Correcting the Equation by Interchanging Signs

This problem asks us to identify which pair of mathematical signs needs to be swapped in the equation '$99 \div 33 - 6 + 48 \times 26 = 40$' to make it mathematically correct. We need to test each option provided.

Understanding Order of Operations (BODMAS/PEMDAS)

To solve this, we must use the standard order of operations, often remembered by the acronym BODMAS or PEMDAS:

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

Let's check the value of the original equation:

$$99 \div 33 - 6 + 48 \times 26$$

1. Division: $99 \div 33 = 3$
2. Multiplication: $48 \times 26 = 1248$
3. Substitute these values back into the expression: $3 - 6 + 1248$
4. Subtraction (left to right): $3 - 6 = -3$
5. Addition: $-3 + 1248 = 1245$

The original equation evaluates to $1245$, not $40$. Therefore, swapping signs is necessary.

Testing Sign Interchange Options

We will now systematically test each option:

Option 1: Interchanging '+' and 'x'

If '+' and 'x' are swapped, the equation becomes:

$$99 \div 33 - 6 \times 48 + 26$$

1. Division: $99 \div 33 = 3$
2. Multiplication: $6 \times 48 = 288$
3. The expression is now: $3 - 288 + 26$
4. Subtraction: $3 - 288 = -285$
5. Addition: $-285 + 26 = -259$

The result is $-259$, which is not $40$.

Option 2: Interchanging '÷' and 'x'

If '÷' and 'x' are swapped, the equation becomes:

$$99 \times 33 - 6 + 48 \div 26$$

1. Multiplication: $99 \times 33 = 3267$
2. Division: $48 \div 26 \approx 1.846$
3. The expression is now: $3267 - 6 + 1.846...$
4. Subtraction: $3267 - 6 = 3261$
5. Addition: $3261 + 1.846... = 3262.846...$

The result is approximately $3262.85$, which is not $40$.

Option 3: Interchanging '-' and '÷'

If '-' and '÷' are swapped, the equation becomes:

$$99 - 33 \div 6 + 48 \times 26$$

1. Division: $33 \div 6 = 5.5$
2. Multiplication: $48 \times 26 = 1248$
3. The expression is now: $99 - 5.5 + 1248$
4. Subtraction: $99 - 5.5 = 93.5$
5. Addition: $93.5 + 1248 = 1341.5$

The result is $1341.5$, which is not $40$.

Option 4: Interchanging 'x' and '-'

If 'x' and '-' are swapped, the equation becomes:

$$99 \div 33 \times 6 - 48 + 26$$

1. Division: $99 \div 33 = 3$
2. Multiplication: $3 \times 6 = 18$
3. The expression is now: $18 - 48 + 26$
4. Subtraction (left to right): $18 - 48 = -30$
5. Addition: $-30 + 26 = -4$

The result is $-4$.