Weighted average of two groups, fast

Question

One desk of 30 traders averages a score of 80 on an arithmetic test; another desk of 20 traders averages 90. Solve in your head. What is the combined average? Aim: under 10 seconds.

Accepted Answer

Fast route, weights, not totals. The groups are in ratio 30 : 20 = 3 : 2, so the combined average sits 	frac{2}{5} of the way from 80 to 90: 80 + \frac{2}{5}(90 - 80) = 80 + 4 = 84. The slow route (\frac{30 \cdot 80 + 20 \cdot 90}{50} = \frac{2400 + 1800}{50} = 84) confirms it, but requires bigger numbers. Takeaway. A weighted average is a lever: it sits between the two group means, closer to the heavier group, splitting the gap in the reverse ratio of the weights. Working with the 10-point gap instead of raw totals keeps every number tiny.

Weighted average of two groups, fast

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