Finding the Correct Mathematical Signs for Equation Balancing

The problem asks us to find the sequence of mathematical operations ($\times, \div, +, -$) that, when substituted for the asterisks (*) in the equation $16 * 3 * 2 * 11 * 9 = 22$, makes the equation true. We need to test each option provided.

We must follow the standard order of operations (often remembered by acronyms like BODMAS or PEMDAS) when evaluating the expressions: Brackets/Parentheses, Orders/Exponents, Division and Multiplication (from left to right), Addition and Subtraction (from left to right).

Testing Option 1: $\times, \div, +, -$

Let's substitute the signs from Option 1 into the equation:

Equation: $16 \times 3 \div 2 + 11 - 9$

Following the order of operations:

1. Perform Multiplication and Division from left to right:
2. $16 \times 3 = 48$
3. $48 \div 2 = 24$
4. The expression becomes: $24 + 11 - 9$
5. Perform Addition and Subtraction from left to right:
6. $24 + 11 = 35$
7. $35 - 9 = 26$

Result: $26$. This does not equal the target value of $22$. So, Option 1 is incorrect.

Testing Option 2: $\times, \div, -, +$

Let's substitute the signs from Option 2 into the equation:

Equation: $16 \times 3 \div 2 - 11 + 9$

Following the order of operations:

1. Perform Multiplication and Division from left to right:
2. $16 \times 3 = 48$
3. $48 \div 2 = 24$
4. The expression becomes: $24 - 11 + 9$
5. Perform Addition and Subtraction from left to right:
6. $24 - 11 = 13$
7. $13 + 9 = 22$

Result: $22$. This equals the target value of $22$. So, Option 2 is correct.

Testing Option 3: $\div, \times, +, +$

Let's substitute the signs from Option 3 into the equation:

Equation: $16 \div 3 \times 2 + 11 + 9$

Following the order of operations:

1. Perform Multiplication and Division from left to right:
2. $16 \div 3 = \frac{16}{3}$
3. $\frac{16}{3} \times 2 = \frac{32}{3}$
4. The expression becomes: $\frac{32}{3} + 11 + 9$
5. Perform Addition from left to right:
6. $\frac{32}{3} + 11 = \frac{32}{3} + \frac{33}{3} = \frac{65}{3}$
7. $\frac{65}{3} + 9 = \frac{65}{3} + \frac{27}{3} = \frac{92}{3}$

Result: $\frac{92}{3}$. This does not equal the target value of $22$. So, Option 3 is incorrect.

Testing Option 4: $\div, +, \times, \div$

Let's substitute the signs from Option 4 into the equation:

Equation: $16 \div 3 + 2 \times 11 \div 9$

Following the order of operations:

1. Perform Multiplication and Division from left to right:
2. $16 \div 3 = \frac{16}{3}$
3. $2 \times 11 = 22$
4. $22 \div 9 = \frac{22}{9}$
5. The expression becomes: $\frac{16}{3} + \frac{22}{9}$
6. Perform Addition:
7. Find a common denominator (which is 9): $\frac{16 \times 3}{3 \times 3} + \frac{22}{9} = \frac{48}{9} + \frac{22}{9}$
8. $\frac{48}{9} + \frac{22}{9} = \frac{70}{9}$

Result: $\frac{70}{9}$. This does not equal the target value of $22$. So, Option 4 is incorrect.

Conclusion on Mathematical Signs

Based on the tests, only the sequence of signs $\times, \div, -, +$ from Option 2 balances the given equation $16 * 3 * 2 * 11 * 9 = 22$, resulting in $22 = 22$.

Revision Table: Equation Balancing Concepts

Additional Information on Solving Equation Balancing Problems

Equation balancing problems are common in logical reasoning and quantitative aptitude tests. They assess your understanding of basic arithmetic operations and the order of operations.

• Always write down the equation with the substituted signs clearly.
• Perform calculations step-by-step, strictly following the BODMAS/PEMDAS rule.
• Be careful with division, especially if it results in fractions. Sometimes, fractions will simplify later in the calculation.
• Test each option systematically until you find the one that satisfies the equation.
• Practice with different combinations of numbers and operators to improve speed and accuracy.