Converting Units With Conversion Factors - Metric System Review & Dimensional Analysis

Unit Conversion Overview

• Converting between various units is essential for solving many scientific problems. This includes distance, mass, volume, time, and metric system conversions.

Units Associated with Distance

• Common conversion factors include:

  • 12 inches = 1 foot

  • 3 feet = 1 yard

  • 1 inch = 2.54 centimeters

  • 1 kilometer = 1,000 meters

  • 1 meter = 100 centimeters

  • 1 mile = 1.609 kilometers

  • 1 mile = 5,280 feet

Units Associated with Mass or Weight

• Key conversion factors:

  • 1 kilogram = 1,000 grams

  • 1 kilogram ≈ 2.2 pounds (weight)

  • 1 pound = 16 ounces

  • 1 ton = 2,000 pounds

  • 1 metric ton = 1,000 kilograms ≈ 2,204 pounds

Mass vs. Weight

• Mass is a measure of the amount of matter in an object, while weight is the force exerted by gravity on that object.

Units Associated with Volume or Capacity

• Important conversion factors include:

  • 1 liter = 1,000 milliliters

  • 1 milliliter = 1 cubic centimeter

  • 1 cubic meter = 1,000 liters

  • 1 gallon = 3.785 liters

  • 1 gallon = 4 quarts

  • 1 quart = 2 pints

Units of Time

• Key time conversions:

  • 1 hour = 60 minutes

  • 1 minute = 60 seconds

  • 1 day = 24 hours

  • 1 year ≈ 365 days (or 366 days in a leap year)

  • Average year = 365.25 days (considering leap years)

  • 1 century = 100 years

  • 1 millennium = 1,000 years

The Metric System

• Important prefixes in metric conversions:

  • Tera (10^12)

  • Giga (10^9)

  • Mega (10^6)

  • Kilo (10^3)

  • Hecto (10^2)

  • Deca (10^1)

  • Base unit (1)

  • Deci (10^-1)

  • Centi (10^-2)

  • Milli (10^-3)

  • Micro (10^-6)

  • Nano (10^-9)

  • Pico (10^-12)

Example of Conversion Factor Writing

• When converting:

  • 1 kilometer = 1 x 10^3 meters

  • Examples:

    • 1 gigawatt = 1 x 10^9 watts

    • 1 megajoule = 1 x 10^6 joules

    • 1 centimeter = 1 x 10^-2 meters

    • 1 nanosecond = 1 x 10^-9 seconds

    • 1 micrometer = 1 x 10^-6 meters

Example Conversions

• Convert 4 ft to inches:

  • Conversion factor: 1 ft = 12 inches

  • Calculation: (4 ft / 1) x (12 inches / 1 ft) = 48 inches

• Convert 350 grams to kilograms:

  • Conversion: 1 kg = 1,000 g

  • Calculation: (350 g / 1) x (1 kg / 1,000 g) = 0.35 kg

• Convert 0.45 L to mL:

  • Conversion: 1 L = 1,000 mL

  • Calculation: (0.45 L / 1) x (1,000 mL / 1 L) = 450 mL

Example Word Problem 1

• Karen's Cake Problem:

  • Karen can make 7 cakes in 3 hours. How many can she make in 10 hours?

  • Conversion factor: 7 cakes = 3 hours.

  • Calculation: (10 hours / 1) x (7 cakes / 3 hours) = 23.33 cakes.

Example Word Problem 2

• Map Scale Problem:

  • Map scale: 1 inch = 5.5 miles. Distance between cities on map = 12.6 inches.

  • Calculation: (12.6 inches / 1) x (5.5 miles / 1 inch) = 69.3 miles.

Two-Step Conversion Problems

• Convert 180 inches to yards:

  • Conversion Factors: 1 yd = 3 ft, 1 ft = 12 in.

  • Calculate: (180 in / 1) x (1 ft / 12 in) x (1 yd / 3 ft) = 5 yards.

• Convert 9,000 ft to kilometers:

  • Step 1: (9,000 ft / 1) x (1 mile / 5,280 ft) = X miles

  • Step 2: (X miles) x (1.609 km / 1 mile) = Y kilometers

Example Mass Conversion

• Book's weight in kg:

  • 7 lb 12 oz = 3.182 kg (for both conversions combined).

Final Conversion Example: Metric to Metric

• Convert 38.6 mm to m:

  • 1 mm = 1 x 10^-3 m

  • 38.6 x 10^-3 = 3.86 x 10^-2 m (scientific notation).

• Convert 4,950 picometers to nanometers:

  • 1 picometer = 1 x 10^-12 m, 1 nanometer = 1 x 10^-9 m.

  • Result = 4.95 nanometers.