Understanding the Equation Correction Problem

The problem asks us to find which pair of numbers, when swapped within the given equation, makes the equation true. The equation is: $12 \times 9 + 27 \div 3 - 5 = 60$

We need to follow the standard order of operations (PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division from left to right, Addition and Subtraction from left to right) to evaluate the expression.

Evaluating the Original Equation

First, let's calculate the value of the original equation as it is:

• Multiplication: $12 \times 9 = 108$
• Division: $27 \div 3 = 9$
• The equation becomes: $108 + 9 - 5$
• Addition: $108 + 9 = 117$
• Subtraction: $117 - 5 = 112$

The original equation evaluates to $112$, not $60$. So, we need to interchange numbers.

Checking Number Interchange Options

Let's test each option by swapping the specified numbers and re-evaluating the equation.

Option 1: Interchange 27 and 9

The equation becomes: $12 \times 27 + 9 \div 3 - 5 = 60$

• Multiplication: $12 \times 27 = 324$
• Division: $9 \div 3 = 3$
• The equation becomes: $324 + 3 - 5$
• Addition: $324 + 3 = 327$
• Subtraction: $327 - 5 = 322$

This does not equal $60$.

Option 2: Interchange 12 and 3

The equation becomes: $3 \times 9 + 27 \div 12 - 5 = 60$

• Multiplication: $3 \times 9 = 27$
• Division: $27 \div 12 = 2.25$
• The equation becomes: $27 + 2.25 - 5$
• Addition: $27 + 2.25 = 29.25$
• Subtraction: $29.25 - 5 = 24.25$

This does not equal $60$.

Option 3: Interchange 12 and 5

The equation becomes: $5 \times 9 + 27 \div 3 - 12 = 60$

• Multiplication: $5 \times 9 = 45$
• Division: $27 \div 3 = 9$
• The equation becomes: $45 + 9 - 12$
• Addition: $45 + 9 = 54$
• Subtraction: $54 - 12 = 42$

This does not equal $60$.

Option 4: Interchange 5 and 9

The equation becomes: $12 \times 5 + 27 \div 3 - 9 = 60$

• Multiplication: $12 \times 5 = 60$
• Division: $27 \div 3 = 9$
• The equation becomes: $60 + 9 - 9$
• Addition: $60 + 9 = 69$
• Subtraction: $69 - 9 = 60$

This equals $60$. Therefore, interchanging 5 and 9 makes the equation correct.

Conclusion

By testing all the given options, we found that swapping the numbers 5 and 9 in the original equation $12 \times 9 + 27 \div 3 - 5 = 60$ results in a correct equation ($12 \times 5 + 27 \div 3 - 9 = 60$).