Solving Equation by Swapping Numbers

The problem requires us to identify which pair of numbers, when interchanged in the given equation, will make the equation mathematically correct.

The original equation provided is:

\(28 + 49 - 35 \div 7 \times 4 = 68\)

Understanding the Order of Operations (BODMAS/PEMDAS)

To correctly evaluate any mathematical expression, we must strictly follow the standard order of operations. This is commonly known by acronyms like BODMAS or PEMDAS.

• Brackets / Parentheses: Operations inside brackets are done first.
• Orders / Exponents: Powers and square roots are evaluated next.
• Division and Multiplication: These are done from left to right.
• Addition and Subtraction: These are done from left to right.

Evaluating the Original Equation

Let's evaluate the left-hand side of the original equation using the BODMAS rule to see if it equals 68.

\(28 + 49 - 35 \div 7 \times 4\)

Following BODMAS:

1. Perform Division: \(35 \div 7 = 5\)
2. Perform Multiplication: \(5 \times 4 = 20\)
3. Perform Addition and Subtraction from left to right: \(28 + 49 - 20 = 77 - 20 = 57\)

So, the original equation evaluates to:

\(57 = 68\)

This is false. The original equation is incorrect. We need to test the given options by swapping the numbers.

Testing the Options by Swapping Numbers

We will test the option that suggests swapping 28 and 35.

Swapping 28 and 35

Original equation: \(28 + 49 - 35 \div 7 \times 4 = 68\)

After swapping 28 and 35, the new equation becomes:

\(35 + 49 - 28 \div 7 \times 4 = 68\)

Now, let's evaluate the left-hand side of this new equation using BODMAS:

\(35 + 49 - 28 \div 7 \times 4\)

1. Perform Division: \(28 \div 7 = 4\)
2. Perform Multiplication: \(4 \times 4 = 16\)
3. Perform Addition and Subtraction from left to right: \(35 + 49 - 16 = 84 - 16 = 68\)

So, the new equation evaluates to:

\(68 = 68\)

This is true. Interchanging 28 and 35 makes the equation correct.

Conclusion

Swapping the numbers 28 and 35 makes the given mathematical equation correct.

Revision Table: Equation Correction by Swapping

Additional Information: Importance of Order of Operations

The order in which mathematical operations are performed is fundamental to obtaining the correct result in any expression. The BODMAS (or PEMDAS) rule provides a standard framework for evaluating expressions consistently.

Problems like this one, where numbers are swapped, highlight how changing the position of numbers can significantly alter the outcome of an equation, especially when different operations (like division, multiplication, addition, and subtraction) are involved.

When tackling such questions, it's essential to carefully write down the new equation after the swap and then meticulously apply the BODMAS rule step-by-step to the altered equation.