ONE-STEP CONVERSIONS

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You have seen multiplying by [beautiful math coming... please be patient] $\,1\,$ in an appropriate form
used to get a common denominator when adding fractions;
this is only one of a multitude of uses for this important technique.

One of the most common applications of multiplying by [beautiful math coming... please be patient] $\,1\,$ in an appropriate form
occurs in the context of unit conversion, which is the subject of this section and the next.

Often, in life, you're required to convert a quantity from one unit to another.
For example, you might need to convert centimeters to inches; miles to feet; tablespoons to teaspoons;
or feet/second (read as “feet per second”) to miles/hour (“miles per hour”).
In such cases, you have a quantity of interest,
but are seeking a new name for that quantity.
The process of finding this new name is called unit conversion.

For example, since $\,1 \text{ meter} = 1000 \text{ millimeters}\,$,
[beautiful math coming... please be patient] $$ \frac{1\text{ meter}}{1000\text{ millimeters}} = \frac{1000\text{ millimeters}}{1\text{ meter}} = 1 $$ Or, since since $\,1 \text{ tablespoon} = 3 \text{ teaspoons}\,$,
[beautiful math coming... please be patient] $$ \frac{1\text{ tablespoon}}{3\text{ teaspoons}} = \frac{3\text{ teaspoons}}{1\text{ tablespoon}} = 1 $$

A “one-step” conversion requires multiplying by the number [beautiful math coming... please be patient] $\,1\,$ only once.

Any unit of length can be converted to any other unit of length. For example,

Or,
Notice the different names for the number [beautiful math coming... please be patient] $\,1\,$ needed in each case!
In the first example, inches was on top, so we also needed inches on the bottom to cancel out.
In the second example, feet was on top, so we also needed feet on the bottom to cancel out.
Units cancel the same way that numbers cancel!

Any unit of time can be converted to any other unit of time. For example,
[beautiful math coming... please be patient] $$ 5\text{ sec} = 5{\text{ sec}\kern{-15px}/}\,\,\,\, \cdot \frac{1\text{ min}}{60\text{ sec}\kern{-15px}/} = \frac{5}{60}\text{ min} \approx 0.083\text{ min} $$

Any unit of volume can be converted to any other unit of volume, and so on.

Feel free to use scrap paper and a calculator to compute your answers.
Use only the conversion information given in the Unit Conversion Tables to compute your answers.