Understanding the Problem: Interchange of Signs

The question asks us to find which interchange of two mathematical signs ($\div$, $+$, $\times$, or $-$) in the given equation would make the equation mathematically correct. The given equation is:

\(\qquad 13 + 26 \times 2 - 5 \div 4 = 6\)

We need to test each option by swapping the specified signs and then evaluating the left side of the equation using the BODMAS/PEMDAS rule (order of operations: Brackets, Orders/Exponents, Division/Multiplication, Addition/Subtraction). If the result equals 6, that option is the correct one.

Evaluating the Original Equation

Let's first evaluate the original equation to see if it's already correct (it isn't, as stated by the need for interchange):

Original equation: $13 + 26 \times 2 - 5 \div 4$

Using BODMAS:

• Division: \(5 \div 4 = 1.25\)
• Multiplication: \(26 \times 2 = 52\)
• Equation becomes: $13 + 52 - 1.25$
• Addition/Subtraction (from left to right): \(13 + 52 = 65\), then \(65 - 1.25 = 63.75\)

So, \(63.75 = 6\), which is false. The equation is indeed incorrect as given.

Testing Each Option for Sign Interchange

Option 1: Interchange $\div$ and +

Swap $+$ and $\div$ signs in the original equation:

Original: \(13 + 26 \times 2 - 5 \div 4 = 6\)

Interchanged: \(13 \div 26 \times 2 - 5 + 4 = 6\)

Evaluate the left side using BODMAS:

• Division: \(13 \div 26 = 0.5\)
• Multiplication: \(0.5 \times 2 = 1\)
• Equation becomes: $1 - 5 + 4$
• Addition/Subtraction (from left to right): \(1 - 5 = -4\), then \(-4 + 4 = 0\)

Result: $0$. Does \(0 = 6\)? No. Option 1 is incorrect.

Option 2: Interchange + and -

Swap $+$ and $-$ signs in the original equation:

Original: \(13 + 26 \times 2 - 5 \div 4 = 6\)

Interchanged: \(13 - 26 \times 2 + 5 \div 4 = 6\)

Evaluate the left side using BODMAS:

• Division: \(5 \div 4 = 1.25\)
• Multiplication: \(26 \times 2 = 52\)
• Equation becomes: $13 - 52 + 1.25$
• Addition/Subtraction (from left to right): \(13 - 52 = -39\), then \(-39 + 1.25 = -37.75\)

Result: $-37.75$. Does \(-37.75 = 6\)? No. Option 2 is incorrect.

Option 3: Interchange $\times$ and -

Swap $\times$ and $-$ signs in the original equation:

Original: \(13 + 26 \times 2 - 5 \div 4 = 6\)

Interchanged: \(13 + 26 - 2 \times 5 \div 4 = 6\)

Evaluate the left side using BODMAS:

• Multiplication: \(2 \times 5 = 10\)
• Division: \(10 \div 4 = 2.5\)
• Equation becomes: $13 + 26 - 2.5$
• Addition/Subtraction (from left to right): \(13 + 26 = 39\), then \(39 - 2.5 = 36.5\)

Result: $36.5$. Does \(36.5 = 6\)? No. Option 3 is incorrect.

Option 4: Interchange $\times$ and $\div$

Swap $\times$ and $\div$ signs in the original equation:

Original: \(13 + 26 \times 2 - 5 \div 4 = 6\)

Interchanged: \(13 + 26 \div 2 - 5 \times 4 = 6\)

Evaluate the left side using BODMAS:

• Division: \(26 \div 2 = 13\)
• Multiplication: \(5 \times 4 = 20\)
• Equation becomes: $13 + 13 - 20$
• Addition/Subtraction (from left to right): \(13 + 13 = 26\), then \(26 - 20 = 6\)

Result: $6$. Does \(6 = 6\)? Yes. Option 4 makes the equation correct.

Summary of Results

The interchange of signs $\times$ and $\div$ is the only one that makes the given equation correct.

Revision Table: Solving Sign Interchange Problems

Additional Information: Order of Operations (BODMAS/PEMDAS)

The order of operations is crucial for solving mathematical expressions correctly. It dictates the sequence in which operations should be performed.

• B/P: Brackets / Parentheses - Solve operations inside brackets first.
• O/E: Orders / Exponents - Solve powers, roots, etc., next.
• D/M: Division / Multiplication - Perform division and multiplication from left to right.
• A/S: Addition / Subtraction - Perform addition and subtraction from left to right.

Following this order ensures that everyone gets the same answer for a given expression, which is essential in mathematics.