Let's solve this step-by-step using the numerical examples provided in your question.
### Part (a)
To determine the relationship between the number you started with and the final number, let's consider the given process:
1. Pick any number: Let's say we start with the number [tex]\( x \)[/tex].
2. Multiply the number by 3: [tex]\( 3x \)[/tex].
3. Add 6 to the product: [tex]\( 3x + 6 \)[/tex].
4. Divide the sum by 3: [tex]\(\frac{3x + 6}{3}\)[/tex].
5. Subtract 2 from the quotient: [tex]\(\frac{3x + 6}{3} - 2\)[/tex].
Let's simplify the expression step-by-step:
[tex]\[
\frac{3x + 6}{3} = x + 2
\][/tex]
Now, subtract 2 from this result:
[tex]\[
(x + 2) - 2 = x
\][/tex]
Therefore, following these steps always returns us to our original number, [tex]\( x \)[/tex].
Thus, the final number is the same as the original number.
The correct answer is:
B. The final number is the same as the original number.
### Part (b)
Let's test this conclusion by picking another number and applying the same steps. We'll select the number 10:
1. Pick any number: [tex]\( 10 \)[/tex].
2. Multiply the number by 3: [tex]\( 10 \times 3 = 30 \)[/tex].
3. Add 6 to the product: [tex]\( 30 + 6 = 36 \)[/tex].
4. Divide the sum by 3: [tex]\( \frac{36}{3} = 12 \)[/tex].
5. Subtract 2 from the quotient: [tex]\( 12 - 2 = 10 \)[/tex].
As we can see, our final number is 10, which is the same as the original number we started with.
Thus, we can confirm that the final number is the same as the original number.
So, the correct answer is:
A. The final number is the same as the original number.
