The question asks us to find the correct sequence of mathematical signs to replace the asterisks (*) in the expression 59 * 12 * 6 * 24 * 2 * 105 such that the equation is balanced. This means the expression on the left side, after replacing the signs, must evaluate to 105.

We need to test each given option, substituting the signs in the order provided, and perform the calculations following the order of operations (BODMAS/PEMDAS). The operations are typically performed in this order: Brackets/Parentheses, Orders/Exponents, Division and Multiplication (from left to right), Addition and Subtraction (from left to right).

Evaluating Option 1: −, +, −, +, =

Substituting the signs into the equation:

59 − 12 + 6 − 24 + 2 = 105

Let's calculate the left side step-by-step:

• 59 - 12 = 47
• 47 + 6 = 53
• 53 - 24 = 29
• 29 + 2 = 31

So, the equation becomes 31 = 105, which is false. Option 1 is incorrect.

Evaluating Option 2: ×, −, +, ×, =

Substituting the signs into the equation:

59 × 12 − 6 + 24 × 2 = 105

Following the order of operations (Multiplication first):

• 59 \times 12 = 708
• 24 \times 2 = 48

Now substitute these values back:

708 − 6 + 48 = 105

Perform subtraction and addition from left to right:

• 708 - 6 = 702
• 702 + 48 = 750

So, the equation becomes 750 = 105, which is false. Option 2 is incorrect.

Evaluating Option 3: +, ×, +, −, =

Substituting the signs into the equation:

59 + 12 × 6 + 24 − 2 = 105

Following the order of operations (Multiplication first):

• 12 \times 6 = 72

Now substitute this value back:

59 + 72 + 24 − 2 = 105

Perform addition and subtraction from left to right:

• 59 + 72 = 131
• 131 + 24 = 155
• 155 - 2 = 153

So, the equation becomes 153 = 105, which is false. Option 3 is incorrect.

Evaluating Option 4: −, ÷, +, ×, =

Substituting the signs into the equation:

59 − 12 ÷ 6 + 24 × 2 = 105

Following the order of operations (Division and Multiplication first, from left to right):

• 12 \div 6 = 2
• 24 \times 2 = 48

Now substitute these values back:

59 − 2 + 48 = 105

Perform subtraction and addition from left to right:

• 59 - 2 = 57
• 57 + 48 = 105

So, the equation becomes 105 = 105, which is true. Option 4 is correct.

Conclusion

By systematically testing each option and applying the correct order of operations, we found that the signs −, ÷, +, ×, = correctly balance the given equation.

Revision Table: Order of Operations

Additional Information: Equation Balancing Practice

Balancing equations by replacing signs is a common type of problem that tests your understanding of mathematical operations and the order of operations. To improve, practice solving similar problems. Pay close attention to the division operation, especially when it might result in fractions or decimals, although in many competitive exams, the numbers are chosen to give integer results at each step when the correct signs are used.

Understanding the BODMAS/PEMDAS rule is crucial for solving these types of problems accurately. A single mistake in the order can lead to an incorrect result. Always perform multiplication and division before addition and subtraction if they appear in the expression without brackets changing the order.