AMC 10 · 2021 · #6

Grade 6 rate-ratio

Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Percent / Fraction Recovery

weighted-averageratio-proportionmean-median-mode-rangefraction-arithmetic easier-related-problemidentify-subproblems ↑ Prerequisites: mean-median-mode-rangeratio-proportion

📏 Short solution 💡 2 insights

Problem

Ms. Blackwell gave one exam to two classes. The morning class averaged 84 and the afternoon class averaged 70. The morning class is smaller than the afternoon class — their student counts are in the ratio 3 : 4. Find the overall (combined) mean of all students' scores.

Pick an answer.

(A)

~74

(B)

~75

(C)

~76

(D)

~77

(E)

~78

AMC 10 2021 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Easier Related Problem

Tool #9 (Easier Related Problem): replace the unspecified class sizes with the smallest whole numbers in ratio 3:4 — pick 3 morning students and 4 afternoon students. The combined mean does not depend on the absolute counts, so this concrete case gives the same answer. Tool #8 (Units / total-over-count) computes the mean as total points ÷ total students. Tool #3 (Eliminate) checks: since the afternoon group is larger, the combined mean must sit below the midpoint (70 + 84)/2 = 77, eliminating (D) and (E) before any arithmetic.

1STEP 1

Reduce the ratio: let the morning class be 3 students, the afternoon class 4 — smallest counts in 3 : 4; the combined mean is unchanged.

#morning = 3, #afternoon = 4

2STEP 2

Each class's points = mean × size: morning 252 = 84 · 3, afternoon 280 = 70 · 4.

252 = 84 · 3, 280 = 70 · 4

3STEP 3

Add across classes: grand total 532 = 252 + 280 points over 7 = 3 + 4 students.

total points = 252 + 280 = 532, total students = 3 + 4 = 7

4STEP 4

Combined mean = grand total ÷ students = 532 ÷ 7 = 76.

x =

\frac{532}{7}

= 76

5STEP 5

Eliminate: the mean sits between 70 and 84, pulled below the midpoint 77 by the bigger class — only 76 fits, choice (C).

76 → (C)

Answer

~76

Sanity: 76 lies between 70 and 84 (good) and is closer to 70 than to 84 (76 - 70 = 6 vs 84 - 76 = 8) — consistent with the afternoon class being the bigger group. Also, scaling the class sizes by 10 (30 morning, 40 afternoon) gives total 84 · 30 + 70 · 40 = 2520 + 2800 = 5320, divided by 70 students = 76 — same answer, confirming the ratio invariance.

💡Key takeaway

This AMC 10 problem only needs Grade 6 ratio thinking you already know — pretend there are just 3 morning students and 4 afternoon students (the ratio is the same), add up 3 · 84 + 4 · 70 = 532 points across 7 students, and divide: 532 ÷ 7 = 76, choice (C).

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