Understanding the Equation and Sign Interchange Problem

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

\(294 + 14 \times 4 \div 16 - 67 = 33\)

We need to evaluate the equation after interchanging signs based on the options provided and check if it equals 33. Remember to follow the order of operations (BODMAS/PEMDAS) when evaluating the expression.

Order of Operations (BODMAS/PEMDAS)

BODMAS or PEMDAS is a rule used to define the correct sequence for evaluating mathematical expressions. It stands for:

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

We will apply this rule after interchanging the signs in each option.

Testing the Sign Interchange Options

Let's test each option by swapping the specified signs in the original equation \(294 + 14 \times 4 \div 16 - 67\).

Option 1: Interchanging × and ÷

Original equation: \(294 + 14 \times 4 \div 16 - 67\)

Swap × and ÷:

\(294 + 14 \div 4 \times 16 - 67\)

Now, let's evaluate using BODMAS:

\(294 + (14 \div 4) \times 16 - 67\)

\(294 + 3.5 \times 16 - 67\)

\(294 + 56 - 67\)

\(350 - 67\)

\(283\)

The result is 283, which is not equal to 33. So, this option is incorrect.

Option 2: Interchanging + and ÷

Original equation: \(294 + 14 \times 4 \div 16 - 67\)

Swap + and ÷:

\(294 \div 14 \times 4 + 16 - 67\)

Now, let's evaluate using BODMAS:

\((294 \div 14) \times 4 + 16 - 67\)

\(21 \times 4 + 16 - 67\)

\(84 + 16 - 67\)

\(100 - 67\)

\(33\)

The result is 33, which matches the required value. So, this option is the correct sign interchange.

Option 3: Interchanging + and -

Original equation: \(294 + 14 \times 4 \div 16 - 67\)

Swap + and -:

\(294 - 14 \times 4 \div 16 + 67\)

Now, let's evaluate using BODMAS:

\(294 - (14 \times 4) \div 16 + 67\)

\(294 - 56 \div 16 + 67\)

\(294 - 3.5 + 67\)

\(290.5 + 67\)

\(357.5\)

The result is 357.5, which is not equal to 33. So, this option is incorrect.

Option 4: Interchanging + and ÷

This is the same as Option 2. We already found that interchanging + and ÷ results in 33. The option text formatting seems slightly different but represents the same swap.

Conclusion on Sign Interchange

By systematically testing each option and applying the BODMAS rule, we found that interchanging the '+' and '÷' signs makes the equation \(294 + 14 \times 4 \div 16 - 67 = 33\) correct. The modified equation becomes \(294 \div 14 \times 4 + 16 - 67\), which evaluates to 33.

Revision Table: Key Concepts

Additional Information: Solving Operator Interchange Problems

Operator interchange problems are common in reasoning and quantitative aptitude tests. They test your ability to apply mathematical rules carefully. Here are some tips for solving them:

• Always write down the original equation clearly.
• For each option, carefully swap the signs.
• Rewrite the equation with the swapped signs.
• Apply the BODMAS/PEMDAS rule step-by-step to evaluate the new expression.
• Perform calculations accurately. Even small errors can lead to the wrong answer.
• Check the result against the target value given in the original equation.
• If the first option doesn't work, move to the next, repeating the process.
• Practice different types of operator interchange problems to become faster and more accurate.

These problems emphasize the importance of understanding operator hierarchy and careful calculation.