How to Solve Sign Interchange Problems in Equations

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

\(312 \text{ – } 13 \text{ × } 14 + 207 \text{ ÷ } 4 = 539\)

First, let's evaluate the original equation using the BODMAS/PEMDAS rule (Brackets, Orders/Exponents, Division and Multiplication (from left to right), Addition and Subtraction (from left to right)).

\(312 - (13 \times 14) + (207 \div 4)\)

\(312 - 182 + 51.75\)

\(130 + 51.75 = 181.75\)

Since \(181.75 \neq 539\), the original equation is incorrect. We need to test each option by interchanging the specified signs and re-evaluating the equation.

Analyzing Each Option by Swapping Signs

Let's systematically test each given option:

Option 1: Interchange ÷ and –

Swap the ÷ and – signs in the original equation. The new equation becomes:

\(312 \text{ ÷ } 13 \text{ × } 14 + 207 \text{ – } 4\)

Now, evaluate this equation using BODMAS/PEMDAS:

Perform Division: \(312 \div 13 = 24\)

The equation is now: \(24 \text{ × } 14 + 207 \text{ – } 4\)

Perform Multiplication: \(24 \times 14 = 336\)

The equation is now: \(336 + 207 \text{ – } 4\)

Perform Addition and Subtraction from left to right:

\(336 + 207 = 543\)

The equation is now: \(543 \text{ – } 4\)

Perform Subtraction: \(543 - 4 = 539\)

The result is 539, which matches the right side of the original equation. Therefore, interchanging the ÷ and – signs makes the equation correct.

Let's quickly look at other options to confirm they don't yield the correct result, even though we found the solution in Option 1.

Option 2: Interchange – and +

Swap the – and + signs: \(312 + 13 \text{ × } 14 \text{ – } 207 \text{ ÷ } 4\)

Evaluate:

\(312 + (13 \times 14) \text{ – } (207 \div 4)\)

\(312 + 182 \text{ – } 51.75\)

\(494 \text{ – } 51.75 = 442.25\)

\(442.25 \neq 539\).

Option 3: Interchange ÷ and +

Swap the ÷ and + signs: \(312 \text{ – } 13 \text{ × } 14 \text{ ÷ } 207 + 4\)

Evaluate:

\(312 \text{ – } (13 \times 14 \div 207) + 4\)

\(312 \text{ – } (182 \div 207) + 4\)

\(312 \text{ – } 0.879... + 4 \approx 311.12 + 4 = 315.12\)

\(315.12 \neq 539\).

Option 4: Interchange + and ×

Swap the + and × signs: \(312 \text{ – } 13 + 14 \text{ × } 207 \text{ ÷ } 4\)

Evaluate:

\(312 \text{ – } 13 + (14 \times 207 \div 4)\)

\(312 \text{ – } 13 + (2898 \div 4)\)

\(312 \text{ – } 13 + 724.5\)

\(299 + 724.5 = 1023.5\)

\(1023.5 \neq 539\).

Correct Signs to Interchange

Based on the evaluation, interchanging the ÷ and – signs results in the correct equation.

Revision Table: Summary of Sign Swaps

The analysis confirms that only swapping ÷ and – yields the target value of 539.

Additional Information: Understanding BODMAS/PEMDAS

To solve equations involving multiple operations, we follow a specific order. This order is commonly known as BODMAS or PEMDAS.

• BODMAS: Brackets, Orders (powers and square 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).

Both acronyms represent the same order of operations. It's crucial to follow this order to get the correct result. For example, in \(10 + 2 \times 5\), you must do the multiplication first (\(2 \times 5 = 10\)), then the addition (\(10 + 10 = 20\)), not \(10 + 2 = 12\) then \(12 \times 5 = 60\).