Lesson 4: Conversions

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

Units of length:

A conversion factor is a fraction equal to 1, such as $\frac{3 \text{ feet}}{1 \text{ yard}}$. We choose the factor that cancels the old units and leaves the new ones. Such conversion factors are also called unit fractions.

Examples

• A person is 5 feet 4 inches tall. To express this in inches, calculate $5 \times 12 + 4 = 64$ inches.

• A piece of fabric is 90 inches long. To find its length in feet, you divide: $\frac{90}{12} = 7.5$ feet.