Lesson 3: Conversion Factors

Property

To make unit conversions, we use the Identity Property of Multiplication. For any real number $a$, $a \cdot 1 = a$ and $1 \cdot a = a$. We write 1 as a fraction to change the units without changing the value.

To make unit conversions:

1. Multiply the measurement to be converted by 1; write 1 as a fraction relating the units given and the units needed.
2. Multiply.
3. Simplify the fraction, performing the indicated operations and removing the common units.

Examples

• To convert 72 inches to feet, you multiply by a fraction that cancels inches: $72 \text{ in} \cdot \frac{1 \text{ ft}}{12 \text{ in}} = 6 \text{ ft}$.
• An African elephant weighs 4.5 tons. To find its weight in pounds, you use the conversion $1 \text{ ton} = 2000 \text{ lbs}$: $4.5 \text{ tons} \cdot \frac{2000 \text{ lbs}}{1 \text{ ton}} = 9000 \text{ lbs}$.
• To find how many minutes are in 3 weeks, you chain conversions together: $\frac{3 \text{ wk}}{1} \cdot \frac{7 \text{ days}}{1 \text{ wk}} \cdot \frac{24 \text{ hr}}{1 \text{ day}} \cdot \frac{60 \text{ min}}{1 \text{ hr}} = 30,240 \text{ min}$.

Property

Multiple conversion factors can be chained together by multiplying them in sequence: $\text{original quantity} \times \frac{\text{unit}_2}{\text{unit}_1} \times \frac{\text{unit}_3}{\text{unit}_2} \times \frac{\text{unit}_4}{\text{unit}_3} = \text{final quantity in unit}_4$

Examples