Equation Solving: Interchange Mathematical Signs

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

\(16 - 18 \times 216 \div 432 + 40 = 20\)

We need to check each option by interchanging the specified signs and then evaluating the modified equation using the order of operations (BODMAS/PEMDAS: Brackets, Orders, Division and Multiplication (from left to right), Addition and Subtraction (from left to right)).

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

\(16 - 18 \times 216 \div 432 + 40\)

First, perform multiplication and division from left to right:

• Divide: \(216 \div 432 = 0.5\)
• Multiply: \(18 \times 0.5 = 9\)

Now substitute these back into the equation:

\(16 - 9 + 40\)

Next, perform addition and subtraction from left to right:

• Subtract: \(16 - 9 = 7\)
• Add: \(7 + 40 = 47\)

So, the original equation is \(47 = 20\), which is incorrect.

Checking Options for Sign Interchange

We will now check each option provided by swapping the signs and re-evaluating the equation.

Option 1: Interchange × and ÷

Swap the multiplication (×) and division (÷) signs in the original equation:

\(16 - 18 \div 216 \times 432 + 40\)

Now, evaluate using the order of operations:

• Division (from left): \(18 \div 216 = \frac{18}{216}\)
• Simplify the fraction: \(\frac{18}{216} = \frac{1}{12}\)
• Multiplication (from left): \(\frac{1}{12} \times 432\)
• Calculate the multiplication: \(\frac{432}{12} = 36\)

Substitute this back into the equation:

\(16 - 36 + 40\)

Perform subtraction and addition from left to right:

• Subtract: \(16 - 36 = -20\)
• Add: \(-20 + 40 = 20\)

The equation becomes \(20 = 20\), which is correct.

This option successfully makes the equation correct.

Option 2: Interchange ÷ and −

Swap the division (÷) and subtraction (−) signs:

\(16 \div 18 \times 216 - 432 + 40\)

Evaluate using the order of operations:

• Division (from left): \(16 \div 18 = \frac{16}{18} = \frac{8}{9}\)
• Multiplication (from left): \(\frac{8}{9} \times 216\)
• Calculate multiplication: \(\frac{8 \times 216}{9} = 8 \times 24 = 192\)

Substitute back:

\(192 - 432 + 40\)

Perform subtraction and addition from left to right:

• Subtract: \(192 - 432 = -240\)
• Add: \(-240 + 40 = -200\)

The equation becomes \(-200 = 20\), which is incorrect.

Option 3: Interchange × and +

Swap the multiplication (×) and addition (+) signs:

\(16 - 18 + 216 \div 432 \times 40\)

Evaluate using the order of operations:

• Division (from left): \(216 \div 432 = 0.5\)
• Multiplication (from left): \(0.5 \times 40 = 20\)

Substitute back:

\(16 - 18 + 20\)

Perform subtraction and addition from left to right:

• Subtract: \(16 - 18 = -2\)
• Add: \(-2 + 20 = 18\)

The equation becomes \(18 = 20\), which is incorrect.

Option 4: Interchange + and −

Swap the addition (+) and subtraction (−) signs:

\(16 + 18 \times 216 \div 432 - 40\)

Evaluate using the order of operations:

• Division (from left): \(216 \div 432 = 0.5\)
• Multiplication (from left): \(18 \times 0.5 = 9\)

Substitute back:

\(16 + 9 - 40\)

Perform addition and subtraction from left to right:

• Add: \(16 + 9 = 25\)
• Subtract: \(25 - 40 = -15\)

The equation becomes \(-15 = 20\), which is incorrect.

Conclusion on Sign Interchange

Based on our evaluation, only interchanging the multiplication (×) and division (÷) signs results in a correct equation.

\(16 - 18 \div 216 \times 432 + 40 = 20\)

Which simplifies to \(20 = 20\).

Revision Table: Checking Equation with Interchanged Signs

Additional Information: Understanding Order of Operations

The order of operations is a crucial concept in solving mathematical expressions. It ensures that everyone gets the same answer when evaluating an expression. A common acronym used is BODMAS or PEMDAS.

• BODMAS: Brackets, Orders (powers, roots), Division, Multiplication, Addition, Subtraction.
• PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.

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

In this problem, we applied this rule correctly after interchanging the signs to verify if the equation holds true.