Solving Sign Interchange Mathematical Equations

The question asks us to find which two mathematical signs, when interchanged in the given equation, will make the equation correct. The given equation is:

\$38 \div 19 \times 11 - 357 + 17 = 226\$

To solve this type of sign interchange problem, we need to test each option provided. For each option, we will swap the specified signs and then evaluate the new equation using the BODMAS/PEMDAS rule (Brackets, Orders/Exponents, Division and Multiplication, Addition and Subtraction) to see if it equals 226.

Let's evaluate the original equation first:

\$38 \div 19 \times 11 - 357 + 17\$

According to BODMAS:

• Division: \$38 \div 19 = 2\$
• Equation becomes: \$2 \times 11 - 357 + 17\$
• Multiplication: \$2 \times 11 = 22\$
• Equation becomes: \$22 - 357 + 17\$
• Addition/Subtraction (from left to right): \$22 - 357 = -335\$
• Equation becomes: \$-335 + 17 = -318\$

So, the original equation evaluates to \$-318\$. Since \$-318 \neq 226\$, the original equation is incorrect.

Testing the Options for Sign Interchange

Now let's test each option by interchanging the signs and re-evaluating the equation.

Option 1: Interchange × and +

The new equation becomes:

\$38 \div 19 + 11 - 357 \times 17\$

Let's evaluate this using BODMAS:

• Division: \$38 \div 19 = 2\$
• Equation becomes: \$2 + 11 - 357 \times 17\$
• Multiplication: \$357 \times 17 = 6069\$
• Equation becomes: \$2 + 11 - 6069\$
• Addition: \$2 + 11 = 13\$
• Equation becomes: \$13 - 6069\$
• Subtraction: \$13 - 6069 = -6056\$

Since \$-6056 \neq 226\$, this option is incorrect.

Option 2: Interchange - and ÷

The new equation becomes:

\$38 - 19 \times 11 \div 357 + 17\$

Let's evaluate this using BODMAS:

• Multiplication: \$19 \times 11 = 209\$
• Equation becomes: \$38 - 209 \div 357 + 17\$
• Division: \$209 \div 357 = \frac{209}{357}\$ (This will result in a fraction/decimal, unlikely to be correct for an integer target)
• Let's calculate: \$38 - \frac{209}{357} + 17\$
• \$38 + 17 - \frac{209}{357} = 55 - \frac{209}{357}\$

This will not result in 226. So, this option is incorrect.

Option 3: Interchange ÷ and +

The new equation becomes:

\$38 + 19 \times 11 - 357 \div 17\$

Let's evaluate this using BODMAS:

• Multiplication: \$19 \times 11 = 209\$
• Equation becomes: \$38 + 209 - 357 \div 17\$
• Division: \$357 \div 17 = 21\$
• Equation becomes: \$38 + 209 - 21\$
• Addition: \$38 + 209 = 247\$
• Equation becomes: \$247 - 21\$
• Subtraction: \$247 - 21 = 226\$

Since \$226 = 226\$, this option makes the equation correct.

Option 4: Interchange × and ÷

The new equation becomes:

\$38 \times 19 \div 11 - 357 + 17\$

Let's evaluate this using BODMAS:

• Multiplication: \$38 \times 19 = 722\$
• Equation becomes: \$722 \div 11 - 357 + 17\$
• Division: \$722 \div 11 = \frac{722}{11}\$ (This will result in a fraction/decimal)
• Let's calculate: \$\frac{722}{11} - 357 + 17\$
• \$\frac{722}{11} + 17 - 357\$

This will not result in 226. So, this option is incorrect.

Based on the evaluation, interchanging the signs ÷ and + makes the equation correct.

Revision Table: Sign Interchange Results

Additional Information on BODMAS/PEMDAS

The order of operations is crucial in solving mathematical expressions. BODMAS (or PEMDAS) dictates the sequence:

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

When operations are at the same level (like Division and Multiplication, or Addition and Subtraction), you should perform them from left to right as they appear in the equation. Understanding BODMAS is fundamental to solving mathematical reasoning problems, including sign interchange questions.