Equation Correction by Symbol Interchange

The problem requires us to find which pair of mathematical symbols, when interchanged in the given equation, makes the equation correct. The given equation is:

$(35 – 2) × 3 = (56 × 8 ÷ 3 + 100) × 11$

Let's evaluate the original equation to see if it is true:

• Left Hand Side (LHS): $(35 - 2) \times 3 = 33 \times 3 = 99$
• Right Hand Side (RHS): $(56 \times 8 \div 3 + 100) \times 11$

Let's calculate the RHS step-by-step following the order of operations (BODMAS/PEMDAS):

• First, multiplication/division from left to right inside the bracket: $56 \times 8 = 448$
• Next, division: $448 \div 3 \approx 149.33$
• Then, addition: $149.33 + 100 = 249.33$
• Finally, multiplication outside the bracket: $249.33 \times 11 \approx 2742.63$

Since $99 \neq 2742.63$, the original equation is incorrect.

We will now test each option by swapping the specified symbols and re-evaluating the equation.

Testing Symbol Interchange Options for Equation Correction

Let's analyze each option:

Option 1: Interchange ÷ and +

Original equation: $(35 - 2) \times 3 = (56 \times 8 \div 3 + 100) \times 11$

Swap ÷ and +:

$(35 - 2) \times 3 = (56 \times 8 + 3 \div 100) \times 11$

Evaluate the modified equation:

• LHS: $(35 - 2) \times 3 = 33 \times 3 = 99$ (LHS remains unchanged as the swapped symbols are only on the RHS)
• RHS: $(56 \times 8 + 3 \div 100) \times 11 = (448 + 0.03) \times 11 = 448.03 \times 11 = 4928.33$

Since $99 \neq 4928.33$, this interchange does not make the equation correct.

Option 2: Interchange × and ÷

Original equation: $(35 - 2) × 3 = (56 × 8 ÷ 3 + 100) × 11$

Swap × and ÷:

$(35 - 2) \div 3 = (56 \div 8 × 3 + 100) \div 11$

Evaluate the modified equation:

• LHS: $(35 - 2) \div 3 = 33 \div 3 = 11$
• RHS: $(56 \div 8 \times 3 + 100) \div 11$

Let's calculate the RHS step-by-step:

• First, division inside the bracket: $56 \div 8 = 7$
• Next, multiplication inside the bracket: $7 \times 3 = 21$
• Then, addition inside the bracket: $21 + 100 = 121$
• Finally, division outside the bracket: $121 \div 11 = 11$

Since LHS = 11 and RHS = 11, LHS = RHS. The equation is correct after this interchange.

Therefore, interchanging the symbols × and ÷ makes the given equation correct.

Revision Table: Symbol Interchange Analysis

Additional Information: Rules of Operations (BODMAS/PEMDAS)

When evaluating mathematical expressions, we follow a specific order of operations to ensure consistency. This order is often remembered using acronyms like BODMAS or PEMDAS.

• BODMAS:
  • B: Brackets (Parentheses)
  • O: Order (Powers, Roots)
  • D: Division
  • M: Multiplication
  • A: Addition
  • S: Subtraction
• PEMDAS:
  • P: Parentheses
  • E: Exponents (Powers, Roots)
  • M: Multiplication
  • D: Division
  • A: Addition
  • S: Subtraction

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

In this symbol interchange problem, correctly applying the order of operations is crucial after swapping symbols to check if the equation holds true.