Making Unit Conversions in the U.S. System

In this topic, we will see how to convert among different types of units, such as feet to miles or kilograms to pounds. The basic idea in all of the unit conversions will be to use a form of \(1,\) the multiplicative identity, to change the units but not the value of a quantity. \(\require{cancel}\)

Making Unit Conversions in the U.S. System

There are two systems of measurement commonly used around the world. Most countries use the metric system. The United States uses a different system of measurement, usually called the U.S. system. We will look at the U.S. system first.

The U.S. system of measurement uses units of inch, foot, yard, and mile to measure length and pound and ton to measure weight. For capacity, the units used are cup, pint, quart and gallons. Both the U.S. system and the metric system measure time in seconds, minutes, or hours.

The equivalencies among the basic units of the U.S. system of measurement are listed in the table below. The table also shows, in parentheses, the common abbreviations for each measurement.

In many real-life applications, we need to convert between units of measurement. We will use the identity property of multiplication to do these conversions. We’ll restate the Identity Property of Multiplication here for easy reference.

To use the identity property of multiplication, we write \(1\) in a form that will help us convert the units. For example, suppose we want to convert inches to feet. We know that \(1\) foot is equal to \(12\) inches, so we can write \(1\) as the fraction \(\frac{\text{1 ft}}{\text{12 in}}.\) When we multiply by this fraction, we do not change the value but just change the units.

But \(\frac{\text{12 in}}{\text{1 ft}}\) also equals \(1.\) How do we decide whether to multiply by \(\frac{\text{1 ft}}{\text{12 in}}\) or \(\frac{\text{12 in}}{\text{1 ft}}?\) We choose the fraction that will make the units we want to convert from divide out. For example, suppose we wanted to convert \(60\) inches to feet. If we choose the fraction that has inches in the denominator, we can eliminate the inches.

On the other hand, if we wanted to convert \(5\) feet to inches, we would choose the fraction that has feet in the denominator.

We treat the unit words like factors and ‘divide out’ common units like we do common factors.

When we use the Identity Property of Multiplication to convert units, we need to make sure the units we want to change from will divide out. Usually this means we want the conversion fraction to have those units in the denominator.

Example

Ndula, an elephant at the San Diego Safari Park, weighs almost \(3.2\) tons. Convert her weight to pounds.

(credit: Guldo Da Rozze, Flickr)

Solution

We will convert \(3.2\) tons into pounds, using the equivalencies in the table below. We will use the Identity Property of Multiplication, writing \(1\) as the fraction \(\frac{\text{2000 pounds}}{\text{1 ton}}.\)

	\(\text{3.2 tons}\)
Multiply the measurement to be converted by 1.	\(\text{3.2 tons}·1\)
Write 1 as a fraction relating tons and pounds.	\(\text{3.2 tons}·\frac{\text{2000 lbs}}{\text{1 ton}}\)
Simplify.	\(\frac{3.2\phantom{\rule{0.2em}{0ex}}\cancel{\text{tons}}·\text{2000 lbs}}{1\phantom{\rule{0.2em}{0ex}}\cancel{\text{ton}}}\)
Multiply.	\(\text{6400 lbs}\)
	Ndula weighs almost 6,400 pounds.

Sometimes to convert from one unit to another, we may need to use several other units in between, so we will need to multiply several fractions.

Example

Juliet is going with her family to their summer home. She will be away for \(9\) weeks. Convert the time to minutes.

Solution

To convert weeks into minutes, we will convert weeks to days, days to hours, and then hours to minutes. To do this, we will multiply by conversion factors of \(1.\)

	\(\text{9 weeks}\)
Write 1 as \(\frac{7\phantom{\rule{0.2em}{0ex}}\text{days}}{1\phantom{\rule{0.2em}{0ex}}\text{week}},\frac{24\phantom{\rule{0.2em}{0ex}}\text{hours}}{1\phantom{\rule{0.2em}{0ex}}\text{day}},\frac{60\phantom{\rule{0.2em}{0ex}}\text{minutes}}{1\phantom{\rule{0.2em}{0ex}}\text{hour}}\).
Cancel common units.
Multiply.	\(\frac{9·7·24·60\phantom{\rule{0.2em}{0ex}}\text{min}}{1·1·1·1}=90,720\phantom{\rule{0.2em}{0ex}}\text{min}\)
	Juliet will be away for 90,720 minutes.

Example

How many fluid ounces are in \(1\) gallon of milk?

(credit: www.bluewaikiki.com, Flickr)

Solution

Use conversion factors to get the right units: convert gallons to quarts, quarts to pints, pints to cups, and cups to fluid ounces.

	1 gallon
Multiply the measurement to be converted by 1.	\(\frac{\text{1 gal}}{1}·\frac{\text{4 qt}}{\text{1 gal}}·\frac{\text{2 pt}}{\text{1 qt}}·\frac{\text{2 C}}{\text{1 pt}}·\frac{\text{8 fl oz}}{\text{1 C}}\)
Simplify.	\(\frac{1\phantom{\rule{0.2em}{0ex}}\cancel{\text{gal}}}{1}·\frac{4\phantom{\rule{0.2em}{0ex}}\cancel{\text{qt}}}{1\phantom{\rule{0.2em}{0ex}}\cancel{\text{gal}}}·\frac{2\phantom{\rule{0.2em}{0ex}}\cancel{\text{pt}}}{1\phantom{\rule{0.2em}{0ex}}\cancel{\text{qt}}}·\frac{2\phantom{\rule{0.2em}{0ex}}\cancel{\text{C}}}{1\phantom{\rule{0.2em}{0ex}}\cancel{\text{pt}}}·\frac{\text{8 fl oz}}{1\phantom{\rule{0.2em}{0ex}}\cancel{\text{C}}}\)
Multiply.	\(\frac{1·4·2·2·\text{8 fl oz}}{1·1·1·1·1}\)
Simplify.	128 fluid ounces
	There are 128 fluid ounces in a gallon.

Optional Video: American Unit Conversion