Understanding the Equation Balancing Problem

The question asks us to find the correct sequence of mathematical signs to replace the asterisks (*) in the given expression so that the equation is balanced. The expression is: 86 * 54 * 34 * 68 * 2 * 33. We need to select one of the provided options, each offering a different sequence of signs, and test if it makes the left side of the equation equal to 33.

To solve this type of problem, we must apply the BODMAS or PEMDAS rule (Order of Operations) correctly while performing the calculations for each option. The order is Parentheses/Brackets, Orders/Exponents, Division and Multiplication (from left to right), and Addition and Subtraction (from left to right).

Testing the Options for Balancing the Equation

Let's test each option by substituting the signs into the expression and calculating the result.

Option 1: -, +, ÷, ×, =

Substituting these signs sequentially, the equation becomes:

\(86 - 54 + 34 \div 68 \times 2 = 33\)

Now, we follow the BODMAS rule:

1. Division: \(34 \div 68 = \frac{34}{68} = \frac{1}{2} = 0.5\)
2. Multiplication: \(0.5 \times 2 = 1\)
3. Addition and Subtraction (from left to right): \(86 - 54 + 1\)
4. \(86 - 54 = 32\)
5. \(32 + 1 = 33\)

So, the equation becomes \(33 = 33\). This option successfully balances the equation.

Option 2: ÷, -, +, ×, =

Substituting these signs, the equation is:

\(86 \div 54 - 34 + 68 \times 2 = 33\)

Applying BODMAS:

1. Division: \(86 \div 54 \approx 1.5926\)
2. Multiplication: \(68 \times 2 = 136\)
3. Addition and Subtraction (from left to right): \(1.5926 - 34 + 136\)
4. \(1.5926 - 34 = -32.4074\)
5. \(-32.4074 + 136 = 103.5926\)

So, \(103.5926 \neq 33\). This option does not balance the equation.

Option 3: +, ÷, ×, -, =

Substituting these signs, the equation is:

\(86 + 54 \div 34 \times 68 - 2 = 33\)

Applying BODMAS:

1. Division: \(54 \div 34 \approx 1.5882\)
2. Multiplication: \(1.5882 \times 68 \approx 107.9976\)
3. Addition and Subtraction (from left to right): \(86 + 107.9976 - 2\)
4. \(86 + 107.9976 = 193.9976\)
5. \(193.9976 - 2 = 191.9976\)

So, \(191.9976 \neq 33\). This option does not balance the equation.

Option 4: ×, ÷, ×, -, =

Substituting these signs, the equation is:

\(86 \times 54 \div 34 \times 68 - 2 = 33\)

Applying BODMAS:

1. Multiplication/Division (from left to right):
2. \(86 \times 54 = 4644\)
3. \(4644 \div 34 \approx 136.5882\)
4. \(136.5882 \times 68 \approx 9287.9976\)
5. Subtraction: \(9287.9976 - 2 = 9285.9976\)

So, \(9285.9976 \neq 33\). This option does not balance the equation.

Conclusion

Based on the step-by-step testing, only the combination of signs in Option 1 balances the given equation.

Revision Table: Key Concepts

Additional Information: Solving Operator Problems

Problems involving finding the correct mathematical operators are common in logical reasoning and quantitative aptitude tests. Success depends on careful application of the order of operations and systematically testing each option. Here are some tips:

• Always write down the equation clearly after substituting the signs.
• Break down the calculation according to BODMAS/PEMDAS.
• Perform operations step by step to avoid errors.
• Pay close attention to division by zero (although not applicable in this specific problem, it's a common pitfall).
• If dealing with large numbers or potential fractions, approximate where reasonable or keep as fractions until necessary. In this case, \(34 \div 68\) resulting in 0.5 or 1/2 was key.
• Practice with different combinations of numbers and operators.

These equation balancing problems test your arithmetic skills and your ability to follow mathematical rules precisely.