Balancing Mathematical Equations with Signs

The problem asks us to find the correct sequence of mathematical signs (\(\times\), \(\div\), \(+\), \(-\)) that, when substituted for the asterisks (*) in the expression \(6 * 10 * 55 * 162 * 9\), makes the equation equal to 23. We are given four options, each providing a different sequence of signs.

Finding the Correct Sequence of Signs

To solve this problem, we need to test each option by replacing the asterisks with the given sequence of signs and then evaluate the resulting mathematical expression following the standard order of operations (BODMAS/PEMDAS). The order of operations is:

1. Brackets/Parentheses
2. Orders/Exponents
3. Division and Multiplication (from left to right)
4. Addition and Subtraction (from left to right)

We will evaluate each option to see which sequence of mathematical signs balances the given equation \(6 * 10 * 55 * 162 * 9 = 23\).

Step-by-Step Evaluation of Options

Option 1: ×, −, +, ÷

Let's substitute the signs \(\times, -, +, \div\) into the equation:

\(6 \times 10 - 55 + 162 \div 9\)

Now, we evaluate this expression using the order of operations:

1. First, perform the division: \(162 \div 9 = 18\)
2. Substitute the result back: \(6 \times 10 - 55 + 18\)
3. Next, perform the multiplication: \(6 \times 10 = 60\)
4. Substitute the result back: \(60 - 55 + 18\)
5. Now, perform addition and subtraction from left to right:
6. \(60 - 55 = 5\)
7. \(5 + 18 = 23\)

The result of evaluating the expression with this sequence of mathematical signs is 23. This matches the value on the right side of the equation.

Option 2: ÷, +, ×, −

Let's substitute the signs \(\div, +, \times, -\) into the equation:

\(6 \div 10 + 55 \times 162 - 9\)

Evaluate using the order of operations:

1. First, perform division and multiplication from left to right:
2. \(6 \div 10 = 0.6\)
3. \(55 \times 162 = 8910\)
4. Substitute the results back: \(0.6 + 8910 - 9\)
5. Now, perform addition and subtraction from left to right:
6. \(0.6 + 8910 = 8910.6\)
7. \(8910.6 - 9 = 8901.6\)

The result is 8901.6, which is not equal to 23.

Option 3: ×, ÷, −, +

Let's substitute the signs \(\times, \div, -, +\) into the equation:

\(6 \times 10 \div 55 - 162 + 9\)

Evaluate using the order of operations:

1. First, perform multiplication and division from left to right:
2. \(6 \times 10 = 60\)
3. \(60 \div 55 = \frac{60}{55} = \frac{12}{11}\)
4. Substitute the result back: \(\frac{12}{11} - 162 + 9\)
5. Now, perform addition and subtraction from left to right:
6. \(\frac{12}{11} - 162 = \frac{12 - 162 \times 11}{11} = \frac{12 - 1782}{11} = \frac{-1770}{11}\)
7. \(\frac{-1770}{11} + 9 = \frac{-1770 + 9 \times 11}{11} = \frac{-1770 + 99}{11} = \frac{-1671}{11} \approx -151.9\)

The result is approximately -151.9, which is not equal to 23.

Option 4: +, −, ÷, ×

Let's substitute the signs \(+, -, \div, \times\) into the equation:

\(6 + 10 - 55 \div 162 \times 9\)

Evaluate using the order of operations:

1. First, perform division and multiplication from left to right:
2. \(55 \div 162 = \frac{55}{162}\)
3. \(\frac{55}{162} \times 9 = \frac{55 \times 9}{162} = \frac{495}{162}\)
4. This fraction can be simplified by dividing the numerator and denominator by their greatest common divisor, which is 27. \(495 \div 27 = 18.33\) and \(162 \div 27 = 6\). Let's recheck. GCD(495, 162). \(162 = 2 \times 3^4\), \(495 = 3^2 \times 5 \times 11\). GCD is \(3^2 = 9\).
5. \(\frac{495}{162} = \frac{495 \div 9}{162 \div 9} = \frac{55}{18}\)
6. Substitute the result back: \(6 + 10 - \frac{55}{18}\)
7. Now, perform addition and subtraction from left to right:
8. \(6 + 10 = 16\)
9. \(16 - \frac{55}{18} = \frac{16 \times 18}{18} - \frac{55}{18} = \frac{288 - 55}{18} = \frac{233}{18} \approx 12.94\)

The result is approximately 12.94, which is not equal to 23.

Conclusion

Based on the evaluations, only the sequence of mathematical signs \(\times, -, +, \div\) results in the equation being balanced.

Revision Table: Arithmetic Signs and Order

Additional Information: Understanding the Order of Operations

The order of operations is a set of rules that tells us the correct sequence for performing calculations in a mathematical expression. It ensures that everyone gets the same answer for the same expression. Common acronyms like BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction) or PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) help remember this order.

Multiplication and Division have the same priority. When both appear in an expression, you perform them from left to right. Similarly, Addition and Subtraction have the same priority, and you perform them from left to right after completing all multiplications and divisions.