7.5 Systems of measurement

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By the end of this section, you will be able to:
• Make unit conversions in the U.S. system
• Use mixed units of measurement in the U.S. system
• Make unit conversions in the metric system
• Use mixed units of measurement in the metric system
• Convert between the U.S. and the metric systems of measurement
• Convert between Fahrenheit and Celsius temperatures

Before you get started, take this readiness quiz.

1. Multiply: $4.29\left(1000\right).$
If you missed this problem, review Decimal Operations .
2. Simplify: $\frac{30}{54}.$
If you missed this problem, review Multiply and Divide Fractions .
3. Multiply: $\frac{7}{15}·\frac{25}{28}.$
If you missed this problem, review Multiply and Divide Fractions .

In this section 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.

Make 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 [link] . The table also shows, in parentheses, the common abbreviations for each measurement.

U.S. System Units
Length Volume
$1$ foot (ft) = $12$ inches (in)
$1$ yard (yd) = $3$ feet (ft)
$1$ mile (mi) = $5280$ feet (ft)
$3$ teaspoons (t) = $1$ tablespoon (T)
$16$ Tablespoons (T) = $1$ cup (C)
$1$ cup (C) = $8$ fluid ounces (fl oz)
$1$ pint (pt) = $2$ cups (C)
$1$ quart (qt) = $2$ pints (pt)
$1$ gallon (gal) = $4$ quarts (qt)
Weight Time
$1$ pound (lb) = $16$ ounces (oz)
$1$ ton = $2000$ pounds (lb)
$1$ minute (min) = $60$ seconds (s)
$1$ hour (h) = $60$ minutes (min)
$1$ day = $24$ hours (h)
$1$ week (wk) = $7$ days
$1$ year (yr) = $365$ days

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.

$\text{For any real number}\phantom{\rule{0.2em}{0ex}}a,\phantom{\rule{2em}{0ex}}a·1=a\phantom{\rule{2em}{0ex}}1·a=a$

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.

$60\phantom{\rule{0.2em}{0ex}}\overline{)\text{in}}·\frac{\text{1 ft}}{12\phantom{\rule{0.2em}{0ex}}\overline{)\text{in}}}=\text{5 ft}$

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

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