<Question>
A class of 30 students recently took a test.  If 20 students scored 80, 8 students scored 90, and 2 students scored 100, then what was the class average (the mean) on this test?
</Question>

<think>
<Plan_1>
Okay, let's see. The problem is asking for the class average, which is the mean score, right? There are 30 students in total. The scores are divided into three groups: 20 students scored 80, 8 scored 90, and 2 scored 100. Hmm, so I need to find the average of all their scores.

First, I remember that the mean is calculated by adding up all the individual scores and then dividing by the number of students. So, the formula should be (Total Sum of Scores) divided by (Number of Students).
The number of students is given as 30, so that part is straightforward.

Now, the tricky part is calculating the total sum. Let's break it down. There are three groups of students with different scores. For each group, I can multiply the number of students by their respective score to get the total points for that group. Then, add all those group totals together to get the overall total.

Starting with the first group: 20 students scored 80.
So, 20 times 80. Let me calculate that. 20*80 is 1600.

Next, the second group: 8 students scored 90.
So, 8*90. Hmm, 8 times 90 is 720.

Then, the third group: 2 students scored 100.
That would be 2*100, which is 200.

Now, add all those together: 1600 + 720 + 200. Let me do that step by step.
1600 + 720 is 2320, and then 2320 + 200 is 2520. So the total sum of all scores is 2520.

Now, to find the average, divide this total by the number of students, which is 30.
So, 2520 divided by 30.

Let me compute that. 2520 ÷ 30.

Well, 30 times 80 is 2400, and 2520 - 2400 is 120.
Then, 120 divided by 30 is 4. So, 80 + 4 = 84. Therefore, the average is 84.

Wait, let me verify that division again. 2520 divided by 30.
Alternatively, I can simplify the division by dividing numerator and denominator by 10 first, which gives 252 ÷ 3. 252 divided by 3.

3 goes into 25 eight times (3*8=24), remainder 1. Bring down the 2 to make 12. 3 goes into 12 four times. So that's 84.

Let me cross-check using another method. Let's see, each student's score can be considered as follows:

<prompt_1.1> Use a weighted average approach to verify the class average. </prompt_1.1>

Alternatively, maybe I can use decimal division. 2520 ÷ 30.

<prompt_1.2> Verify the class average using decimal division. </prompt_1.2>

Alternatively, thinking in terms of fractions: 2520/30. Both numbers are divisible by 10, so 252/3.
Which is the same as 84.

Another way to think about it: If each student's score is multiplied by the number of students, then summed up, that's the total.

<prompt_1.3> Reiterate the calculation of the class average using the total points and number of students. </prompt_1.3>

Alternatively, maybe use a different approach. Suppose I calculate the average per group and then find the weighted average.

<prompt_1.4> Verify the class average using a weighted average approach. </prompt_1.4>
</Plan_1>

<Execution_1>
<execution_1.1>
20 students scored 80. So, the proportion is 20/30, which simplifies to 2/3. Then, 8 students scored 90, which is 8/30, or 4/15. And 2 students scored 100, which is 2/30, or 1/15.

Then, the average would be (2/3)*80 + (4/15)*90 + (1/15)*100.
</execution_1.1>

<execution_1.2>
30 goes into 2520 how many times? 30*80=2400, 2520-2400=120. 30 goes into 120 4 times. So 80+4=84.
</execution_1.2>

<execution_1.3>
Wait, yes, that's exactly what I did earlier. So, 20*80 + 8*90 + 2*100 = 1600 + 720 + 200 = 2520. Then 2520/30 = 84.
</execution_1.3>

<execution_1.4>
Wait, that's essentially what I did before. The first group's average is 80, weighted by 20 students; the second group 90 weighted by 8; the third group 100 weighted by 2. So, the weighted average is (80*20 + 90*8 + 100*2)/(20+8+2). Which is exactly the same as before.
</execution_1.4>
</Execution_1>

<Plan_2>
Based on execution_1.1, execution_1.2, execution_1.3, execution_1.4:
Therefore, after multiple methods of calculation and verification, I'm confident that the class average is 84.
</Plan_2>

<Execution_2>
No prompt to execute.
</Execution_2>

<Final_answer>
Therefore, the class average (the mean) on this test is $\boxed{84}$.
</Final_answer>
</think>