Solving Operator Interchange Puzzles

The question asks us to find which pair of mathematical operators, when swapped in the given equation, will make the equation balanced or true. The given equation is:

$14 - 96 \times 24 + 16 \div 4 = 74$

To solve this, we need to test each option by interchanging the specified operators and then evaluating the resulting expression using the standard order of operations (BODMAS/PEMDAS).

Analyzing the Original Equation

First, let's evaluate the original equation without any operator interchange to see if it is already balanced:

$14 - 96 \times 24 + 16 \div 4$

Applying BODMAS/PEMDAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction):

• Multiplication and Division first (from left to right):
  $= 14 - (96 \times 24) + (16 \div 4)$
  $= 14 - 2304 + 4$
• Addition and Subtraction next (from left to right):
  $= (14 - 2304) + 4$ or $= 14 + 4 - 2304$
  $= -2290 + 4$ or $= 18 - 2304$
  $= -2286$

So, $-2286 = 74$ is false. The original equation is not balanced.

Testing Operator Interchange Options

We will now test each option provided by swapping the operators as indicated and evaluating the new expression.

Interchanging '-' and '÷'

If we interchange '-' and '÷', the equation becomes:

$14 \div 96 \times 24 + 16 - 4 = 74$

Evaluating the left side:

• Division and Multiplication (from left to right):
  $= (14 \div 96) \times 24 + 16 - 4$
  $= \frac{14}{96} \times 24 + 16 - 4$
  $= \frac{7}{48} \times 24 + 16 - 4$
  $= \frac{7 \times 24}{48} + 16 - 4$
  $= \frac{7 \times 1}{2} + 16 - 4$
  $= \frac{7}{2} + 16 - 4$
  $= 3.5 + 16 - 4$
• Addition and Subtraction (from left to right):
  $= (3.5 + 16) - 4$
  $= 19.5 - 4$
  $= 15.5$

$15.5 = 74$ is false. This interchange does not balance the equation.

Interchanging '+' and '-'

If we interchange '+' and '-', the equation becomes:

$14 + 96 \times 24 - 16 \div 4 = 74$

Evaluating the left side:

• Multiplication and Division (from left to right):
  $= 14 + (96 \times 24) - (16 \div 4)$
  $= 14 + 2304 - 4$
• Addition and Subtraction (from left to right):
  $= (14 + 2304) - 4$
  $= 2318 - 4$
  $= 2314$

$2314 = 74$ is false. This interchange does not balance the equation.

Interchanging '÷' and '×'

If we interchange '÷' and '×', the equation becomes:

$14 - 96 \div 24 + 16 \times 4 = 74$

Evaluating the left side:

• Division and Multiplication (from left to right):
  $= 14 - (96 \div 24) + (16 \times 4)$
  $= 14 - 4 + 64$
• Addition and Subtraction (from left to right):
  $= (14 - 4) + 64$
  $= 10 + 64$
  $= 74$

$74 = 74$ is true. This interchange balances the equation.

Interchanging '+' and '×'

If we interchange '+' and '×', the equation becomes:

$14 - 96 + 24 \times 16 \div 4 = 74$

Evaluating the left side:

• Multiplication and Division (from left to right):
  $= 14 - 96 + (24 \times 16 \div 4)$
  $= 14 - 96 + (384 \div 4)$
  $= 14 - 96 + 96$
• Addition and Subtraction (from left to right):
  $= (14 - 96) + 96$
  $= -82 + 96$
  $= 14$

$14 = 74$ is false. This interchange does not balance the equation.

Conclusion: Balancing the Equation

Based on our evaluation of each option, interchanging the '÷' and '×' operators successfully balances the given equation, making the left side equal to the right side ($74 = 74$).

Operator Interchange Revision Table

Understanding how operator interchange affects the value of a mathematical expression is key to solving these types of problems. It requires careful application of the order of operations rule after performing the swap.

Additional Information: Order of Operations

The order of operations is a set of rules that dictates the sequence in which operations should be performed in a mathematical expression. A common acronym used is BODMAS or PEMDAS.

• BODMAS: Brackets, Orders (powers/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).

Following this order consistently is crucial to arrive at the correct result when evaluating expressions.