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Averages & Mixtures

When simple averages aren't enough. Master grouped data and variable speeds.

Heart of Algebra

Weighted Average

\[ \frac{\sum (w \cdot x)}{\sum w} \]

Essential for mixture problems where groups have different sizes.

📝 Example: Class Average

Class A (10 students) has average 80. Class B (20 students) has average 95. Overall average?

\(\text{Avg} = \frac{10(80) + 20(95)}{30} = \frac{800 + 1900}{30} = 90\)

Result: 90.

The #1 Trap
Heart of Algebra

Harmonic Mean (Avg Speed)

\[ \text{Avg Speed} = \frac{2ab}{a+b} \]

When distances are equal, your average speed is NOT the simple average of speeds \(a\) and \(b\).

🧪 The Proof: Why 2ab/(a+b)?

Average Speed = Total Distance / Total Time. Let Distance = \(D\). Time to go = \(D/a\). Time to return = \(D/b\).

\(\text{Avg} = \frac{2D}{D/a + D/b} = \frac{2D}{D(1/a + 1/b)} = \frac{2}{1/a + 1/b} = \frac{2ab}{a+b}\)