Balancing the Mathematical Equation with Correct Signs

The problem requires us to insert the correct sequence of mathematical signs into a given expression to form a balanced equation. The equation is presented with asterisks (*) where the signs should be placed:

\(11 * 4 * 12 * 14 * 7 * 530\)

We are given options containing sequences of signs. We need to replace the asterisks from left to right with the signs from an option and check if the resulting equation is true. Since the last element is \(530\) and the options end with \(=\), the last asterisk before \(530\) should be replaced by \(=\).

Let's test the sequence of signs provided in the first option: \( \times, \times, +, \div, = \).

Replacing the * symbols sequentially with these signs gives us the equation:

\(11 \times 4 \times 12 + 14 \div 7 = 530\)

Evaluating the Equation using Order of Operations

To determine if this equation is balanced, we must evaluate the left side following the standard order of mathematical operations, often remembered by acronyms like BODMAS or PEMDAS.

The order of operations is:

1. Brackets (Parentheses)
2. Orders (Exponents, roots)
3. Division and Multiplication (from left to right)
4. Addition and Subtraction (from left to right)

Let's apply this order to the equation \(11 \times 4 \times 12 + 14 \div 7 = 530\):

Step 1: Perform Division

First, we calculate the division:

\(14 \div 7 = 2\)

Substitute this back into the equation:

\(11 \times 4 \times 12 + 2 = 530\)

Step 2: Perform Multiplication (from left to right)

Next, we calculate the multiplications. Starting from the left:

\(11 \times 4 = 44\)

The equation becomes:

\(44 \times 12 + 2 = 530\)

Now, perform the remaining multiplication:

\(44 \times 12 = 528\)

The equation is now:

\(528 + 2 = 530\)

Step 3: Perform Addition

Finally, perform the addition:

\(528 + 2 = 530\)

The left side of the equation simplifies to \(530\). The right side is also \(530\).

\(530 = 530\)

Since the left side equals the right side, the equation is balanced with the sequence of mathematical signs \( \times, \times, +, \div, = \).

Evaluating the other options would show that they do not balance the equation to \(530\).

Revision Table: Key Mathematical Operations

Additional Information: Importance of Order of Operations

Understanding the correct order of operations is crucial for solving mathematical expressions accurately. Without a standard order, the same expression could yield different results depending on which operation is performed first. Mnemonics like BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) or PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) help remember the sequence. Division and Multiplication have equal priority and are evaluated from left to right as they appear. Similarly, Addition and Subtraction have equal priority and are evaluated from left to right.