaluate this:

• Division: \(4 \div 60 = 4/60 = 1/15\)
• Multiplication: \((1/15) \times 9 = 9/15 = 3/5 = 0.6\)
• Addition/Subtraction (from left to right): \(264 + 24 = 288\)
• Subtraction: \(288 - 0.6 = 287.4\)

So, the left side is 287.4. Since \(287.4 \neq 630\), this option is incorrect.

Conclusion on Interchange

Based on our evaluation, interchanging the numbers 4 and 24, and the signs $\div$ and + makes the original equation correct.

Revision Table: Key Concepts

Additional Information on BODMAS/PEDMAS

The order of operations is essential for solving mathematical expressions consistently. It ensures that everyone gets the same answer for a given expression.

• B/P: Brackets or Parentheses - Evaluate expressions inside brackets first.
• O/E: Orders or Exponents - Calculate powers, square roots, etc.
• DM: Division and Multiplication - Perform these operations from left to right as they appear.
• AS: Addition and Subtraction - Perform these operations from left to right as they appear.

In this problem, applying BODMAS correctly after interchanging the signs and numbers is key to verifying which option makes the equation correct.