Solving Equations by Number Interchange

The problem asks us to find which pair of numbers, when interchanged in the given equation, makes the equation correct. The target value for the equation is 0.

The original equation is:

$(4 \times 12 + 15) \div 8 – 9 = 0$

Let's evaluate the original equation to see if it is already correct:

$(4 \times 12 + 15) \div 8 – 9$

$= (48 + 15) \div 8 – 9$

$= 63 \div 8 – 9$

Since 63 is not perfectly divisible by 8, the original equation does not equal 0.

Now, let's test each option by interchanging the specified numbers and calculating the result.

Testing Number Interchange Options

Option 1: Interchange 12 and 15

If we interchange the numbers 12 and 15, the equation becomes:

$(4 \times 15 + 12) \div 8 – 9$

Now, let's evaluate this modified equation:

$(4 \times 15 + 12) \div 8 – 9$

$= (60 + 12) \div 8 – 9$ (First, perform the multiplication inside the parentheses)

$= 72 \div 8 – 9$ (Next, perform the addition inside the parentheses)

$= 9 – 9$ (Then, perform the division)

$= 0$ (Finally, perform the subtraction)

The result is 0, which matches the right side of the original equation. So, interchanging 12 and 15 makes the equation correct.

Option 2: Interchange 4 and 8

If we interchange the numbers 4 and 8, the equation becomes:

$(8 \times 12 + 15) \div 4 – 9$

Let's evaluate this modified equation:

$(8 \times 12 + 15) \div 4 – 9$

$= (96 + 15) \div 4 – 9$

$= 111 \div 4 – 9$

Since 111 is not divisible by 4, the result will not be an integer, and thus not 0.

Option 3: Interchange 12 and 8

If we interchange the numbers 12 and 8, the equation becomes:

$(4 \times 8 + 15) \div 12 – 9$

Let's evaluate this modified equation:

$(4 \times 8 + 15) \div 12 – 9$

$= (32 + 15) \div 12 – 9$

$= 47 \div 12 – 9$

Since 47 is not divisible by 12, the result will not be an integer, and thus not 0.

Option 4: Interchange 8 and 9

If we interchange the numbers 8 and 9, the equation becomes:

$(4 \times 12 + 15) \div 9 – 8$

Let's evaluate this modified equation:

$(4 \times 12 + 15) \div 9 – 8$

$= (48 + 15) \div 9 – 8$

$= 63 \div 9 – 8$

$= 7 – 8$

$= -1$

The result is -1, which is not 0.

Based on the evaluation of each option, interchanging the numbers 12 and 15 is the only way to make the given equation correct.

Revision Table: Equation Solving and Number Swapping

This problem requires careful application of the order of operations (BODMAS/PEMDAS) and testing different possibilities through number swapping.

• Understand the goal: Make the equation equal to the target value (0 in this case).
• Identify the numbers to be swapped for each option.
• Substitute the numbers in the equation according to the swap.
• Recalculate the equation step-by-step using the correct order of operations: Parentheses/Brackets first, then Orders/Exponents, then Division and Multiplication (from left to right), and finally Addition and Subtraction (from left to right).
• Compare the final result with the target value.

Additional Information: Order of Operations

The order of operations is a rule used to clarify which procedures should be performed first in a given mathematical expression. Common acronyms are BODMAS or PEMDAS.

• BODMAS:
  • B - Brackets (Parentheses)
  • O - Order (Exponents, roots, etc.)
  • D - Division
  • M - Multiplication
  • A - Addition
  • S - Subtraction
• PEMDAS:
  • P - Parentheses
  • E - Exponents
  • M - Multiplication
  • D - Division
  • A - Addition
  • S - Subtraction

Multiplication and Division have the same priority and are done from left to right. Similarly, Addition and Subtraction have the same priority and are done from left to right.

In the given equation, we first calculate the expression inside the parentheses $(4 \times 12 + 15)$, then perform the division by 8, and finally the subtraction of 9.