Two classes have average test scores 70 (m students) and 91 (n students). Combined, the average is 80. What is n/m?

Introduction / Context:
Weighted averages yield linear equations in group sizes when subgroup means and combined mean are known. Solving for the ratio avoids needing absolute headcounts.

Given Data / Assumptions:

• Group 1: mean 70 over m students.
• Group 2: mean 91 over n students.
• Combined mean = 80.

Concept / Approach:
Equation: (70m + 91n) / (m + n) = 80 ⇒ 70m + 91n = 80m + 80n ⇒ 11n = 10m ⇒ n / m = 10 / 11.

Step-by-Step Solution:

Verification / Alternative check:
Choose m = 11, n = 10: combined mean = (70*11 + 91*10) / 21 = (770 + 910) / 21 = 1680 / 21 = 80, confirming the ratio.

Why Other Options Are Wrong:
11/10, 13/10, 10/13: Do not satisfy the linear relation derived from the weighted mean 80.

Common Pitfalls:
Taking an unweighted mean of 70 and 91 or inverting the ratio by mistake.

Final Answer:
10/11