Unit conversion with ratios Knowledge-Based Study

Study notes

1. A conversion factor is a ratio that expresses how two different measurements of the same quantity relate to each other. Because the numerator and denominator describe equivalent amounts, the value of the ratio is equal to 1.

   For example, knowing that 1 meter is equal to 100 centimeters allows us to write two reciprocal unit ratios:

   • $\frac{100\text{ cm}}{1\text{ m}}$ (used when converting meters to centimeters)
   • $\frac{1\text{ m}}{100\text{ cm}}$ (used when converting centimeters to meters)

   Multiplying a measurement by a conversion ratio changes its unit without altering its actual physical size.

   When converting units using ratios, arrange the ratio so that the unit you want to eliminate is in the opposite position (numerator vs. denominator) of your original quantity.

   Step-by-Step Procedure

   1. Write the given quantity as a fraction over 1.
   2. Identify the relationship between the current unit and the desired unit.
   3. Form a ratio placing the current unit in the denominator so that units cancel diagonally.
   4. Multiply across the numerators and denominators, then simplify the numerical value.

   Example: Convert 4.5 kilometers into meters.

   $4.5\text{ km} \times \frac{1000\text{ m}}{1\text{ km}} = \frac{4.5 \times 1000}{1}\text{ m} = 4500\text{ m}$

   When no single direct conversion ratio is available, you can chain multiple ratios together. Each intermediate unit cancels out until only the target unit remains.

   Example: Chaining Ratios

   Convert 2.5 days to minutes:

   $2.5\text{ days} \times \frac{24\text{ hours}}{1\text{ day}} \times \frac{60\text{ minutes}}{1\text{ hour}} = 2.5 \times 24 \times 60\text{ minutes} = 3600\text{ minutes}$

   Converting Rates (Compound Units)

   To convert rates such as speed, convert the numerator and denominator units independently using separate ratios.

   Convert 90 kilometers per hour to meters per second:

   $\frac{90\text{ km}}{1\text{ h}} \times \frac{1000\text{ m}}{1\text{ km}} \times \frac{1\text{ h}}{3600\text{ s}} = \frac{90 \times 1000}{3600}\text{ m/s} = 25\text{ m/s}$

   To avoid errors in unit conversion, keep these key checks in mind:

   • Check Unit Orientation: Ensure units cancel algebraically. If you multiply by $\frac{1\text{ km}}{1000\text{ m}}$ when converting from kilometers, you would end up with $\text{km}^2/\text{m}$, indicating an inverted ratio.
   • Reasonableness Check: Converting to a smaller unit (e.g., meters to centimeters) must yield a larger numerical value, while converting to a larger unit (e.g., seconds to hours) must yield a smaller numerical value.
   • Multiply Fractions Correctly: Multiply all values in the numerator together, multiply all values in the denominator together, and then divide the two totals.