Combined average of two unequal classes

Question

A class of 40 students averages 70 on a test; a second class of 60 students averages 85. Solve in your head. What is the combined average? Aim: under 10 seconds.

Accepted Answer

Fast route, work with the gap and the weights. The two classes are in ratio 40 : 60 = 2 : 3, so the combined average sits 	frac{3}{5} of the way from 70 toward 85 (pulled toward the larger, heavier class): 70 + \frac{3}{5}(85 - 70) = 70 + \frac{3}{5}(15) = 70 + 9 = 79. The slow route confirms it: \frac{40 \cdot 70 + 60 \cdot 85}{100} = \frac{2800 + 5100}{100} = 79, but it drags four-digit products through your head. Takeaway. A combined average lands between the two group means, and the fraction of the way it moves toward a group equals that group's weight. Sliding along the 15-point gap kee…

Combined average of two unequal classes

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