You have seen multiplying by $\,1\,$ 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 $\,1\,$ 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 $\,1 \text{ meter} = 1000 \text{ millimeters}\,$,
$$
\frac{1\text{ meter}}{1000\text{ millimeters}}
=
\frac{1000\text{ millimeters}}{1\text{ meter}}
= 1
$$
Or, since
since $\,1 \text{ tablespoon} = 3 \text{ teaspoons}\,$,
$$
\frac{1\text{ tablespoon}}{3\text{ teaspoons}}
=
\frac{3\text{ teaspoons}}{1\text{ tablespoon}}
= 1
$$
A “one-step” conversion requires multiplying by the number $\,1\,$ only once.
Any unit of length can be converted to any other unit of length.
For example,
Any unit of time can be converted to any other unit of time.
For example,
$$
5\text{ sec}
= 5{\text{ sec}\kern{-15px}/}\,\,\,\, \cdot
\frac{1\text{ min}}{60\text{ sec}\kern{-15px}/}
= \frac{5}{60}\text{ min}
\approx 0.083\text{ min}
$$
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.