Question

repeat the following procedure for the four given numbers. multiply the number by 12. add 10 to the product. divide this sum by 2. subtract 5 from the quotient. the 1st number is 1. the result is 6. the 2nd number is 6. the result is 36. the 3rd number is 9. the result is 54. the 4th number is 10. the result is 60. 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 \square. (simplify your answer.)

Explanation:

Step1: Define the procedure with \( n \)

Let the original number be \( n \). Multiply by 12: \( 12n \). Add 10: \( 12n + 10 \). Divide by 2: \( \frac{12n + 10}{2}=6n + 5 \). Subtract 5: \( 6n + 5 - 5 = 6n \). Wait, but looking at examples: for \( n = 1 \), result is 6 (\( 6\times1 \)); \( n = 6 \), result 36 (\( 6\times6 \)); \( n = 9 \), result 54 (\( 6\times9 \)); \( n = 10 \), result 60 (\( 6\times10 \)). Wait, maybe miscalculation. Wait, original procedure: Multiply by 12, add 10, divide by 2, subtract 5. Let's redo: \( \frac{12n + 10}{2}-5 = 6n + 5 - 5 = 6n \). But the examples: 16=6, 66=36, 96=54, 106=60. So the result is 6 times the original number. Wait, but the question is to write a conjecture. Wait, maybe I misread. Wait, the problem says "Write a conjecture that relates the result of the process to the original number. Represent the original number as \( n \)." Let's check each step with \( n \):

1. Multiply \( n \) by 12: \( 12n \)
2. Add 10: \( 12n + 10 \)
3. Divide by 2: \( \frac{12n + 10}{2}=6n + 5 \)
4. Subtract 5: \( 6n + 5 - 5 = 6n \)

Wait, but the first example: \( n = 1 \), result is 6, which is \( 6\times1 \). Second: \( n = 6 \), result 36, \( 6\times6 \). Third: \( n = 9 \), 54, \( 6\times9 \). Fourth: \( n = 10 \), 60, \( 6\times10 \). So the result is \( 6n \), meaning the result is 6 times the original number. Wait, but maybe the conjecture is that the result is 6 times the original number. Wait, but let's check the calculation again. Wait, \( \frac{12n + 10}{2}-5 = 6n + 5 - 5 = 6n \). Yes, so the result is \( 6n \), so the conjecture is that when you perform the procedure on a number \( n \), the result is \( 6n \) (or 6 times the original number). But maybe the problem is to find the result in terms of \( n \), but the question's part (a) is to write the conjecture, but the box is for "The result is \(\square\)"? Wait, maybe I misread. Wait, the problem says: "Write a conjecture... Represent the original number as \( n \). The result is \(\square\)." Wait, maybe the conjecture is that the result is 6 times the original number, so if we take \( n \) as the original number, the result is \( 6n \). But maybe the problem is to see the pattern. Let's check the examples:

• Original 1, result 6: \( 6 = 6\times1 \)
• Original 6, result 36: \( 36 = 6\times6 \)
• Original 9, result 54: \( 54 = 6\times9 \)
• Original 10, result 60: \( 60 = 6\times10 \)

So the pattern is result = 6 × original number. So the conjecture is that the result of the procedure is 6 times the original number \( n \), so the result is \( 6n \). But maybe the problem is to express the result in terms of \( n \), so the result is \( 6n \), meaning that for any original number \( n \), the result after the procedure is \( 6n \).

Step2: Confirm with examples

For \( n = 1 \): \( 6\times1 = 6 \) (matches). For \( n = 6 \): \( 6\times6 = 36 \) (matches). For \( n = 9 \): \( 6\times9 = 54 \) (matches). For \( n = 10 \): \( 6\times10 = 60 \) (matches). So the conjecture is that the result is 6 times the original number \( n \), so the result is \( 6n \). But the problem's box is to "Simplify your answer" for the result when writing the conjecture? Wait, maybe the question is to find the relationship, so the result is 6 times the original number, so if we represent the original number as \( n \), the result is \( 6n \). But maybe the problem is to see that the result is 6 times the original, so the conjecture is result = 6n.

Answer:

The result is \( 6n \) (or, based on the pattern, the result is 6 times the original number \( n \), so the result is \( 6n \)). If we take the original number \( n \), the result after the procedure is \( \boldsymbol{6n} \). But looking at the examples, for \( n = 1 \), result 6; \( n = 6 \), 36; etc., so the result is 6 times the original number. So the answer (the result in terms of \( n \)) is \( 6n \), but if we consider the conjecture, the relationship is result = 6 × original number. So the box should have \( 6n \), but maybe the problem expects recognizing that the result is 6 times the original, so for a general \( n \), result is \( 6n \).