Solving Mathematical Equations by Interchanging Signs and Numbers

The problem asks us to determine which interchange of two signs and two numbers will make the given mathematical equation correct. We are given the equation: \(10 \times 11 + 19 - 323 \div 3 = 50\).

To solve this type of logical puzzle, we need to test the interchanged equation based on the options provided and check if it evaluates to the right-hand side value, which is 50.

Testing the Interchange of 3 and 19, and - and ×

Let's consider the interchange proposed: swapping the number 3 with the number 19, and swapping the subtraction sign (-) with the multiplication sign (×). This means:

• Every '3' in the equation becomes '19'.
• Every '19' in the equation becomes '3'.
• Every '-' sign becomes a '×' sign.
• Every '×' sign becomes a '-' sign.

The original equation is:

\(10 \times 11 + 19 - 323 \div 3 = 50\)

Applying the proposed interchanges (3 <-> 19 and - <-> ×), the equation becomes:

\(10 \mathbf{-} 11 + \mathbf{3} \mathbf{\times} 323 \div \mathbf{19} = ?\)

Now, let's evaluate this new equation using the order of operations (BODMAS/PEMDAS).

Evaluating the Modified Equation Using BODMAS

The BODMAS (or PEMDAS) rule helps us determine the correct sequence of operations:

• Brackets (Parentheses)
• Orders (Exponents, Roots)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

Our modified equation is: \(10 - 11 + 3 \times 323 \div 19\)

Let's perform the operations step-by-step:

Step 1: Perform Division and Multiplication from left to right.

First, the Division: \(323 \div 19\)

\(323 \div 19 = 17\)

The equation becomes: \(10 - 11 + 3 \times 17\)

Next, the Multiplication: \(3 \times 17\)

\(3 \times 17 = 51\)

The equation becomes: \(10 - 11 + 51\)

Step 2: Perform Addition and Subtraction from left to right.

First, the Subtraction: \(10 - 11\)

\(10 - 11 = -1\)

The equation becomes: \(-1 + 51\)

Next, the Addition: \(-1 + 51\)

\(-1 + 51 = 50\)

The result of the modified equation is 50. This matches the right-hand side of the original equation.

Therefore, interchanging the numbers 3 and 19, and the signs - and × makes the equation correct.

Summary of the Interchange

The modified equation is \(10 - 11 + 3 \times 323 \div 19 = 50\).

Revision Table: Key Concepts Used

Additional Information: Understanding Order of Operations

The order of operations is crucial in evaluating mathematical expressions unambiguously. Without a standard order, the same expression could yield different results. BODMAS and PEMDAS are mnemonics used worldwide to remember this order:

• 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. Similarly, Addition and Subtraction have the same priority and are performed from left to right.

For example, in \(10 - 11 + 51\), subtraction comes before addition when reading left to right, so \(10 - 11\) is calculated first.