Understanding the Problem: Equation Sign Interchange

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

\(32 + 24 \times 4 \div 16 – 64 = 90\)

We need to test each given option by interchanging the specified signs and then evaluating the modified equation using the BODMAS (or PEMDAS) rule to see if it equals 90.

Applying the BODMAS Rule

Remember the order of operations according to the BODMAS rule:

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

Testing the Options

Option 1: Interchanging '+' and '×'

If we interchange '+' and '×', the equation becomes:

\(32 \times 24 + 4 \div 16 – 64\)

Let's evaluate this:

• First, Division: \(4 \div 16 = 0.25\)
• Next, Multiplication: \(32 \times 24 = 768\)
• Now, Addition and Subtraction (from left to right): \(768 + 0.25 – 64 = 768.25 – 64 = 704.25\)

\(704.25 \neq 90\). So, this option is incorrect.

Option 2: Interchanging '÷' and '+'

If we interchange '÷' and '+', the equation becomes:

\(32 \div 24 \times 4 + 16 – 64\)

Let's evaluate this:

• First, Division/Multiplication (from left to right): \(32 \div 24 = \frac{32}{24} = \frac{4}{3}\)
• Next: \(\frac{4}{3} \times 4 = \frac{16}{3}\)
• Now, Addition and Subtraction (from left to right): \(\frac{16}{3} + 16 – 64 = \frac{16}{3} - 48\)
• \(\frac{16}{3} - \frac{48 \times 3}{3} = \frac{16 - 144}{3} = \frac{-128}{3}\)

\(\frac{-128}{3} \neq 90\). So, this option is incorrect.

Option 3: Interchanging '÷' and '×'

If we interchange '÷' and '×', the equation becomes:

\(32 + 24 \div 4 \times 16 – 64\)

Let's evaluate this:

• First, Division/Multiplication (from left to right): \(24 \div 4 = 6\)
• Next: \(6 \times 16 = 96\)
• Now, Addition and Subtraction (from left to right): \(32 + 96 – 64 = 128 – 64 = 64\)

\(64 \neq 90\). So, this option is incorrect.

Option 4: Interchanging '+' and '–'

If we interchange '+' and '–', the equation becomes:

\(32 – 24 \times 4 \div 16 + 64\)

Let's evaluate this:

• First, Multiplication/Division (from left to right): \(24 \times 4 = 96\)
• Next: \(96 \div 16 = 6\)
• Now, Addition and Subtraction (from left to right): \(32 – 6 + 64 = 26 + 64 = 90\)

\(90 = 90\). This makes the equation correct.

Conclusion

Interchanging the signs '+' and '–' makes the given equation correct. Therefore, Option 4 is the correct answer.

Revision Table: Sign Interchange Results

Additional Information: Solving Math Operations Questions

Solving problems that involve interchanging signs or numbers in an equation requires careful application of the order of operations (BODMAS/PEMDAS). Here are some tips:

• Always write down the new equation clearly after interchanging the signs.
• Perform operations step-by-step following the BODMAS/PEMDAS rule.
• Be careful with fractions and decimals during calculations.
• Test each option systematically until you find the one that satisfies the equation.
• Double-check your calculations, especially when dealing with multiple operations.