Balancing Mathematical Equations: Finding the Correct Operator Sequence

The problem asks us to find the correct combination of mathematical signs from the given options that, when placed sequentially in place of the asterisk (*) symbols, will balance the equation. Balancing an equation means making sure the value of the expression on the left side of the equals sign is equal to the value of the expression on the right side.

The given equation with asterisks is:

\(15 * 1411 * 83 * 137 * 218 * 100\)

We need to test each option by replacing the asterisks from left to right with the operators provided in the option's sequence. The last operator in each sequence is the equals sign (=).

Checking Option 1: The Correct Sequence

Option 1 provides the sequence: ×, ÷, =, +, −

Let's substitute these signs into the equation:

\(15 \times 1411 \div 83 = 137 + 218 - 100\)

Now, we evaluate both sides of the equation following the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Evaluating the Left Side (LHS):

\(15 \times 1411 \div 83\)

First, perform the multiplication:

\(15 \times 1411 = 21165\)

Next, perform the division:

\(21165 \div 83\)

To calculate \(21165 \div 83\), we can perform long division or use a calculator:

\(21165 \div 83 = 255\)

So, the Left Side (LHS) equals 255.

Evaluating the Right Side (RHS):

\(137 + 218 - 100\)

First, perform the addition:

\(137 + 218 = 355\)

Next, perform the subtraction:

\(355 - 100 = 255\)

So, the Right Side (RHS) equals 255.

Comparing LHS and RHS:

LHS = 255

RHS = 255

Since LHS = RHS, the equation is balanced with the signs from Option 1.

\(255 = 255\)

Why Other Options Are Incorrect

Let's briefly look at the other options to understand why they wouldn't balance the equation:

• Option 2: ×, =, −, ÷, +
  \(15 \times 1411 = 83 - 137 \div 218 + 100\)
  The division \(137 \div 218\) does not result in a whole number, making it unlikely for the equation to balance with whole numbers elsewhere. Also, \(15 \times 1411\) is a large number (21165), while the right side involves smaller numbers, so equality is improbable.
• Option 3: +, −, ×, =, ÷
  \(15 + 1411 - 83 \times 137 = 218 \div 100\)
  The right side \(218 \div 100 = 2.18\). The left side involves multiplication of larger numbers (\(83 \times 137 = 11371\)), resulting in a large negative number (\(15 + 1411 - 11371 = 1426 - 11371 = -9945\)). Clearly, the sides are not equal.
• Option 4: ×, +, ÷, −, =
  \(15 \times 1411 + 83 \div 137 - 218 = 100\)
  The division \(83 \div 137\) does not result in a whole number. The calculation on the left side would likely result in a large number (\(15 \times 1411 = 21165\)) plus or minus smaller values, making it impossible to equal 100.

Only Option 1 provides a sequence of operators that makes the equation true.

Final Balanced Equation

Substituting the signs from Option 1 gives the balanced equation:

\(15 \times 1411 \div 83 = 137 + 218 - 100\)

Both sides evaluate to 255.

Revision Table: Equation Balancing Concepts

Additional Information: Mathematical Operators and Their Usage

Understanding basic mathematical operators is fundamental to solving equation balancing problems. Here's a quick overview:

• Addition (\(+\)): Combines two numbers to find their sum.
• Subtraction (\(-\)): Finds the difference between two numbers.
• Multiplication (\(\times\)): Repeated addition of a number by itself a certain number of times, finds the product.
• Division (\(\div\)): Splits a number into equal parts, finds the quotient.
• Equals (\(=\)): Indicates that the expression on the left side has the same value as the expression on the right side.

When evaluating an expression with multiple operators, always remember to follow the established order of operations to arrive at the correct result.