Average Speed Calculation for Round Trip
The problem requires calculating the average speed for a journey covering the same distance twice, but at different speeds.
Key Concepts
- Average Speed Formula: When the distance is the same for two parts of a journey, the average speed is the harmonic mean of the two speeds.
- Formula: Average Speed = \(\frac{2}{\frac{1}{v_1} + \frac{1}{v_2}}\), where \(v_1\) and \(v_2\) are the speeds for the two parts.
Given Information
- Speed from A to B (\(v_1\)) = 13 km/h
- Speed from B to A (\(v_2\)) = 10 km/h
Step-by-Step Calculation
- Apply the Average Speed Formula: Substitute the given speeds into the formula.
Average Speed = \(\frac{2}{\frac{1}{13 \, \text{km/h}} + \frac{1}{10 \, \text{km/h}}}\)
- Find a Common Denominator for the Speeds in the Denominator: The common denominator for 13 and 10 is \(13 \times 10 = 130\).
Average Speed = \(\frac{2}{\frac{10}{130} + \frac{13}{130}}\)
Average Speed = \(\frac{2}{\frac{10 + 13}{130}}\)
Average Speed = \(\frac{2}{\frac{23}{130}}\)
- Simplify the Expression: Divide 2 by the fraction \(\frac{23}{130}\).
Average Speed = \(2 \times \frac{130}{23}\)
Average Speed = \(\frac{260}{23}\) km/h
- Convert to Mixed Fraction: Divide 260 by 23.
\(260 \div 23 = 11\) with a remainder of \(7\).
Therefore, the average speed is \(11\frac{7}{23}\) km/h.
Conclusion
The average speed for the whole journey is \(\frac{260}{23}\) km/h, which is equivalent to \(11\frac{7}{23}\) km/h.
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