Balancing Mathematical Equations with Signs

The problem asks us to find the correct sequence of mathematical signs (−, +, ÷, ×, =) to replace the asterisks (*) in the given expression so that the equation becomes balanced. The expression is:

\(32 * 14 * 18 * 9 * 12 * 22\)

We need to test the options provided to see which sequence of signs makes the left side of the equation equal to the right side (which will be the number after the '=' sign).

Testing the Options to Balance the Equation

Let's evaluate each option by substituting the signs into the equation and following the order of operations (BODMAS/PEMDAS: Brackets, Orders/Exponents, Division and Multiplication (from left to right), Addition and Subtraction (from left to right)).

Option 1: −, +, ÷, ×, =

Substituting these signs, the equation becomes:

\(32 - 14 + 18 \div 9 \times 12 = 22\)

Let's calculate the left side:

1. Perform Division: \(18 \div 9 = 2\)
2. The expression becomes: \(32 - 14 + 2 \times 12 = 22\)
3. Perform Multiplication: \(2 \times 12 = 24\)
4. The expression becomes: \(32 - 14 + 24 = 22\)
5. Perform Subtraction and Addition from left to right: \(32 - 14 = 18\)
6. The expression becomes: \(18 + 24 = 22\)
7. Perform Addition: \(18 + 24 = 42\)

So, the equation becomes \(42 = 22\), which is false. Option 1 is incorrect.

Option 2: +, −, ÷, ×, =

Substituting these signs, the equation becomes:

\(32 + 14 - 18 \div 9 \times 12 = 22\)

Let's calculate the left side following the order of operations:

1. Perform Division: \(18 \div 9 = 2\)
2. The expression becomes: \(32 + 14 - 2 \times 12 = 22\)
3. Perform Multiplication: \(2 \times 12 = 24\)
4. The expression becomes: \(32 + 14 - 24 = 22\)
5. Perform Addition and Subtraction from left to right: \(32 + 14 = 46\)
6. The expression becomes: \(46 - 24 = 22\)
7. Perform Subtraction: \(46 - 24 = 22\)

So, the equation becomes \(22 = 22\), which is true. Option 2 balances the equation.

Option 3: ×, +, ÷, −, =

Substituting these signs, the equation becomes:

\(32 \times 14 + 18 \div 9 - 12 = 22\)

Let's calculate the left side:

1. Perform Division: \(18 \div 9 = 2\)
2. The expression becomes: \(32 \times 14 + 2 - 12 = 22\)
3. Perform Multiplication: \(32 \times 14 = 448\)
4. The expression becomes: \(448 + 2 - 12 = 22\)
5. Perform Addition and Subtraction from left to right: \(448 + 2 = 450\)
6. The expression becomes: \(450 - 12 = 22\)
7. Perform Subtraction: \(450 - 12 = 438\)

So, the equation becomes \(438 = 22\), which is false. Option 3 is incorrect.

Option 4: ÷, −, +, ×, =

Substituting these signs, the equation becomes:

\(32 \div 14 - 18 + 9 \times 12 = 22\)

Let's calculate the left side:

1. Perform Division: \(32 \div 14\). This does not result in an integer, suggesting this option is unlikely to balance to a whole number like 22. \(32 \div 14 = \frac{32}{14} = \frac{16}{7}\)
2. The expression becomes: \(\frac{16}{7} - 18 + 9 \times 12 = 22\)
3. Perform Multiplication: \(9 \times 12 = 108\)
4. The expression becomes: \(\frac{16}{7} - 18 + 108 = 22\)
5. Combine integers: \( - 18 + 108 = 90\)
6. The expression becomes: \(\frac{16}{7} + 90 = 22\)
7. Perform Addition: \(\frac{16}{7} + \frac{90 \times 7}{7} = \frac{16 + 630}{7} = \frac{646}{7}\)

So, the equation becomes \(\frac{646}{7} = 22\), which is false (\(\frac{646}{7} \approx 92.28\)). Option 4 is incorrect.

Conclusion for Balancing Equation

Option 2 is the only combination of mathematical signs that correctly balances the given equation.

The correct sequence of signs is +, −, ÷, ×, =.

Revision Table: Mathematical Signs and Order of Operations

Additional Information: Solving Mathematical Sign Puzzles

Mathematical sign puzzles test your ability to apply the correct order of operations (BODMAS/PEMDAS) when different arithmetic operations are present in an expression. Here are some tips for solving these puzzles:

• Understand the Order of Operations: Remember BODMAS/PEMDAS (Brackets, Orders/Exponents, Division and Multiplication (left-to-right), Addition and Subtraction (left-to-right)).
• Substitute and Calculate: Replace the placeholders (*) with the given signs for each option and evaluate the expression carefully.
• Work Step-by-Step: Perform operations in the correct order. First division and multiplication, then addition and subtraction. If there are multiple operations of the same level (like division and multiplication), perform them from left to right.
• Check for Integer Results: If the numbers in the puzzle are integers and the target result is an integer, look for division operations that result in whole numbers early in the process, as this can sometimes quickly eliminate options.
• Be Careful with Signs: Pay close attention to negative signs, especially when subtracting or multiplying/dividing negative numbers.

Practicing different types of these puzzles helps improve speed and accuracy in applying mathematical operations.