Unit Conversions: Quick Notes

Railroad Tracks Method for Unit Conversions

  • Concept: treat unit conversions as joining railroad tracks; cancel units by placing fractions so units on opposite sides cancel out.
  • Key base conversions:
    • 1 km=103 m1\ \text{km} = 10^3\ \text{m}
    • 1 furlong=200 m1\ \text{furlong} = 200\ \text{m}
  • Example: Mongol Derby length conversion
    • Start with 1000 km1000\ \text{km}.
    • Use fractions that keep kilometers and meters consistent:
    • 103 m1 km\frac{10^3\ \text{m}}{1\ \text{km}} and 1 furlong200 m\frac{1\ \text{furlong}}{200\ \text{m}}
    • Multiply top and bottom: 1000×103200=5×103\frac{1000\times 10^3}{200} = 5\times 10^3
    • Result: 1000 km=5×103 furlongs1000\ \text{km} = 5\times 10^3\ \text{furlongs}
  • Context: distance of the Kentucky Derby is 10 furlongs10\ \text{furlongs}

Nanoliter to Fluid Ounce via Flow Chart

  • Principle: go from nL→L→mL→fl oz\text{nL} \rightarrow \text{L} \rightarrow \text{mL} \rightarrow \text{fl oz} using fractions that cancel intermediate units.
  • Key conversions:
    • 1 nL=10−9 L1\ \text{nL} = 10^{-9}\ \text{L}
    • 1 mL=10−3 L1\ \text{mL} = 10^{-3}\ \text{L}
    • 1 fl oz=29.57 mL1\ \text{fl oz} = 29.57\ \text{mL}
  • Example with 2.09 nL:
    • 2.09 nL×10−9 L1 nL×1 mL10−3 L×1 fl oz29.57 mL2.09\ \text{nL} \times \frac{10^{-9}\ \text{L}}{1\ \text{nL}} \times \frac{1\ \text{mL}}{10^{-3}\ \text{L}} \times \frac{1\ \text{fl oz}}{29.57\ \text{mL}}
    • Unit cancellations: nL → L → mL → fl oz
    • Numeric top: 2.09×10−92.09\times 10^{-9}; bottom: 2.957×10−22.957\times 10^{-2}
    • Result: 7.07×10−8 fl oz7.07\times 10^{-8}\ \text{fl oz}
  • Takeaway: flow charts simplify multi-step conversions and reduce errors

Quick Reference: Key Distances and Conversions

  • Mongol Derby length: 1000 km≈600 mi1000\ \text{km} \approx 600\ \text{mi}
  • Race structure: horses rotated roughly every ≈30 mi\approx 30\ \text{mi}
  • Kentucky Derby distance: 10 furlongs10\ \text{furlongs}
  • Common equivalences to memorize (for quick recall):
    • 1 km=103 m1\ \text{km} = 10^3\ \text{m}
    • 1 furlong=200 m1\ \text{furlong} = 200\ \text{m}
    • 1 nL=10−9 L1\ \text{nL} = 10^{-9}\ \text{L}
    • 1 mL=10−3 L1\ \text{mL} = 10^{-3}\ \text{L}
    • 1 fl oz=29.57 mL1\ \text{fl oz} = 29.57\ \text{mL}