Finding the Correct Mathematical Signs to Balance an Equation

The problem asks us to find the correct sequence of mathematical signs to replace the asterisks (*) in the expression \(18 \ast 12 \ast 4 \ast 5 \ast 6 \ast 53\) so that the equation becomes balanced. We are given options, each providing a specific order of signs.

To solve this, we need to substitute the signs from each option into the expression and evaluate the resulting equation. We must follow the standard order of operations (BODMAS or PEDMAS: Brackets/Parentheses, Orders/Exponents, Division and Multiplication (from left to right), Addition and Subtraction (from left to right)). Since the last sign in all options is '=', it indicates that the expression before the last asterisk should evaluate to the number after it.

Let's test each option provided:

Testing Option 1: \(\times, \div, =, +, \minus\)

Substituting the signs into the expression, we get:

\(18 \times 12 \div 4 = 5 + 6 - 53\)

First, evaluate the left side:

• Division: \(12 \div 4 = 3\)
• Multiplication: \(18 \times 3 = 54\)

So, the left side is \(54\).

Now, evaluate the right side:

• Addition: \(5 + 6 = 11\)
• Subtraction: \(11 - 53 = -42\)

So, the right side is \(-42\).

Comparing both sides: \(54 \neq -42\). This option does not balance the equation.

Testing Option 2: \(\times, \div, +, \minus, =\)

Substituting the signs into the expression, we get:

\(18 \times 12 \div 4 + 5 - 6 = 53\)

Evaluate the left side following the order of operations:

• Division: \(12 \div 4 = 3\)
• Multiplication: \(18 \times 3 = 54\)
• Addition: \(54 + 5 = 59\)
• Subtraction: \(59 - 6 = 53\)

So, the left side is \(53\).

The right side is \(53\).

Comparing both sides: \(53 = 53\). This option balances the equation.

Testing Option 3: \(\times, \div, \minus, +, =\)

Substituting the signs into the expression, we get:

\(18 \times 12 \div 4 - 5 + 6 = 53\)

Evaluate the left side following the order of operations:

• Division: \(12 \div 4 = 3\)
• Multiplication: \(18 \times 3 = 54\)
• Subtraction: \(54 - 5 = 49\)
• Addition: \(49 + 6 = 55\)

So, the left side is \(55\).

The right side is \(53\).

Comparing both sides: \(55 \neq 53\). This option does not balance the equation.

Testing Option 4: \(\times, \div, +, =, \minus\)

Substituting the signs into the expression, we get:

\(18 \times 12 \div 4 + 5 = 6 - 53\)

Evaluate the left side following the order of operations:

• Division: \(12 \div 4 = 3\)
• Multiplication: \(18 \times 3 = 54\)
• Addition: \(54 + 5 = 59\)

So, the left side is \(59\).

Evaluate the right side:

• Subtraction: \(6 - 53 = -47\)

So, the right side is \(-47\).

Comparing both sides: \(59 \neq -47\). This option does not balance the equation.

Based on the evaluation of each option, only Option 2 results in a balanced equation (\(53 = 53\)). Therefore, the correct combination of mathematical signs is \(\times, \div, +, \minus, =\).

Revision Table: Mathematical Sign Placement

Understanding how to place mathematical signs correctly is key to balancing equations. This problem specifically tested the order of operations alongside sign substitution.

• Key Skill: Substituting given operators (\(\times, \div, +, \minus, =\)) sequentially.
• Key Concept: Following the BODMAS/PEDMAS rule strictly during evaluation.
• Verification: Checking if the Left Hand Side (LHS) equals the Right Hand Side (RHS) after calculation.

Additional Information: Order of Operations (BODMAS/PEDMAS)

The order of operations is a set of rules that dictate the sequence in which operations should be performed in a mathematical expression to ensure a unique result.

• BODMAS: Brackets, Orders (powers/roots), Division, Multiplication, Addition, Subtraction.
• PEDMAS: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.

Division and Multiplication have the same priority and should be done from left to right. Similarly, Addition and Subtraction have the same priority and should be done from left to right. In the equation \(18 \times 12 \div 4 + 5 - 6 = 53\), we first perform \(12 \div 4\), then \(18 \times 3\), then \(+5\), and finally \(-6\).