2021 AMC 10B Problem 6

1. The problem gives only the ratio of the class sizes, $3$ to $4$, so the answer cannot depend on the actual sizes. Pick the simplest sizes that fit: $3$ students in the morning and $4$ in the afternoon.
2. An average is a total divided by a count, so turn each average back into a total:
- Morning: $3 \times 84 = 252$ points.
- Afternoon: $4 \times 70 = 280$ points.
3. All together: $252 + 280 = 532$ points for $3 + 4 = 7$ students.
4. The overall average is
$$\frac{532}{7} = 76.$$
5. Sense check: $76$ is closer to $70$ than to $84$, which makes sense because the afternoon class is larger.

The answer is $\boxed{\textbf{(C)}\ 76}$.

When two groups are combined, the bigger group counts for more, so you cannot just average the two averages. Go back to totals: add the totals, add the counts, divide. If only a ratio of sizes is given, choose sizes that match it. Doubling both classes would double both totals and leave the average the same.

Think of the $3 + 4 = 7$ students as $7$ equal "votes." The gap between the two averages is $84 - 70 = 14$. Starting from $70$, the $3$ morning votes out of $7$ pull the average up by $\frac{3}{7}$ of that gap:
$$70 + \tfrac{3}{7} \times 14 = 70 + 6 = 76.$$

Only the ratio of class sizes matters, so suppose the morning class has $3$ students and the afternoon class has $4$.

Total of the morning scores: $3 \cdot 84 = 252$. Total of the afternoon scores: $4 \cdot 70 = 280$. Combined total $252 + 280 = 532$ over $7$ students:
$$\frac{532}{7} = 76.$$

The answer is $\boxed{\textbf{(C)}\ 76}$.