1.10 Systems of Measurement

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 U.S. 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, and hours.

The equivalencies of measurements are shown in Table 1.58. 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, such as feet and yards, minutes and seconds, quarts and gallons, etc. 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 change inches to feet. We know that 1 foot is equal to 12 inches, so we will write 1 as the fraction $\frac{1\text{foot}}{12\text{inches}}.$ When we multiply by this fraction we do not change the value, but just change the units.

But $\frac{12\text{inches}}{1\text{foot}}$ also equals 1. How do we decide whether to multiply by $\frac{1\text{foot}}{12\text{inches}}$ or $\frac{12\text{inches}}{1\text{foot}}?$ We choose the fraction that will make the units we want to convert from divide out. Treat the unit words like factors and “divide out” common units like we do common factors. If we want to convert $66$ inches to feet, which multiplication will eliminate the inches?

The inches divide out and leave only feet. The second form does not have any units that will divide out and so will not help us.

Example 1.140

How to Make Unit Conversions

MaryAnne is 66 inches tall. Convert her height into feet.

Solution

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 1.141

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

Solution

We will convert 3.2 tons into pounds. We will use the identity property of multiplication, writing 1 as the fraction $\frac{2000\text{pounds}}{1\text{ton}}.$

	$\text{3.2 tons}$
Multiply the measurement to be converted, by 1.	$\text{3.2 tons}\cdot 1$
Write 1 as a fraction relating tons and pounds.	$\text{3.2 tons}\cdot \frac{\text{2,000 pounds}}{\text{1 ton}}$
Simplify.
Multiply.	$\text{6,400 pounds}$
	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 1.142

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

Solution

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

	9 weeks
Write $1$ as $\frac{\text{7 days}}{\text{1 week}}$, $\frac{\text{24 hours}}{\text{1 day}}$, and $\frac{\text{60 minutes}}{\text{1 hour}}$.	$\frac{\text{9 wk}}{1}\cdot \frac{\text{7 days}}{\text{1 wk}}\cdot \frac{\text{24 hr}}{\text{1 day}}\cdot \frac{\text{60 min}}{\text{1 hr}}$
Divide out the common units.
Multiply.	$\frac{9\cdot 7\cdot 24\cdot \text{60 min}}{1\cdot 1\cdot 1\cdot 1}$
Multiply.	$\text{90,720 min}$

Juliet and her boyfriend will be apart for 90,720 minutes (although it may seem like an eternity!).

Example 1.143

How many ounces are in 1 gallon?

Solution

We will convert gallons to ounces by multiplying by several conversion factors. Refer to Table 1.58.

	1 gallon
Multiply the measurement to be converted by 1.	$\frac{\text{1 gallon}}{1}\cdot \frac{\text{4 quarts}}{\text{1 gallon}}\cdot \frac{\text{2 pints}}{\text{1 quart}}\cdot \frac{\text{2 cups}}{\text{1 pint}}\cdot \frac{\text{8 ounces}}{\text{1 cup}}$
Use conversion factors to get to the right unit. Simplify.
Multiply.	$\frac{1\cdot 4\cdot 2\cdot 2\cdot \text{8 ounces}}{1\cdot 1\cdot 1\cdot 1\cdot 1}$
Simplify.	$\text{128 ounces}$ There are 128 ounces in a gallon.

Use Mixed Units of Measurement in the U.S. System

We often use mixed units of measurement in everyday situations. Suppose Joe is 5 feet 10 inches tall, stays at work for 7 hours and 45 minutes, and then eats a 1 pound 2 ounce steak for dinner—all these measurements have mixed units.

Performing arithmetic operations on measurements with mixed units of measures requires care. Be sure to add or subtract like units!

Example 1.144

Seymour bought three steaks for a barbecue. Their weights were 14 ounces, 1 pound 2 ounces and 1 pound 6 ounces. How many total pounds of steak did he buy?

Solution

We will add the weights of the steaks to find the total weight of the steaks.

Add the ounces. Then add the pounds.
Convert 22 ounces to pounds and ounces.	$$2 pounds + 1 pound, 6 ounces
Add the pounds.	$$3 pounds, 6 ounces
	Seymour bought 3 pounds 6 ounces of steak.

Example 1.145

Anthony bought four planks of wood that were each 6 feet 4 inches long. What is the total length of the wood he purchased?

Solution

We will multiply the length of one plank to find the total length.

Multiply the inches and then the feet.
Convert the 16 inches to feet. Add the feet.
	Anthony bought 25 feet and 4 inches of wood.

Make Unit Conversions in the Metric System

In the metric system, units are related by powers of 10. The roots words of their names reflect this relation. For example, the basic unit for measuring length is a meter. One kilometer is 1,000 meters; the prefix kilo means thousand. One centimeter is $\frac{1}{100}$ of a meter, just like one cent is $\frac{1}{100}$ of one dollar.

The equivalencies of measurements in the metric system are shown in Table 1.59. The common abbreviations for each measurement are given in parentheses.

To make conversions in the metric system, we will use the same technique we did in the US system. Using the identity property of multiplication, we will multiply by a conversion factor of one to get to the correct units.

Have you ever run a 5K or 10K race? The length of those races are measured in kilometers. The metric system is commonly used in the United States when talking about the length of a race.

Example 1.146

Nick ran a 10K race. How many meters did he run?

Solution

We will convert kilometers to meters using the identity property of multiplication.

	10 kilometers
Multiply the measurement to be converted by 1.
Write 1 as a fraction relating kilometers and meters.
Simplify.
Multiply.	10,000 meters
	Nick ran 10,000 meters.

Example 1.147

Eleanor’s newborn baby weighed 3,200 grams. How many kilograms did the baby weigh?

Solution

We will convert grams into kilograms.

Multiply the measurement to be converted by 1.
Write 1 as a function relating kilograms and grams.
Simplify.
Multiply.	$\frac{\text{3,200 kilograms}}{\mathrm{1,000}}$
Divide.	3.2 kilograms The baby weighed 3.2 kilograms.

As you become familiar with the metric system you may see a pattern. Since the system is based on multiples of ten, the calculations involve multiplying by multiples of ten. We have learned how to simplify these calculations by just moving the decimal.

To multiply by 10, 100, or 1,000, we move the decimal to the right one, two, or three places, respectively. To multiply by 0.1, 0.01, or 0.001, we move the decimal to the left one, two, or three places, respectively.

We can apply this pattern when we make measurement conversions in the metric system. In Example 1.147, we changed 3,200 grams to kilograms by multiplying by $\frac{1}{1000}$ (or 0.001). This is the same as moving the decimal three places to the left.

Use Mixed Units of Measurement in the Metric System

Performing arithmetic operations on measurements with mixed units of measures in the metric system requires the same care we used in the US system. But it may be easier because of the relation of the units to the powers of 10. Make sure to add or subtract like units.

Convert Between the U.S. and the Metric Systems of Measurement

Many measurements in the United States are made in metric units. Our soda may come in 2-liter bottles, our calcium may come in 500-mg capsules, and we may run a 5K race. To work easily in both systems, we need to be able to convert between the two systems.

Table 1.62 shows some of the most common conversions.

Figure 1.22 shows how inches and centimeters are related on a ruler.

Figure 1.22 This ruler shows inches and centimeters.

Figure 1.23 shows the ounce and milliliter markings on a measuring cup.

Figure 1.23 This measuring cup shows ounces and milliliters.

Figure 1.24 shows how pounds and kilograms marked on a bathroom scale.

Figure 1.24 This scale shows pounds and kilograms.

We make conversions between the systems just as we do within the systems—by multiplying by unit conversion factors.

Convert between Fahrenheit and Celsius Temperatures

Have you ever been in a foreign country and heard the weather forecast? If the forecast is for $22\text{°}\text{C},$ what does that mean?

The U.S. and metric systems use different scales to measure temperature. The U.S. system uses degrees Fahrenheit, written $\text{°}\text{F}\text{.}$ The metric system uses degrees Celsius, written $\text{°}\text{C}\text{.}$ Figure 1.25 shows the relationship between the two systems.

Figure 1.25 The diagram shows normal body temperature, along with the freezing and boiling temperatures of water in degrees Fahrenheit and degrees Celsius.

Example 1.153

Convert $50\text{°}$ Fahrenheit into degrees Celsius.

Solution

We will substitute $50\text{°}\text{F}$ into the formula to find C.

Simplify in parentheses.
Multiply.
	So we found that 50°F is equivalent to 10°C.

Example 1.154

While visiting Paris, Woody saw the temperature was $20\text{°}$ Celsius. Convert the temperature into degrees Fahrenheit.

Solution

We will substitute $20\text{°}\text{C}$ into the formula to find F.

Multiply.
Add.
	So we found that 20°C is equivalent to 68°F.