Unit conversions (metric and customary) Knowledge-Based Study

Study notes

1. Systems of Measurement

   In mathematics and science, measurements quantify physical properties such as length, mass, and liquid capacity. Two primary systems of measurement are commonly used:

   • Metric System (SI): A decimal-based system built on powers of 10, using standard base units like the meter (length), gram (mass), and liter (capacity), combined with prefixes such as milli- ($0.001$), centi- ($0.01$), and kilo- ($1000$).
   • US Customary System: A system based on historical English units, using varied conversion ratios such as inches, feet, pounds, and gallons.

   Converting between units requires understanding whether the target unit is larger or smaller than the starting unit, as well as applying the correct conversion factor.

   Converting Units of Length

   Length measures the distance between two points.

   Metric Length Conversions

   Because the metric system is decimal, conversions involve multiplying or dividing by powers of 10:

   • $1\text{ centimeter (cm)} = 10\text{ millimeters (mm)}$
   • $1\text{ meter (m)} = 100\text{ centimeters (cm)}$
   • $1\text{ kilometer (km)} = 1000\text{ meters (m)}$

   Customary Length Conversions

   Customary conversions rely on specific fixed ratios:

   • $1\text{ foot (ft)} = 12\text{ inches (in)}$
   • $1\text{ yard (yd)} = 3\text{ feet (ft)} = 36\text{ inches (in)}$
   • $1\text{ mile (mi)} = 5280\text{ feet (ft)} = 1760\text{ yards (yd)}$

   Example: To convert $4.5\text{ m}$ to centimeters, multiply by 100: $4.5 \times 100 = 450\text{ cm}$.

   Converting Units of Weight and Mass

   Mass refers to the amount of matter in an object, while weight measures the gravitational pull on that mass. In everyday measurement, these terms are often used interchangeably.

   Metric Mass Conversions

   • $1\text{ gram (g)} = 1000\text{ milligrams (mg)}$
   • $1\text{ kilogram (kg)} = 1000\text{ grams (g)}$
   • $1\text{ metric ton (t)} = 1000\text{ kilograms (kg)}$

   Customary Weight Conversions

   • $1\text{ pound (lb)} = 16\text{ ounces (oz)}$
   • $1\text{ ton (T)} = 2000\text{ pounds (lb)}$

   Example: To convert $48\text{ ounces}$ to pounds, divide by the conversion factor 16: $48 \div 16 = 3\text{ lb}$.

   Converting Units of Capacity

   Capacity measures the amount of liquid a container can hold.

   Metric Capacity Conversions

   • $1\text{ liter (L)} = 1000\text{ milliliters (mL)}$
   • $1\text{ kiloliter (kL)} = 1000\text{ liters (L)}$

   Customary Capacity Conversions

   • $1\text{ cup (c)} = 8\text{ fluid ounces (fl oz)}$
   • $1\text{ pint (pt)} = 2\text{ cups (c)} = 16\text{ fluid ounces (fl oz)}$
   • $1\text{ quart (qt)} = 2\text{ pints (pt)} = 4\text{ cups (c)}$
   • $1\text{ gallon (gal)} = 4\text{ quarts (qt)} = 128\text{ fluid ounces (fl oz)}$

   Example: To convert $3.5\text{ gallons}$ to quarts, multiply by 4: $3.5 \times 4 = 14\text{ qt}$.

   Conversion Strategy: Dimensional Analysis

   The unit-factor method (dimensional analysis) prevents errors by treating units algebraically:

   1. Identify the starting quantity and desired unit.
   2. Form a conversion ratio equivalent to 1, placing the unit you want to eliminate in the opposite position (numerator vs. denominator).
   3. Multiply and cancel identical units.

   Common Cross-System Equivalencies

   • Length: $1\text{ inch} \approx 2.54\text{ cm}$; $1\text{ mile} \approx 1.609\text{ km}$
   • Mass: $1\text{ pound} \approx 0.454\text{ kg}$; $1\text{ kg} \approx 2.205\text{ lb}$
   • Capacity: $1\text{ gallon} \approx 3.785\text{ liters}$; $1\text{ liter} \approx 33.814\text{ fl oz}$

   Example: Convert $15\text{ inches}$ to centimeters using $1\text{ in} = 2.54\text{ cm}$:

   $$15\text{ in} \times \frac{2.54\text{ cm}}{1\text{ in}} = 15 \times 2.54\text{ cm} = 38.1\text{ cm}$$