1.10 Systems of Measurement

The US Customary and Metric systems are two major measurement systems used worldwide. Understanding how to convert between units within each system and between the two systems is crucial for practical applications in everyday life and various fields.

Mastering these conversions allows for seamless communication across different measurement standards. From cooking to construction, knowing how to switch between inches and centimeters or Fahrenheit and Celsius can make a big difference in accurately interpreting and applying measurements.

US Customary System

Conversions within US customary system

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• Length
  • 12 inches (in) equal 1 foot (ft) basic unit of length in US customary system
  • 3 feet equal 1 yard (yd) commonly used for measuring fabric or distances in sports (football field)
  • 5280 feet equal 1 mile (mi) used for longer distances like travel
• Weight
  • 16 ounces (oz) equal 1 pound (lb) basic unit of weight often used for measuring food ingredients (flour, sugar)
  • 2000 pounds equal 1 ton (t) used for measuring very heavy objects (vehicles, large quantities of materials)
• Volume
  • 8 fluid ounces (fl oz) equal 1 cup (c) commonly used in cooking and baking recipes
  • 2 cups equal 1 pint (pt)
  • 2 pints equal 1 quart (qt) often used for larger liquid volumes (milk)
  • 4 quarts equal 1 gallon (gal) used for even larger volumes like fuel or paint

Calculations with mixed US units

• Add or subtract mixed units by converting to the smallest common unit first
  • Convert each measurement to the smallest unit (inches for length)
  • Add or subtract the converted values
  • Convert the result back to mixed units if needed
  • Example: 2 ft 5 in + 3 ft 8 in
    1. Convert to inches: (2 ft × 12 in/ft + 5 in) + (3 ft × 12 in/ft + 8 in)
    2. Add: 24 in + 5 in + 36 in + 8 in = 73 in
    3. Convert back: 73 in = 6 ft 1 in
• Multiply or divide mixed units by converting to a single unit first
  • Convert the mixed unit measurement to a single unit (feet for length)
  • Perform the multiplication or division
  • Convert the result back to mixed units if needed
  • Example: 2 yd 1 ft × 4
    1. Convert to feet: (2 yd × 3 ft/yd + 1 ft) = 6 ft + 1 ft = 7 ft
    2. Multiply: 7 ft × 4 = 28 ft
    3. Convert back: 28 ft = 9 yd 1 ft

Metric System

Conversions within metric system

• Metric prefixes: kilo- (k), hecto- (h), deka- (da), base unit, deci- (d), centi- (c), milli- (m)
  • Each prefix represents a power of 10 (kilo = 1000, centi = 1/100)
  • Base units are the fundamental units of measurement in the metric system (e.g., meter for length, gram for mass)
• Move the decimal point to the right when converting from larger to smaller units
  • Divide by powers of 10 to go from larger to smaller (km to m, divide by 1000)
  • Example: 3.5 kilometers (km) = 3500 meters (m)
• Move the decimal point to the left when converting from smaller to larger units
  • Multiply by powers of 10 to go from smaller to larger (mL to L, multiply by 1/1000)
  • Example: 450 milliliters (mL) = 0.45 liters (L)

Problems with mixed metric units

• Convert mixed units to a single unit before performing calculations
  • Choose the most appropriate unit for the situation (meters for length)
  • Convert each measurement to the chosen unit
  • Perform the calculation
  • Example: 2.3 m + 45 cm
    1. Convert to meters: 2.3 m + 0.45 m
    2. Add: 2.3 m + 0.45 m = 2.75 m

Converting Between Systems

US customary vs metric translations

• Length
  • 1 inch ≈ 2.54 centimeters
  • 1 foot ≈ 0.3048 meters
  • 1 mile ≈ 1.609 kilometers
• Volume
  • 1 quart ≈ 0.946 liters
  • 1 gallon ≈ 3.785 liters
• Use approximate conversion factors to translate between systems
  • Multiply US customary unit by conversion factor to get metric equivalent
  • Divide metric unit by conversion factor to get US customary equivalent
• Unit conversion involves changing from one unit of measurement to another while maintaining the same quantity

Fahrenheit vs Celsius conversions

• Fahrenheit to Celsius: $°C = (°F - 32) × \frac{5}{9}$
  • Subtract 32 from Fahrenheit temperature
  • Multiply the result by $\frac{5}{9}$
  • Example: 77°F = $(77 - 32) × \frac{5}{9}$ = 25°C
• Celsius to Fahrenheit: $°F = °C × \frac{9}{5} + 32$
  • Multiply Celsius temperature by $\frac{9}{5}$
  • Add 32 to the result
  • Example: 30°C = $30 × \frac{9}{5} + 32$ = 86°F

Measurement Precision and Accuracy

• Precision refers to the level of detail in a measurement, often indicated by the number of decimal places
• Accuracy is how close a measurement is to the true value
• Significant figures are the meaningful digits in a measurement that contribute to its precision