Interchanging Mathematical Signs in Equations

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

\begin{equation*} 31 \div 14 \times 9 + 3 - 7 = 66 \end{equation*}

Currently, this equation is not correct. We need to test the options provided by interchanging the specified signs and then evaluate the resulting equation using the order of operations (BODMAS or PEMDAS).

Testing the Options for Sign Interchange

Let's examine each option and see if swapping the signs makes the equation valid.

Option 1: Interchange ÷ and +

If we swap the division (÷) and addition (+) signs, the equation becomes:

\begin{equation*} 31 + 14 \times 9 \div 3 - 7 \end{equation*}

Now, let's evaluate this expression using the BODMAS/PEMDAS rule:

• Brackets / Parentheses (None in this equation)
• Orders / Exponents (None in this equation)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

Following the rule:

1. First, perform the division: $9 \div 3 = 3$. The equation is now: $31 + 14 \times 3 - 7$
2. Next, perform the multiplication: $14 \times 3 = 42$. The equation is now: $31 + 42 - 7$
3. Then, perform the addition: $31 + 42 = 73$. The equation is now: $73 - 7$
4. Finally, perform the subtraction: $73 - 7 = 66$.

The result of the evaluation is 66. This matches the Right Hand Side (RHS) of the original equation (= 66). Therefore, interchanging the ÷ and + signs makes the equation mathematically correct.

The corrected equation is:

\begin{equation*} 31 + 14 \times 9 \div 3 - 7 = 66 \end{equation*}

Testing Other Options (Verification)

While we have found the correct interchange, let's briefly consider why other options would not work. We would follow the same process: swap the signs, then evaluate using BODMAS/PEMDAS.

• Option 2: Interchange ÷ and ×
  Equation becomes: $31 \times 14 \div 9 + 3 - 7$. Calculating $14 \div 9$ will result in a fraction or a non-integer, making it unlikely to reach 66.
• Option 3: Interchange − and ×
  Equation becomes: $31 \div 14 - 9 + 3 \times 7$. Again, $31 \div 14$ results in a fraction.
• Option 4: Interchange − and +
  Equation becomes: $31 \div 14 \times 9 - 3 + 7$. Once more, $31 \div 14$ results in a fraction.

Only the interchange of ÷ and + leads to a straightforward integer calculation that results in 66.

Conclusion on Sign Interchange

Based on the evaluation using the BODMAS/PEMDAS rule, interchanging the division (÷) and addition (+) signs makes the given equation mathematically correct.

Revision Table: Mathematical Signs Interchange

Additional Information: Understanding BODMAS/PEMDAS Rule for Equation Solving

The BODMAS or PEMDAS rule is a set of rules that dictate the order in which mathematical operations should be performed when evaluating an expression. This ensures that everyone gets the same result for a given expression.

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

Division and Multiplication have the same priority, and should be performed from left to right. Similarly, Addition and Subtraction have the same priority and should be performed from left to right.

Applying this rule is crucial for correctly solving problems involving multiple mathematical operations or when interchanging mathematical signs to correct an equation.