Unit Conversion Notes

Distance and Length

  • Common conversion facts: 12 in=1 ft, 3 ft=1 yd, 1 in=2.54 cm12 \text{ in} = 1 \text{ ft}, \ 3 \text{ ft} = 1 \text{ yd}, \ 1 \text{ in} = 2.54 \text{ cm}
  • Distance in metric: 1 km=1000 m, 1 m=100 cm1 \text{ km} = 1000 \text{ m}, \ 1 \text{ m} = 100 \text{ cm}
  • Length relations: 1 mile=1.609 km, 1 mile=5280 ft1 \text{ mile} = 1.609 \text{ km}, \ 1 \text{ mile} = 5280 \text{ ft}

Mass and Weight

  • 1 kg = 1000 g; 1 kg ≈ 2.2 lb (exactly 2.204… lb; rounded 2.2 lb)
  • 1 lb = 16 oz
  • 1 ton = 2000 lb
  • 1 metric ton = 1000 kg ≈ 2200 lb (≈ 2204 lb exact in some contexts)

Volume and Capacity

  • 1 L = 1000 mL; 1 mL = 1 cm^3
  • 1 m^3 = 1000 L
  • 1 gal = 3.785 L; 1 gal = 4 qt; 1 qt = 2 pt (less commonly needed)

Time

  • 1 h = 60 min; 1 min = 60 s
  • 1 day = 24 h; 1 year ≈ 365 days
  • Leap year every 4 years: 366 days; February has 29 days in a leap year
  • Typical calendar month ≈ 30 days
  • 1 century = 100 years; 1 millennium = 1000 years
  • Average year length ≈ 365.25 days (use 365 and 0.25 day depending on problem)

The Metric System Prefixes (memory aid)

  • tera = 10^{12}, giga = 10^{9}, mega = 10^{6}, kilo = 10^{3}
  • hecto = 10^{2}, deca = 10^{1}, base unit
  • deci = 10^{-1}, centi = 10^{-2}, milli = 10^{-3}
  • micro = 10^{-6}, nano = 10^{-9}, pico = 10^{-12}
  • Other prefixes exist (e.g., atto 10^{-18}, femto 10^{-15}); review as needed

Writing a conversion factor (rules)

  • For a prefix on the base unit, write the factor as 1 (prefix)=10n (base unit)1 \text{ (prefix)} = 10^{n} \text{ (base unit)}
  • Place the multiplier next to the base unit; put 1 on the side with the prefix
  • Example: 1 km=103 m1 \text{ km} = 10^{3} \text{ m}
  • Common examples: 1 GW=1×109 W, 1 J=106 µJ1 \text{ GW} = 1 \times 10^{9} \text{ W}, \ 1 \text{ J} = 10^{6} \text{ µJ}, 1 cm=10−2 m1 \text{ cm} = 10^{-2} \text{ m}, 1 ns=10−9 s1 \text{ ns} = 10^{-9} \text{ s}, 1 µm=10−6 m1 \text{ µm} = 10^{-6} \text{ m}

Quick examples (one-step conversions)

  • 4 ft → in: 1 ft=12 in⇒4 ft=4×12=48 in1 \text{ ft} = 12 \text{ in} \quad\Rightarrow\quad 4 \text{ ft} = 4 \times 12 = 48 \text{ in}
  • 350 g → kg: 1 kg=1000 g⇒350 g=350/1000=0.350 kg1 \text{ kg} = 1000 \text{ g} \quad\Rightarrow\quad 350 \text{ g} = 350/1000 = 0.350 \text{ kg}
  • 0.45 L → mL: 1 L=1000 mL⇒0.45 L=0.45×1000=450 mL1 \text{ L} = 1000 \text{ mL} \quad\Rightarrow\quad 0.45 \text{ L} = 0.45 \times 1000 = 450 \text{ mL}
  • 38.6 mm → m: 1 mm=10−3 m⇒38.6 mm=38.6×10−3 m=3.86×10−2 m1 \text{ mm} = 10^{-3} \text{ m} \quad\Rightarrow\quad 38.6 \text{ mm} = 38.6 \times 10^{-3} \text{ m} = 3.86 \times 10^{-2} \text{ m}

Word and multi-step problems (selected methods)

  • Cakes in time: if 7 cakes in 3 hours, then in 10 hours: 10 h×7 cakes3 h=703≈23.3 cakes10 \text{ h} \times \frac{7 \text{ cakes}}{3 \text{ h}} = \frac{70}{3} \approx 23.3 \text{ cakes}
  • Map scale (inches to miles): if 1 in = 5.5 miles and distance = 12.6 in, then distance = 12.6×5.5=69.3 miles12.6 \times 5.5 = 69.3 \text{ miles}
  • Inches to yards (2-step): 180 in → ft → yd: 180 in→ ft→ yd180 \text{ in} \to \text{ ft} \to \text{ yd} with 12 in=1 ft, 3 ft=1 yd12 \text{ in} = 1 \text{ ft}, \ 3 \text{ ft} = 1 \text{ yd}, result 5 yd5 \text{ yd}
  • Feet to kilometers (two-step): 9000 ft → miles → km, using 1 mile=5280 ft, 1 mile=1.609 km1 \text{ mile} = 5280 \text{ ft}, \ 1 \text{ mile} = 1.609 \text{ km}, result 9000/5280×1.609≈2.7426 km9000/5280 \times 1.609 \approx 2.7426 \text{ km}
  • Milliliters to gallons (volume): 7500 mL → L → gal, using 1 L=1000 mL, 1 gal=3.785 L1 \text{ L} = 1000 \text{ mL}, \ 1 \text{ gal} = 3.785 \text{ L}, result 7.5/3.785≈1.98 gal7.5 / 3.785 \approx 1.98 \text{ gal}
  • Pounds and ounces to kilograms: 7 lb 12 oz → kg
    • lb to kg: 1 kg=2.2 lb1 \text{ kg} = 2.2 \text{ lb} ⇒ 7 lb=7/2.2≈3.182 kg7 \text{ lb} = 7/2.2 \approx 3.182 \text{ kg}
    • oz to kg: 12 oz → lb: 1 lb=16 oz1 \text{ lb} = 16 \text{ oz} ⇒ 12/16=0.75 lb ⇒ 0.75/2.2≈0.341 kg12/16 = 0.75 \text{ lb} \, \Rightarrow\, 0.75/2.2 \approx 0.341 \text{ kg}
    • Total ≈ 3.182+0.341=3.523 kg3.182 + 0.341 = 3.523 \text{ kg}
  • Cupcakes in boxes: 12 boxes × 28 cupcakes/box = 336 cupcakes; 5 cups flour per 3 cupcakes → cups flour = 336/3×5=112×5=560 cups336/3 \times 5 = 112 \times 5 = 560 \text{ cups}
  • Metric: mm to m (43). 38.6 mm → m: as above, 3.86×10−2 m3.86 \times 10^{-2} \text{ m}
  • Picometers to nanometers: 4950 pm → nm
    • 1 \text{ pm} = 10^{-12} \text{ m}, 1 \text{ nm} = 10^{-9} \text{ m}
    • 4950 \text{ pm} = 4950 \times 10^{-12} \text{ m} = 4.95 \times 10^{-9} \text{ m} = 4.95 \text{ nm}