Solving Equation by Interchanging Signs and Numbers

The problem asks us to find which two signs and which two numbers need to be swapped in the given equation $14 \times 8 + 64 \div 32 – 2 = 88$ to make it mathematically correct. We need to test each of the given options by applying the proposed interchanges and evaluating the resulting equation.

Analyzing Option 1: Interchange 14 and 2; × and +

Original equation: $14 \times 8 + 64 \div 32 – 2 = 88$

Applying the interchanges (14 $\leftrightarrow$ 2, $\times \leftrightarrow$ +):

Replace 14 with 2 and 2 with 14.

Replace $\times$ with + and + with $\times$.

The new equation becomes: $2 + 8 \times 64 \div 32 – 14$

Now, let's evaluate this expression following the BODMAS/PEMDAS rule (Brackets, Orders/Exponents, Division and Multiplication, Addition and Subtraction):

• Division: $64 \div 32 = 2$
• The equation is now: $2 + 8 \times 2 – 14$
• Multiplication: $8 \times 2 = 16$
• The equation is now: $2 + 16 – 14$
• Addition/Subtraction (from left to right): $2 + 16 = 18$
• The equation is now: $18 – 14$
• Final result: $18 – 14 = 4$

Since $4 \neq 88$, Option 1 is incorrect.

Analyzing Option 2: Interchange 8 and 32; × and +

Original equation: $14 \times 8 + 64 \div 32 – 2 = 88$

Applying the interchanges (8 $\leftrightarrow$ 32, $\times \leftrightarrow$ +):

Replace 8 with 32 and 32 with 8.

Replace $\times$ with + and + with $\times$.

The new equation becomes: $14 + 32 \times 64 \div 8 – 2$

Evaluating the expression:

• Division: $64 \div 8 = 8$
• The equation is now: $14 + 32 \times 8 – 2$
• Multiplication: $32 \times 8 = 256$
• The equation is now: $14 + 256 – 2$
• Addition/Subtraction (from left to right): $14 + 256 = 270$
• The equation is now: $270 – 2$
• Final result: $270 – 2 = 268$

Since $268 \neq 88$, Option 2 is incorrect.

Analyzing Option 3: Interchange 32 and 64; + and ÷

Original equation: $14 \times 8 + 64 \div 32 – 2 = 88$

Applying the interchanges (32 $\leftrightarrow$ 64, + $\leftrightarrow$ $\div$):

Replace 32 with 64 and 64 with 32.

Replace + with $\div$ and $\div$ with +.

The new equation becomes: $14 \times 8 \div 32 + 64 – 2$

Evaluating the expression:

• Multiplication/Division (from left to right): $14 \times 8 = 112$
• The equation is now: $112 \div 32 + 64 – 2$
• Division: $112 \div 32 = 3.5$
• The equation is now: $3.5 + 64 – 2$
• Addition/Subtraction (from left to right): $3.5 + 64 = 67.5$
• The equation is now: $67.5 – 2$
• Final result: $67.5 – 2 = 65.5$

Since $65.5 \neq 88$, Option 3 is incorrect.

Analyzing Option 4: Interchange 8 and 2; – and ÷

Original equation: $14 \times 8 + 64 \div 32 – 2 = 88$

Applying the interchanges (8 $\leftrightarrow$ 2, – $\leftrightarrow$ $\div$):

Replace 8 with 2 and 2 with 8.

Replace – with $\div$ and $\div$ with –.

The new equation becomes: $14 \times 2 + 64 – 32 \div 8$

Evaluating the expression using BODMAS/PEMDAS:

• Division: $32 \div 8 = 4$
• The equation is now: $14 \times 2 + 64 – 4$
• Multiplication: $14 \times 2 = 28$
• The equation is now: $28 + 64 – 4$
• Addition/Subtraction (from left to right): $28 + 64 = 92$
• The equation is now: $92 – 4$
• Final result: $92 – 4 = 88$

Since $88 = 88$, the equation becomes correct after these interchanges.

Therefore, interchanging 8 and 2, and – and $\div$ makes the equation correct.

Revision Table: Equation Solving by Interchange

When solving problems involving interchanging signs and numbers to correct an equation, always follow these steps:

• Identify the proposed changes in each option.
• Apply the changes carefully to the original equation.
• Evaluate the new equation strictly following the order of operations (BODMAS/PEMDAS).
• Compare the result with the target value (in this case, 88).
• The option that yields the correct result is the answer.

Additional Information: Order of Operations (BODMAS/PEMDAS)

The order of operations is crucial for evaluating mathematical expressions correctly. It dictates the sequence in which different operations should be performed.

• Brackets (or Parentheses): Evaluate expressions inside brackets first.
• Orders (or Exponents): Evaluate powers and square roots next.
• Division and Multiplication: Perform division and multiplication from left to right.
• Addition and Subtraction: Perform addition and subtraction from left to right.

Ignoring the order of operations can lead to incorrect results when evaluating equations with multiple operations.