Finding Which Signs to Interchange in the Equation

The question asks us to find which pair of signs should be interchanged in the given mathematical equation to make it correct.

The given equation is:

\(24 \div 4 - 18 + 2 \times 5 = 14\)

First, let's evaluate the original equation using the order of operations (BODMAS/PEMDAS): Brackets, Orders (powers/roots), Division and Multiplication (from left to right), Addition and Subtraction (from left to right).

• Division: \(24 \div 4 = 6\)
• Multiplication: \(2 \times 5 = 10\)
• The equation becomes: \(6 - 18 + 10\)
• Subtraction: \(6 - 18 = -12\)
• Addition: \(-12 + 10 = -2\)

So, the original equation evaluates to \(-2\), which is not equal to \(14\). Thus, the equation is currently incorrect.

Now, let's test each option by interchanging the specified signs and re-evaluating the equation.

Testing Options for Sign Interchange

Option 1: Interchange \(\div\) and \(\times\)

The equation becomes: \(24 \times 4 - 18 + 2 \div 5 = 14\)

Evaluate:

• Multiplication: \(24 \times 4 = 96\)
• Division: \(2 \div 5 = 0.4\)
• The equation becomes: \(96 - 18 + 0.4\)
• Subtraction: \(96 - 18 = 78\)
• Addition: \(78 + 0.4 = 78.4\)

\(78.4 \neq 14\). So, this option is incorrect.

Option 2: Interchange \(\times\) and \(+\)

The equation becomes: \(24 \div 4 - 18 \times 2 + 5 = 14\)

Evaluate:

• Division: \(24 \div 4 = 6\)
• Multiplication: \(18 \times 2 = 36\)
• The equation becomes: \(6 - 36 + 5\)
• Subtraction: \(6 - 36 = -30\)
• Addition: \(-30 + 5 = -25\)

\(-25 \neq 14\). So, this option is incorrect.

Option 3: Interchange \(+\) and \(\div\)

The equation becomes: \(24 + 4 - 18 \div 2 \times 5 = 14\)

Evaluate:

• Division: \(18 \div 2 = 9\)
• Multiplication: \(9 \times 5 = 45\)
• The equation becomes: \(24 + 4 - 45\)
• Addition: \(24 + 4 = 28\)
• Subtraction: \(28 - 45 = -17\)

\(-17 \neq 14\). So, this option is incorrect.

Option 4: Interchange \(+\) and \(-\)

The equation becomes: \(24 \div 4 + 18 - 2 \times 5 = 14\)

Evaluate using the order of operations:

• Division: \(24 \div 4 = 6\)
• Multiplication: \(2 \times 5 = 10\)
• The equation becomes: \(6 + 18 - 10\)
• Addition: \(6 + 18 = 24\)
• Subtraction: \(24 - 10 = 14\)

\(14 = 14\). This makes the equation correct.

Therefore, interchanging the signs \(+\) and \(-\) makes the given equation correct.

Revision Table: Evaluating Sign Interchanges

As shown in the table, only interchanging \(+\) and \(-\) results in the correct value of \(14\).

Additional Information: Order of Operations (BODMAS/PEMDAS)

When solving mathematical expressions, it's crucial to follow a specific order of operations to ensure consistent results. This order is commonly remembered using acronyms like BODMAS or PEMDAS.

• BODMAS:
  • B: Brackets (Parentheses)
  • O: Orders (Exponents, powers, roots)
  • D: Division
  • M: Multiplication
  • A: Addition
  • S: Subtraction
• PEMDAS:
  • P: Parentheses
  • E: Exponents
  • M: Multiplication
  • D: Division
  • A: Addition
  • S: Subtraction

Division and Multiplication have the same priority and should be performed from left to right as they appear in the expression. Similarly, Addition and Subtraction have the same priority and should be performed from left to right.

In this problem, we applied this rule at each step to correctly evaluate the equation after interchanging the signs.