Systems of Measurement

7.5 Systems of Measurement

The topics covered in this section are:

  1. Make unit conversions in the U.S. system
  2. Use mixed units of measurement in the U.S. system
  3. Make unit conversions in the metric system
  4. Use mixed units of measurement in the metric system
  5. Convert between the U.S. and the metric systems of measurement
  6. Convert between Fahrenheit and Celsius temperatures

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 1,1, the multiplicative identity, to change the units but not the value of a quantity.

7.5.1 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 Table 7.2. The table also shows, in parentheses, the common abbreviations for each measurement.

U.S. System Units
LengthVolume
$1$ 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)
WeightTime
$1$ 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
Table 7.2

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.

For any real number $a,$ $a \cdot 1 = a$ and $1 \cdot a = a$

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 $\frac{1 \ \mathrm{ft}}{12 \ \mathrm{in}}$. When we multiply by this fraction, we do not change the value but just change the units.

But $\frac{12 \ \mathrm{in}}{1 \ \mathrm{ft}}$ also equals $1$. How do we decide whether to multiply by $\frac{1 \ \mathrm{ft}}{12 \ \mathrm{in}}$ or $\frac{12 \ \mathrm{in}}{1 \ \mathrm{ft}}$? 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.

$\large 60 \ \cancel{\mathrm{in}} \cdot \frac{1 \ \mathrm{ft}}{12 \cancel{\mathrm{in}}} = 5 \mathrm{ft}$

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

$\large 5 \ \cancel{\mathrm{ft}} \cdot \frac{12 \ \mathrm{in}}{1 \ \cancel{\mathrm{ft}}} = 60 \ \mathrm{in}$

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

HOW TO: Make unit conversions.

  1. Multiply the measurement to be converted by $1$; write $1$ as a fraction relating the units given and the units needed.
  2. Multiply.
  3. Simplify the fraction, performing the indicated operations and removing the common units.

Example 1

Mary Anne is $66$ inches tall. What is her height in feet?

Solution

Convert $66$ inches into feet.
Multiply the measurement to be converted by $1$.$66 \ \mathrm{inches} \cdot 1$
Write $1$ as a fraction relating the units given and the units needed.$66 \ \mathrm{inches} \cdot \frac{1 \ \mathrm{foot}}{12 \ \mathrm{inches}}$
Multiply.$\frac{66 \ \mathrm{inches} \ \cdot 1 \ \mathrm{foot}}{12 \ \mathrm{inches}}$
Simplify the fraction.$\frac{66 \ \cancel{\mathrm{inches}} \cdot 1 \ \mathrm{foot}}{12 \ \cancel{\mathrm{inches}}}$
$\frac{66 \ \mathrm{feet}}{12}$
$5.5$ feet

Notice that the when we simplified the fraction, we first divided out the inches.

Mary Anne is $5.5$ feet tall.

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 2

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

A photograph of an adult elephant.
Figure 7.5 (credit: Guldo Da Rozze, Flickr)
Solution

We will convert $3.2$ tons into pounds, using the equivalencies in Table 7.2. We will use the Identity Property of Multiplication, writing $1$ as the fracion $\frac{2000 \ \mathrm{pounds}}{1 \ \mathrm{ton}}$.

$3.2 \ \mathrm{tons}$
Multiply the measurement to be converted by $1$.$3.2 \ \mathrm{tons} \ \cdot 1$
Write $1$ as a fraction relating tons and pounds.$3.2 \ \mathrm{tons} \ \cdot \frac{2000 \ \mathrm{lbs}}{1 \ \mathrm{ton}}$
Simplify.$\frac{3.2 \ \cancel{\mathrm{tons}} \ \cdot 2000 \ \mathrm{lbs}}{1 \ \cancel{\mathrm{ton}}}$
Multiply.$6400 \ \mathrm{lbs}$
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 3

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 $1$.

$9$ weeks
Write $1$ as $\frac{7 \ \mathrm{days}}{1 \ \mathrm{week}}, \frac{24 \ \mathrm{hours}}{1 \ \mathrm{day}}, \frac{60 \ \mathrm{minutes}}{1 \ \mathrm{hour}}$..
Cancel common units..
Multiply.$\frac{9 \cdot 7 \cdot 24 \cdot 60 \ \mathrm{min}}{1 \cdot 1 \cdot 1 \cdot 1} = 90,720 \ \mathrm{min}$
Juliet will be away for $90,720 \ \mathrm{minutes}$

Example 4

How many fluid ounces are in $1$ gallon of milk?

A photograph of a milk display in a grocery store.
Figure 7.6 (credit: http://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.

$1 \ \mathrm{gallon}$
Multiply the measurement to be converted by $1$.$\frac{1 \ \mathrm{gal}}{1} \ \cdot \frac{4 \ \mathrm{qt}}{1 \ \mathrm{gal}} \ \cdot \frac{2 \ \mathrm{pt}}{1 \ \mathrm{qt}} \ \cdot \frac{2 \ \mathrm{C}}{1 \ \mathrm{pt}} \ \cdot \frac{8 \ \mathrm{fl \ oz}}{1 \ \mathrm{C}}$
Simplify.$\frac{1 \ \mathrm{\cancel{gal}}}{1} \ \cdot \frac{4 \ \mathrm{\cancel{qt}}}{1 \ \mathrm{\cancel{gal}}} \ \cdot \frac{2 \ \mathrm{\cancel{pt}}}{1 \ \mathrm{\cancel{qt}}} \ \cdot \frac{2 \ \mathrm{\cancel{C}}}{1 \ \mathrm{\cancel{pt}}} \ \cdot \frac{8 \ \mathrm{fl \ oz}}{1 \ \mathrm{\cancel{C}}}$
Multiply.$\frac{1 \cdot 4 \cdot 2 \cdot 2 \cdot 8 \ \mathrm{fl \ oz}}{1 \cdot 1 \cdot 1 \cdot 1 \cdot 1}$
Simplify.$128$ fluid ounces
There are $128$ fluid ounces in a gallon.

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

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

Example 5

Charlie 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?

A photograph of meat being cooked on a charcoal grill.
Figure 7.7 (credit:Helen Penjam, Flickr)
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.
Add the pounds.$2$ pounds $+$ $1$ pound, $6$ ounces
$3$ pounds, $6$ ounces
Charlie bought $3$ pounds $6$ ounces of steak.

Example 6

Anthony bought four planks of wood that were each $6$ feet $4$ inches long. If the four planks are placed end-to-end, what is the total length of the wood?

The image shows 4 planks of wood placed end-to-end horizontally. Each plank is labeled 6 feet 4 inches. A line starts at the left of the first plank and runs horizontally to the right of the fourth plank. The line is labeled with the letter l to represent length.
Solution

We will multiply the length of one plank by $4$ to find the total length.

Multiply the inches and then the feet..
Convert $16$ inches to feet.$24$ feet $+$ $1$ foot $4$ inches
Add the feet.$25$ feet $4$ inches
Anthony bought $25$ feet $4$ inches of wood.

7.5.3 Make Unit Conversions in the Metric System

In the metric system, units are related by powers of $1$. The root words of their names reflect this relation. For example, the basic unit for measuring length is a meter. One kilometer is $1000$ meters; the prefix kilo- means thousand. One centimeter is $\frac{1}{100}$ of a meter, because the prefix centi- means one one-hundredth (just like one cent is $\frac{1}{100}$ of one dollar).

The equivalencies of measurements in the metric system are shown in Table 7.3. The common abbreviations for each measurement are given in parentheses.

Metric Measurements
LengthMassVolume/Capacity
$1$ kilometer (km) $=$ $1000$ m
$1$ hectometer (hm) $=$ $100$ m
$1$ dekameter (dam) $=$ $10$ m
$1$ meter (m) $=$ $1$ m
$1$ decimeter (dm) $=$ $0.1$ m
$1$ centimeter (cm) $=$ $0.01$ m
$1$ millimeter (mm) $=$ $0.01$ m
$1$ kilogram (kg) $=$ $1000$ g
$1$ hectogram (hg) $=$ $100$ g
$1$ dekagram (dag) $=$ $10$ g
$1$ gram (g) $=$ $1$ g
$1$ decigram (dg) $=$ $0.1$ g
$1$ centigram (cg) $=$ $0.01$ g
$1$ milligram (mg) $=$ $0.001$ g
$1$ kiloliter (kL) $=$ $1000$ L
$1$ hectoliter (hL) $=$ $100$ L
$1$ dekaliter (daL) $=$ $10$ L
$1$ liter (L) $=$ $1$ L
$1$ deciliter (dL) $=$ $0.1$ L
$1$ centiliter (cL) $=$ $0.01$ L
$1$ milliliter (mL) $=$ $0.01$ L
$1$ meter $=$ $100$ centimeters
$1$ meter $=$ $1000$ millimeters
$1$ gram $=$ $100$ centigrams
$1$ gram $=$ $1000$ milligrams
$1$ liter $=$ $100$ centiliters
$1$ liter $=$ $1000$ milliliters
Table 7.3

To make conversions in the metric system, we will use the same technique we did in the U.S. 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 $5$k or $10$k race? The lengths 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 7

Nick ran a $10$-kilometer race. How many meters did he run?

A photograph of 8 male runners in a track race.
Figure 7.8 (credit: William Warby, Flickr)
Solution

We will convert kilometers to meters using the Identity Property of Multiplication and the equivalencies in Table 7.3.

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

Example 8

Eleanor’s newborn baby weighed $3200$ grams. How many kilograms did the baby weigh?

Solution

We will convert grams to kilograms.

.
Multiply the measurement to be converted by $1$..
Write $1$ as a fraction relating kilograms and grams..
Simplify..
Multiply..
Divide.$3.2$ kilograms
The baby weighed $3.2$ kilograms.

Since the metric system is based on multiples of ten, conversions involve multiplying by multiples of ten. In Decimal Operations, we learned how to simplify these calculations by just moving the decimal.

To multiply by $10, 100,$ or $1000,$, we move the decimal to the right $1, 2,$ or $3$ places, respectively. To multiply by $0.1, 0.01,$ or $0.001$ we move the decimal to the left $1, 2,$ or $3$ places respectively.

We can apply this pattern when we make measurement conversions in the metric system.

In Example 8, we changed $3200$ grams to kilograms by multiplying by $\frac{1}{1000}$ (or $0.001$). This is the same as moving the decimal $3$ places to the left.

Multiplying 3200 by 1 over 1000 gives 3.2. Notice that the answer, 3.2, is similar to the original value, 3200, just with the decimal moved three places to the left.

Example 9

Convert:

  • $350$ liters to kiloliters
  • $4.1$ liters to milliters.
Solution

Part 1. We will convert liters to kiloliters. In Table 7.3, we see that $1$ kiloliter $=$ $1000$ liters.

$350$ L
Multiply by $1$, writing $1$ as a fraction relating liters to kiloliters..
Simplify..
Move the decimal $3$ units to the left..
$0.35$ kL

Part 2. We will convert liters to milliliters. In Table 7.3, we see that $1$ liter $=$ $1000$ milliliters.

$4.1$ L
Multiply by $1$, writing $1$ as a fraction relating milliliters to liters..
Simplify..
Move the decimal $3$ units to the left..
$4100$ mL

7.5.4 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 U.S. system. But it may be easier because of the relation of the units to the powers of $10$. We still must make sure to add or subtract like units.

Example 10

Ryland is $1.6$ meters tall. His younger brother is $85$ centimeters tall. How much taller is Ryland than his younger brother?

Solution

We will subtract the lengths in meters. Convert $85$ centimeters to meters by moving the decimal $2$ places to the left; $85$ cm is the same as $0.85$ m.

Now that both measurements are in meters, subtract to find out how much taller Ryland is than his brother.

$\begin{array}{c@{\,}c} & 1.60 \mathrm{m} \\ – & 0.85 \mathrm{m} \\ \hline & 0.75 \mathrm{m} \\ \end{array} %$

Ryland is $0.75$ meters taller than his brother.

Example 11

Dena’s recipe for lentil soup calls for $150$ milliliters of olive oil. Dena wants to triple the recipe. How many liters of olive oil will she need?

Solution

We will find the amount of olive oil in milliliters then convert to liters.

Triple $150$ mL
Translate to algebra$3 \cdot 150$ mL
Multiply.$450$ mL
Convert to liters.$0.45$ L
Dena needs $0.45$ liter of olive oil.

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

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

Table 7.4 shows some of the most common conversions.

Conversion Factors Between U.S. and Metric Systems
LengthWeightVolume
$1$ in $=$ $2.54$ cm
$1$ ft $=$ $0.305$ m
$1$ yd $=$ $0.914$ m
$1$ mi $=$ $1.61$ km

$1$ m $=$ $3.28$ ft
$1$ lb $=$ $0.45$ kg
$1$ oz $=$ $28$ g



$1$ kg $=$ $2.2$ lb
$1$ qt $=$ $0.95$ L
$1$ fl oz $=$ $30$ mL



$1$ L $=$ $1.06$ qt
Table 7.4

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

Example 12

Lee’s water bottle holds $500$ mL of water. How many fluid ounces are in the bottle? Round to the nearest tenth of an ounce.

Solution

$500$ mL
Multiply by a unit conversion factor relating mL and ounces.$500$ mL $\cdot \frac{1 \ \mathrm{fl \ oz}}{30 \ \mathrm{mL}}$
Simplify.$\frac{500 \ \mathrm{fl \ oz}}{30}$
Divide.$16.7$ fl. oz.
The water bottle holds $16.7$ fluid ounces.

The conversion factors in Table 7.4 are not exact, but the approximations they give are close enough for everyday purposes. In Example 12, we rounded the number of fluid ounces to the nearest tenth.

Example 13

Soleil lives in Minnesota but often travels in Canada for work. While driving on a Canadian highway, she passes a sign that says the next rest stop is in $100$ kilometers. How many miles until the next rest stop? Round your answer to the nearest mile.

Solution

$100$ kilometers
Multiply by a unit conversion factor relating kilometers and miles.$100$ kilometers $\cdot \frac{1 \ \mathrm{mile}}{1.61 \ \mathrm{kilometers}}$
$100 \cdot \frac{1 \ \mathrm{mi}}{1.61 \ \mathrm{km}}$
Simplify.$\frac{100 \ \mathrm{mi}}{1.61}$
Divide.$62$ mi
It is about 62 miles to the next rest stop.

7.5.6 Convert Between Fahrenheit and Celsius Temperatures

Have you ever been in a foreign country and heard the weather forecast? If the forecast is for $22^{\circ} C$. What does that mean?

The U.S. and metric systems use different scales to measure temperature. The U.S. system uses degrees Fahrenheit, written $^{\circ} F$. The metric system uses degrees Celsius, written $^{\circ} C$. Figure 7.9 shows the relationship between the two systems.

On the left side of the figure is a thermometer marked in degrees Celsius. The bottom of the thermometer begins with negative 20 degrees Celsius and ranges up to 100 degrees Celsius. There are tick marks on the thermometer every 5 degrees with every 10 degrees labeled. On the right side is a thermometer marked in degrees Fahrenheit. The bottom of the thermometer begins with negative 10 degrees Fahrenheit and ranges up to 212 degrees Fahrenheit. There are tick marks on the thermometer every 2 degrees with every 10 degrees labeled. Between the thermometers there is an arrow pointing on the left to 0 degrees Celsius and on the right to 32 degrees Fahrenheit. This is the temperature at which water freezes. Another arrow points on the left to 37 degrees Celsius and on the right to 98.6 degrees Fahrenheit. This is normal body temperature. A third arrow points on the left to 100 degrees Celsius and on the right to 212 degrees Fahrenheit. This is the temperature at which water boils.
Figure 7.9 A temperature of $37^{\circ} C$ is equivalent to $98.6^{\circ} F$.

If we know the temperature in one system, we can use a formula to convert it to the other system.

TEMPERATURE CONVERSION

To convert from Fahrenheit temperature, $F$, to Celsius temperature, $C$, use the formula

$\large C= \frac{5}{9} (F-32)$

To convert from Celsius temperature, $C$, to Fahrenheit temperature, $F$, use the formula

$\large F= \frac{9}{5} C +32$

Example 14

Convert $50^{\circ} F$ into degrees Celsius.

Solution

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

Use the formula for converting $^{\circ} F$ to $^{\circ} C$$C= \frac{5}{9} (F-32)$
Substitute $50$ for $F$.$C= \frac{5}{9} (50-32)$
Simplify in parentheses.$C= \frac{5}{9} (18)$
Multiply.$C=10$
A temperature of $50^{\circ} F$ is equivalent to $10^{\circ} C$.

Example 15

The weather forcast for Paris predicts a high of $20^{\circ} C$. Convert the temperature into degrees Fahrenheit.

Solution

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

Use the formula for converting $^{\circ} F$ to $^{\circ} C$$F= \frac{9}{5} C+32$
Substitute $20$ for $C$.$F= \frac{9}{5} (20)+32$
Multiply.$F=36+32$
Add.$F=68$
So $20^{\circ} C$ is equivalent to $68^{\circ} F$.
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