Solving Equation by Sign Interchange

The question asks us to find which pair of mathematical signs in the equation \(13 \times 6 + 24 \div 6 - 17 = 91\) should be swapped to make the equation mathematically correct. We need to test each given option by interchanging the specified signs and then evaluating the resulting equation using the standard order of operations (BODMAS/PEMDAS).

The original equation is: \(13 \times 6 + 24 \div 6 - 17\)

Testing Option 1: Interchange ÷ and +

Swap the division (\(\div\)) and addition (\(+\)) signs in the equation.

New equation: \(13 \times 6 \div 24 + 6 - 17\)

Let's evaluate this:

• First, multiplication and division from left to right: \(13 \times 6 = 78\).
• Equation becomes: \(78 \div 24 + 6 - 17\).
• Next, division: \(78 \div 24 = \frac{78}{24} = \frac{13}{4} = 3.25\).
• Equation becomes: \(3.25 + 6 - 17\).
• Now, addition and subtraction from left to right: \(3.25 + 6 = 9.25\).
• Finally, subtraction: \(9.25 - 17 = -7.75\).

The result is -7.75, which is not equal to 91. So, Option 1 is incorrect.

Testing Option 2: Interchange + and ×

Swap the addition (\(+\)) and multiplication (\(\times\)) signs in the equation.

New equation: \(13 + 6 \times 24 \div 6 - 17\)

Let's evaluate this:

• First, multiplication and division from left to right: \(6 \times 24 = 144\).
• Equation becomes: \(13 + 144 \div 6 - 17\).
• Next, division: \(144 \div 6 = 24\).
• Equation becomes: \(13 + 24 - 17\).
• Now, addition and subtraction from left to right: \(13 + 24 = 37\).
• Finally, subtraction: \(37 - 17 = 20\).

The result is 20, which is not equal to 91. So, Option 2 is incorrect.

Testing Option 3: Interchange ÷ and ×

Swap the division (\(\div\)) and multiplication (\(\times\)) signs in the equation.

New equation: \(13 \div 6 + 24 \times 6 - 17\)

Let's evaluate this:

• First, multiplication and division from left to right: \(13 \div 6 \approx 2.166\).
• Next, multiplication: \(24 \times 6 = 144\).
• Equation becomes: \(13 \div 6 + 144 - 17\).
• Using the precise fraction for division: \(\frac{13}{6} + 144 - 17\).
• Now, addition and subtraction from left to right: \(\frac{13}{6} + 144 = \frac{13 + 144 \times 6}{6} = \frac{13 + 864}{6} = \frac{877}{6} \approx 146.166\).
• Finally, subtraction: \(\frac{877}{6} - 17 = \frac{877 - 17 \times 6}{6} = \frac{877 - 102}{6} = \frac{775}{6} \approx 129.166\).

The result is approximately 129.166, which is not equal to 91. So, Option 3 is incorrect.

Testing Option 4: Interchange + and -

Swap the addition (\(+\)) and subtraction (\(-\)) signs in the equation.

New equation: \(13 \times 6 - 24 \div 6 + 17\)

Let's evaluate this step-by-step using the order of operations:

• Step 1: Perform multiplication and division from left to right.
• First, multiplication: \(13 \times 6 = 78\). The equation becomes \(78 - 24 \div 6 + 17\).
• Next, division: \(24 \div 6 = 4\). The equation becomes \(78 - 4 + 17\).
• Step 2: Perform addition and subtraction from left to right.
• First, subtraction: \(78 - 4 = 74\). The equation becomes \(74 + 17\).
• Next, addition: \(74 + 17 = 91\).

The result is 91, which is equal to the right side of the original equation (\(91\)).

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

The signs that should be interchanged are + and -.

Revision Table: Sign Interchange Analysis

Additional Information: Understanding Operator Precedence (BODMAS/PEMDAS)

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

• BODMAS: Brackets, Orders (powers/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).

In the given problem, after interchanging the signs, we used this rule. For example, in the correct option (\(13 \times 6 - 24 \div 6 + 17\)), we first performed the multiplication (\(13 \times 6\)) and division (\(24 \div 6\)) before moving on to subtraction (\(78 - 4\)) and addition (\(74 + 17\)). The order of operations is fundamental in solving mathematical and reasoning problems involving multiple operators.