Unit Conversion Notes

Distance and Length

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

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 $1 \text{ (prefix)} = 10^{n} \text{ (base unit)}$
• Place the multiplier next to the base unit; put 1 on the side with the prefix
• Example: $1 \text{ km} = 10^{3} \text{ m}$
• Common examples: $1 \text{ GW} = 1 \times 10^{9} \text{ W}, \ 1 \text{ J} = 10^{6} \text{ µJ}$, $1 \text{ cm} = 10^{-2} \text{ m}$, $1 \text{ ns} = 10^{-9} \text{ s}$, $1 \text{ µm} = 10^{-6} \text{ m}$

Quick examples (one-step conversions)

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

Word and multi-step problems (selected methods)

• Cakes in time: if 7 cakes in 3 hours, then in 10 hours: $10 \text{ h} \times \frac{7 \text{ cakes}}{3 \text{ h}} = \frac{70}{3} \approx 23.3 \text{ cakes}$
• Map scale (inches to miles): if 1 in = 5.5 miles and distance = 12.6 in, then distance = $12.6 \times 5.5 = 69.3 \text{ miles}$
• Inches to yards (2-step): 180 in → ft → yd: $180 \text{ in} \to \text{ ft} \to \text{ yd}$ with $12 \text{ in} = 1 \text{ ft}, \ 3 \text{ ft} = 1 \text{ yd}$, result $5 \text{ yd}$
• Feet to kilometers (two-step): 9000 ft → miles → km, using $1 \text{ mile} = 5280 \text{ ft}, \ 1 \text{ mile} = 1.609 \text{ km}$, result $9000/5280 \times 1.609 \approx 2.7426 \text{ km}$
• Milliliters to gallons (volume): 7500 mL → L → gal, using $1 \text{ L} = 1000 \text{ mL}, \ 1 \text{ gal} = 3.785 \text{ L}$, result $7.5 / 3.785 \approx 1.98 \text{ gal}$
• Pounds and ounces to kilograms: 7 lb 12 oz → kg
  • lb to kg: $1 \text{ kg} = 2.2 \text{ lb}$ ⇒ $7 \text{ lb} = 7/2.2 \approx 3.182 \text{ kg}$
  • oz to kg: 12 oz → lb: $1 \text{ lb} = 16 \text{ oz}$ ⇒ $12/16 = 0.75 \text{ lb} \, \Rightarrow\, 0.75/2.2 \approx 0.341 \text{ kg}$
  • Total ≈ $3.182 + 0.341 = 3.523 \text{ kg}$
• Cupcakes in boxes: 12 boxes × 28 cupcakes/box = 336 cupcakes; 5 cups flour per 3 cupcakes → cups flour = $336/3 \times 5 = 112 \times 5 = 560 \text{ cups}$
• Metric: mm to m (43). 38.6 mm → m: as above, $3.86 \times 10^{-2} \text{ m}$
• 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}