Questions: Repeat the following procedure for the four given numbers. Multiply the number by 10. Add 10 to the product. Divide this sum by 2. Subtract 5 from the quotient. The 1st number is 1. The result is 5 The 2nd number is 6. The result is The 3rd number is 9. The result is . The 4th number is 10. The result is 55 a. Write a conjecture that relates the result of the process to the original number selected. Represent the original number as n. The result is . (Simplify your answer.) b. Represent the original number as n, and use deductive reasoning to prove the conjecture in part (a). Multiply the number by 10. Add 10 to the product. Divide the sum by 2. (Simplify your answer.) Subtract 5 from the quotient.

Solution Steps

To solve the given problem, we need to follow the specified procedure for each number and then generalize the result to form a conjecture. We will then use deductive reasoning to prove the conjecture.

1. For each number, perform the following steps:

   • Multiply the number by 10.
   • Add 10 to the product.
   • Divide the sum by 2.
   • Subtract 5 from the quotient.

2. Form a conjecture based on the results obtained from the above steps.

3. Use algebraic manipulation to prove the conjecture.

1. Apply the given procedure to each number.
2. Form a conjecture based on the results.
3. Use algebra to prove the conjecture.

We start by applying the given procedure to each of the four numbers:

1. Multiply the number by 10.
2. Add 10 to the product.
3. Divide the sum by 2.
4. Subtract 5 from the quotient.

For the numbers \(1\), \(6\), \(9\), and \(10\), the results are:

• For \(n = 1\): \[ \left(\frac{1 \times 10 + 10}{2}\right) - 5 = \left(\frac{20}{2}\right) - 5 = 10 - 5 = 5.0 \]

• For \(n = 6\): \[ \left(\frac{6 \times 10 + 10}{2}\right) - 5 = \left(\frac{70}{2}\right) - 5 = 35 - 5 = 30.0 \]

• For \(n = 9\): \[ \left(\frac{9 \times 10 + 10}{2}\right) - 5 = \left(\frac{100}{2}\right) - 5 = 50 - 5 = 45.0 \]

• For \(n = 10\): \[ \left(\frac{10 \times 10 + 10}{2}\right) - 5 = \left(\frac{110}{2}\right) - 5 = 55 - 5 = 50.0 \]

Based on the results obtained, we observe that the result is always \(5\) times the original number \(n\). Thus, we form the conjecture:

\[ \text{Result} = 5n \]

To prove the conjecture, we represent the original number as \(n\) and follow the steps algebraically:

1. Multiply the number by 10: \[ 10n \]

2. Add 10 to the product: \[ 10n + 10 \]

3. Divide the sum by 2: \[ \frac{10n + 10}{2} = 5n + 5 \]

4. Subtract 5 from the quotient: \[ 5n + 5 - 5 = 5n \]

Thus, the conjecture is proven to be correct.

Final Answer