0.3 Unit Conversions

Jared Eusea; Karen Perilloux; Kristina VanDusen; Lauren Johnson; Prakash Ghimire

0.3 Unit Conversions

Learning Objectives

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

Scientists make observations and ask basic questions. For example, how big is an object? How much mass does it have? How far did it travel? To answer these questions, they make measurements with various instruments (e.g., meter stick, balance, stopwatch, etc.).

The measurements of physical quantities are expressed in terms of units, which are standardized values. For example, the length of a race, which is a physical quantity, can be expressed in meters (for sprinters) or miles (for marathon runners). Without standardized units, it would be extremely difficult for scientists to express and compare measured values in a meaningful way (Figure 1).

A person is standing in front of a map that has cable as units. The person is confused and is wondering how big a cable is. — Figure 1. Distances given in unknown units are maddeningly useless.

In this section, we will see how to convert 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 gallon. 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.

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

Table 1 U.S. System Units

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 [latex]a[/latex], [latex]a \cdot 1 = a[/latex] and [latex]1 \cdot a = a[/latex].

To use the identity property of multiplication, we write [latex]1[/latex] in a form that will help us convert the units. For example, suppose we want to convert inches to feet. We know that [latex]1[/latex] foot is equal to [latex]12[/latex] inches, so we can write [latex]1[/latex] as the fraction [latex]\frac{1\text{ ft}}{12\text{ in}}[/latex]. When we multiply by this fraction, we do not change the value but just change the units. But [latex]\frac{12\text{ in}}{1\text{ ft}}[/latex] also equals [latex]1[/latex]. How do we decide whether to multiply by [latex]\frac{1\text{ ft}}{12\text{ in}}[/latex] or [latex]\frac{12\text{ in}}{1\text{ ft}}[/latex] ? We choose the fraction that will make the units we want to convert from divide out. For example, suppose we wanted to convert [latex]60[/latex] inches to feet. If we choose the fraction that has inches in the denominator, we can eliminate the inches.

[latex]60 \text{ in} \cdot \frac{1 \text{ ft}}{12 \text{ in}} = 5 \text{ ft}[/latex]

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

[latex]5 \text{ ft} \cdot \frac{12 \text{ in}}{1\text{ ft}} = 60 \text{ in}[/latex]

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

How To

Convert Between Units

Step 1. Identify the starting unit and the target unit.

Step 2. Find a unit equivalence between the two units. (Sometimes using an intermediate unit is helpful.)

Step 3. Multiply the measurement by 1: write 1 as a fraction relating the given units and the desired units.

Step 4. Simplify the result, performing the indicated operations and removing the common units.

Converting Inches to Feet

Mary Anne is [latex]66\text{ inches}[/latex] tall. What is her height in feet?

Show Solution

We will convert grams to kilograms.

Explanation

Steps

Convert 66 inches into feet.

Multiply the measurement to be converted by 1.

[latex]66 \text{ inches} \cdot 1[/latex]

Write 1 as a fraction relating the units given and the units needed.

[latex]66 \text{ inches} \cdot \frac{1 \text{ foot}}{12 \text{ inches}}[/latex]

Multiply.

[latex]\frac{66 \text{ inches} \cdot 1 \text{ foot}}{12 \text{ inches}}[/latex]

Simplify the fraction.

[latex]\frac{66\cancel{\text{ inches}} \cdot 1 \text{ ft}}{12 \cancel{\text{ inches}}}[/latex]

Multiply.

[latex]\frac{66\cdot 1 \text{ feet}}{12}[/latex]

Simplify the fraction.

[latex]5.5 \text{ feet}[/latex]

Notice that when we simplified the fraction, we first divided out the inches. Mary Anne is [latex]5.5 \text{ feet}[/latex] tall.

Try It

Lexie is [latex]30\text{ inches}[/latex] tall. Convert her height to feet.

Show Solution

[latex]2.5\text{ feet}[/latex]

Try It

Rene bought a hose that is [latex]18\text{ yards}[/latex] long. Convert the length to feet.

Show Solution

[latex]54\text{ feet}[/latex]

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.

Converting Tons to Pounds

Ndula, an elephant at the San Diego Safari Park, weighs almost [latex]3.2\text{ tons}[/latex]. Convert her weight to pounds.

A photograph of an adult elephant. — Figure 2 ( Credit: Guldo Da Rozze, Flickr)

Show Solution

We will convert [latex]3.2[/latex] tons into pounds. We will use the Identity Property of multiplication, writing [latex]1[/latex] as the fraction [latex]\frac{2000 \text{ lbs} }{1 \text{ ton}}[/latex].

Explanation

Steps

[latex]3.2\text{ tons}[/latex]

Multiply the measurement to be converted by 1.

[latex]3.2\text{ tons}\cdot 1[/latex]

Write 1 as a fraction relating tons and pounds.

[latex]3.2\text{ tons}\cdot\frac{2000\text{ lbs}}{1\text{ ton}}[/latex]

Simplify.

[latex]3.2\cancel{\text{ tons}}\cdot\frac{2000\text{ lbs}}{1\cancel{\text{ ton}}}[/latex]

Multiply.

[latex]6400\text{ lbs}[/latex]

Ndula weighs almost 6,400 pounds.

Try It

Arnold's SUV weighs about [latex]4.3\text{ tons}[/latex]. Convert the weight to pounds.

Show Solution

[latex]8600\text{ pounds}[/latex]

Try It

A cruise ship weighs [latex]51{,}000\text{ tons}[/latex]. Convert the weight to pounds.

Show Solution

[latex]102{,}000{,}000\text{ pounds}[/latex]

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.

Converting Weeks to Minutes

Juliet is going with her family to their summer home. She will be away for [latex]9\text{ weeks}[/latex]. Convert the time to minutes.

Show 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.

Explanation

Steps

[latex]9\text{ weeks}[/latex]

Write 1 as [latex]\frac{7\text{ days}}{1\text{ week}}, \frac{24\text{ hours}}{1\text{ day}}, \text{ and }\frac{60\text{ minutes}}{1\text{ hour}}[/latex].

[latex]\frac{9\text{ weeks}}{1}\cdot\frac{7\text{ days}}{1\text{ week}}\cdot\frac{24\text{ hours}}{1\text{ day}}\cdot\frac{60\text{ minutes}}{1\text{ hour}}[/latex]

Cancel common units.

[latex]\frac{9\cancel{\text{ weeks}}}{1}\cdot\frac{7\cancel{\text{ days}}}{1\cancel{\text{ week}}}\cdot\frac{24\cancel{\text{ hours}}}{1\cancel{\text{ day}}}\cdot\frac{60\text{ minutes}}{1\cancel{\text{ hour}}}[/latex]

Multiply.

[latex]\frac{9\cdot 7\cdot 24\cdot 60\text{ min}}{1\cdot 1\cdot1\cdot 1}=90{,}720\text{ min}[/latex]

Juliet will be away for 90,720 minutes.

Try It

The distance between Earth and the moon is about [latex]250{,}000\text{ miles}[/latex]. Convert this length to yards.

Show Solution

[latex]440{,}000{,}000\text{ yards}[/latex]

Try It

A team of astronauts spends [latex]15\text{ weeks}[/latex] in space. Convert the time to minutes.

Show Solution

[latex]151{,}200\text{ minutes}[/latex]

Converting Gallons to Fluid Ounces

How many fluid ounces are in [latex]1\text{ gallon}[/latex] of milk?

A milk display in a grocery store. — Figure 3 (credit: www.bluewaikiki.com, Flickr)

Show Solution

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

Explanation

Steps

[latex]1\text{ gallon}[/latex]

Multiply the measurement to be converted by 1.

[latex]\frac{1\text{ gal}}{1}\cdot\frac{4\text{ qt}}{1\text{ gal}}\cdot\frac{2\text{ pt}}{1\text{ qt}}\cdot\frac{2\text{ C}}{1\text{ pt}}\cdot\frac{8\text{ fl oz}}{1\text{ C}}[/latex]

Simplify.

[latex]\frac{1\cancel{\text{ gal}}}{1}\cdot\frac{4\cancel{\text{ qt}}}{1\cancel{\text{ gal}}}\cdot\frac{2\cancel{\text{ pt}}}{1\cancel{\text{ qt}}}\cdot\frac{2\cancel{\text{ C}}}{1\cancel{\text{ pt}}}\cdot\frac{8\text{ fl oz}}{1\cancel{\text{ C}}}[/latex]

Multiply.

[latex]\frac{1\cdot 4\cdot 2\cdot 2\cdot 8\text{ fl oz}}{1\cdot 1\cdot 1\cdot 1\cdot 1}=128\text{ fl oz}[/latex]

There are 128 fluid ounces in a gallon.

Try It

How many cups are in [latex]1\text{ gallon}[/latex]?

Show Solution

[latex]16\text{ cups}[/latex]

Try It

How many teaspoons are in [latex]1\text{ cup}[/latex]?

Show Solution

[latex]48\text{ teaspoons}[/latex]

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.

Working With Pounds and Ounces

Charlie bought three steaks for a barbecue. Their weights were [latex]14\text{ ounces}[/latex]; [latex]1\text{ pound, }2\text{ ounces}[/latex]; and [latex]1\text{ pound, }6\text{ ounces}[/latex]. How many total pounds of steak did he buy?

Meat being cooked on a charcoal grill. — Figure 4 (credit: Helen Penjam, Flickr)

Show Solution

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

Explanation

Steps

Add the ounces. Then add the pounds.

Adding 1 pound and 1 pound, and then adding 2 ounces and 6 ounces

Convert 22 ounces to pounds and ounces.

22 ounces is equal to 1 pound, 6 ounces.

Add the pounds.

2 pounds + 1 pound, 6 ounces
3 pounds, 6 ounces

Charlie bought 3 pounds 6 ounces of steak.

Try It

Laura gave birth to triplets weighing [latex]3\text{ pounds, }12\text{ ounces}[/latex]; [latex]3\text{ pounds, }3\text{ ounces}[/latex]; and [latex]2\text{ pounds, }9\text{ ounces}[/latex]. What was the total birth weight of the three babies?

Show Solution

[latex]9\text{ lbs., } 8\text{ oz.}[/latex]

Try It

Seymour cut two pieces of crown molding for his family room that were [latex]8\text{ feet, }7\text{ inches}[/latex] and [latex]12\text{ feet, }11\text{ inches}[/latex]. What was the total length of the molding?

Show Solution

[latex]21\text{ ft., } 6\text{ in.}[/latex]

Working With Feet and Inches

Anthony bought four planks of wood that were each[latex]6\text{ feet, } 4\text{ inches}[/latex] 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, labeled 'l' for length. Each plank is labeled 6 feet 4 inches. — Figure 5

Show Solution

We will multiply the length of one plank by [latex]4[/latex] to find the total length.

Explanation

Steps

Multiply the inches and then the feet.

Multiplying 6 feet and 4 inches by 4 to get 24 feet and 16 inches

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.

Try It

Henri wants to triple his spaghetti sauce recipe, which calls for [latex]1\text{ pound, }8 \text{ ounces}[/latex] of ground turkey. How many pounds of ground turkey will he need?

Show Solution

[latex]4\text{ lbs., }8 \text{ oz.}[/latex]

Try It

Joellen wants to double a solution of [latex]5\text{ gallons, }3\text{ quarts}[/latex]. How many gallons of solution will she have in all?

Show Solution

[latex]11\text{ gal., }2\text{ qts.}[/latex]

Mathispower4u. American Unit Conversion. YouTube. https://www.youtube.com/watch?v=ozSnWr4do5o

Make Unit Conversions in the Metric System

In the metric system, units are related by powers of [latex]10[/latex]. The root words of their names reflect this relation. For example, the basic unit for measuring length is a meter. One kilometer is [latex]1000[/latex] meters; the prefix kilo- means thousand. One centimeter is [latex]\frac{1}{100}[/latex] of a meter, because the prefix centi- means one one-hundredth (just like one cent is [latex]\frac{1}{1000}[/latex] of one dollar). Metric units consist of the base unit (meters for length, grams for mass, liters for volume, etc.) and a prefix describing how many base units are in the unit.

Metric Prefixes for Powers of 10 and Their Symbols

Prefix

Symbol

Value

Example Name

Example Symbol

Example Value

Example Description

kilo

k

10³

Kilometer

km

10³ m

About 6/10 mile

hecto

h

10²

Hectoliter

hL

10² L

26 gallons

deka

da

10¹

Dekagram

dag

10¹ g

Teaspoon of butter

10⁰ (=1)

deci

d

10^–1

Deciliter

dL

10^–1 L

Less than half a soda

centi

c

10^–2

Centimeter

cm

10^–2 m

Fingertip thickness

milli

m

10^–3

Millimeter

mm

10^–3 m

Flea at its shoulder

Table 2 Metric Prefixes and Their Symbols

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 [latex]5\text{K}[/latex] or [latex]10\text{K}[/latex] 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.

Converting Metric Units of Length

Nick ran a [latex]10\text{-kilometer}[/latex] race. How many meters did he run?

8 male runners in a track race. — Figure 6 (credit: William Warby, Flickr)

Show Solution

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

Explanation

Steps

[latex]10 \text{ kilometers}[/latex]

Multiply the measurement to be converted by 1.

[latex]10\text{ km}\cdot 1[/latex]

Write 1 as a fraction relating kilometers and meters.

[latex]10\text{ km}\cdot\frac{1000\text{ m}}{1\text{ km}}[/latex]

Simplify.

[latex]10\cancel{\text{ km}}\cdot\frac{1000\text{ m}}{1\cancel{\text{ km}}}[/latex]

Multiply.

[latex]\frac{10\cdot 1000\text{ m}}{1}[/latex]

Nick ran 10,000 meters.

Try It

Sandy completed her first [latex]5\text{-km}[/latex] race. How many meters did she run?

Show Solution

[latex]5000\text{ m}[/latex]

Try It

Herman bought a rug [latex]2.5\text{ meters}[/latex] in length. How many centimeters is the length?

Show Solution

[latex]250\text{ cm}[/latex]

Converting Metric Units of Mass

Eleanor's newborn baby weighed [latex]3200\text{ grams}[/latex]. How many kilograms did the baby weigh?

Show Solution

We will convert grams to kilograms.

Explanation

Steps

[latex]3200\text{ grams}[/latex]

Multiply the measurement to be converted by 1.

[latex]3200\text{ grams}\cdot 1[/latex]

Write 1 as a fraction relating kilograms and grams.

[latex]3200\text{ grams}\cdot \frac{1\text{ kg}}{1000\text{ g}}[/latex]

Simplify.

[latex]3200\cancel{\text{ grams}}\cdot \frac{1\text{ kg}}{1000\cancel{\text{ g}}}[/latex]

Multiply.

[latex]\frac{3200\cdot 1\text{ kg}}{1000}[/latex]

The baby weighed [latex]3.2\text{ kilograms}[/latex].

Try It

Kari's newborn baby weighed [latex]2800\text{ grams}[/latex]. How many kilograms did the baby weigh?

Show Solution

[latex]2.8\text{ kilograms}[/latex]

Try It

Anderson received a package that was marked [latex]4500\text{ grams}[/latex]. How many kilograms did this package weigh?

Show Solution

[latex]4.5 \text{ kilograms}[/latex]

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

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

Below, we changed [latex]3200[/latex] grams to kilograms by multiplying by [latex]\frac{1}{1000}[/latex] or [latex]0.001[/latex]. This is the same as moving the decimal [latex]3[/latex] places to the left.

Multiplying 3200 by 1 over 1000 gives 3.2. The decimal moved three places to the left.

Converting Metric Units of Volume

Convert: (a) 350 liters to kiloliters; (b) 4.1 liters to milliliters.

Show Solution

(a) We will convert liters to kiloliters. We know that 1 kiloliter is equal to 1000 liters.

Explanation

Steps

[latex]350\text{ L}[/latex]

Multiply by 1, writing 1 as a fraction relating liters to kiloliters.

[latex]350\text{ L}\cdot \frac{1\text{ kL}}{1000\text{ L}}[/latex]

Simplify.

[latex]350\cancel{\text{ L}}\cdot \frac{1\text{ kL}}{1000\cancel{\text{ L}}}[/latex]

Move the decimal 3 units to the left.

[latex]350\text{ L}\cdot \frac{350\cdot 1\text{ kL}}{1000}[/latex]

[latex]0.35\text{ kL}[/latex]

(b) We will convert liters to milliliters. We know that 1 liter is equal to 1000 milliliters.

Explanation

Steps

[latex]4.1 \text{ L}[/latex]

Multiply by 1, writing 1 as a fraction relating milliliters to liters.

[latex]4.1\text{ L}\cdot \frac{1000\text{ mL}}{1\text{ L}}[/latex]

Simplify.

[latex]4.1\cancel{\text{ L}}\cdot \frac{1000\text{ mL}}{1\cancel{\text{ L}}}[/latex]

Move the decimal 3 units to the right.

[latex]\frac{4.1\cdot 1000\text{mL}}{1}[/latex]

[latex]4100 \text{ mL}[/latex]

Try It

Convert: (a) 7.25 L to kL; (b) 6.3 L to mL.

Show Solution

(a) [latex]0.00725\text{ kL};[/latex] (b) [latex]6300\text{ mL}[/latex]

Try It

Convert: (a) 350 hL to L; (a) 4.1 L to cL.

Show Solution

(a) [latex]35,000\text{ L};[/latex] (b) [latex]410\text{ cL}[/latex]

Mathispower4u. Metric Unit Conversion. YouTube. https://www.youtube.com/watch?v=cMFwpxkIFMY

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.

Converting Metric Units of Length

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

Show 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{matrix} 1.60 m \\ \underset{_______}{−0.85 m} \\ 0.75 m \end{matrix}

Ryland is 0.75 meters taller than his brother.

Try It

Mariella is 1.58 meters tall. Her daughter is 75 centimeters tall. How much taller is Mariella than her daughter? Write the answer in centimeters.

Show Solution

[latex]83 \text{ cm}[/latex]

Try It

The fence around Hank’s yard is 2 meters high. Hank is 96 centimeters tall. How much shorter than the fence is Hank? Write the answer in meters.

Show Solution

[latex]1.04\text{ m}[/latex]

Converting Metric Units of Volume

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?

Show Solution

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

Explanation

Steps

Triple 150 mL

Translate to algebra.

3 \cdot 150 mL

Multiply.

450 mL

Convert to liters.

450 mL \cdot \frac{0.001 L}{1 mL}

Simplify.

0.45 L

Dena needs 0.45 liter of olive oil.

Try It

A recipe for Alfredo sauce calls for 250 milliliters of milk. Renata is making pasta with Alfredo sauce for a big party and needs to multiply the recipe amounts by 8. How many liters of milk will she need?

Show Solution

[latex]2 \text{ L}[/latex]

Try It

To make one pan of baklava, Dorothea needs 400 grams of filo pastry. If Dorothea plans to make 6 pans of baklava, how many kilograms of filo pastry will she need?

Show Solution

[latex]2.4 \text{ kg}[/latex]

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 3 below shows some of the most common conversions.

Common Equivalences

Length

Volume

Mass

1 m = 1.0936 yd

1 L = 1.0567 qt

1 kg = 2.2046 lb

1 in. = 2.54 cm (exact)

1 fl oz = 30 mL

1 lb = 453.59 g

1 km = 0.62137 mi

1 ft³ = 28.317 L

1 (avoirdupois) oz = 28.349 g

1 mi = 1609.3 m

1 tbsp = 14.787 mL

1 (troy) oz = 31.103 g

Table 3 Common Equivalences

Converting Between Metric and U.S. Units of Volume

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.

Show Solution

Explanation

Steps

[latex]500\text{ mL}[/latex]

Multiply by a unit conversion factor relating mL and ounces.

[latex]500 \text{mL} \cdot \frac{1 \text{ fl. oz.}}{30 \text{mL}}[/latex]

Simplify.

[latex]\frac{500 \text{ fl. oz.}}{30}[/latex]

Divide.

[latex]16.7 \text{ fl. oz.}[/latex]

The water bottle holds 16.7 fluid ounces.

Try It

How many quarts of soda are in a $2-liter$ bottle?

Show Solution

[latex]2.12\text{ quarts}[/latex]

Try It

How many liters are in $4$ quarts of milk?

Show Solution

[latex]3.8 \text{ liters}[/latex]

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

Converting Between Metric and U.S. Units of Length

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.

Show Solution

Explanation

Steps

[latex]100\text{ kilometers}[/latex]

Multiply by a unit conversion factor relating kilometers and miles.

[latex]100 \text{ kilometers} \cdot \frac{1 \text{ mile}}{1.61 \text{ kilometers}}[/latex]

Simplify.

[latex]\frac{100 \cdot 1\text{ mi}}{1.61}[/latex]

Divide.

[latex]62 \text{ mi}[/latex]

It is about 62 miles to the next rest stop.

Try It

The height of Mount Kilimanjaro is $5,895$ meters. Convert the height to feet. Round to the nearest foot.

Show Solution

[latex]19{,}336 \text{ ft}[/latex]

Try It

The flight distance from New York City to London is $5,586$ kilometers. Convert the distance to miles. Round to the nearest mile.

Show Solution

[latex]3{,}470\text{ mi}[/latex]

Access the online resource below for additional instruction and practice with the American and Metric Unit Conversion:

Mathispower4u. American and Metric Conversions. YouTube. https://www.youtube.com/watch?v=sn8Y7qpYLCY

Convert Between Fahrenheit and Celsius Temperatures

Have you ever been in a foreign country and heard the weather forecast? If the forecast is for $22 °C .$ hat does that mean?

The U.S. and metric systems use different scales to measure temperature. The U.S. system uses degrees Fahrenheit, written $°F .$ The metric system uses degrees Celsius, written $°C .$ Figure 7 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 A temperature of 37°C is equivalent to 98.6°F.

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

Converting Between Fahrenheit and Celsius

To convert from Fahrenheit temperature, [latex]F[/latex], to Celsius temperature, [latex]C[/latex], use the formula

[latex]C = \frac{5}{9} ( F-32)[/latex]

To convert from Celsius temperature, [latex]C[/latex], to Fahrenheit temperature, [latex]F[/latex], use the formula

[latex]F = \frac{9}{5} C + 32[/latex]

Converting Between Fahrenheit and Celsius

Convert $50 °F$ into degrees Celsius.

Show Solution

We will substitute $50 °F$ into the formula to find C.

Explanation

Steps

Use the formula for converting °F to °C

[latex]C= \frac{5}{9} (F-32)[/latex]

Substitute 50 for F.

[latex]C= \frac{5}{9} (50-32)[/latex]

Simplify in parentheses.

[latex]C= \frac{5}{9} (18)[/latex]

Multiply.

[latex]C=10[/latex]

A temperature of 50°F is equivalent to 10°C.

Try It

Convert the Fahrenheit temperatures to degrees Celsius: $59 °F .$

Show Solution

[latex]15°C[/latex]

Try It

Convert the Fahrenheit temperatures to degrees Celsius: $41 °F .$

Show Solution

[latex]5°C[/latex]

Converting Between Celsius and Fahrenheit

The weather forecast for Paris predicts a high of $20 °C.$ Convert the temperature into degrees Fahrenheit.

Show Solution

We will substitute $20 °C$ into the formula to find $F .$

Explanation

Steps

Use the formula for converting °F to °C.

[latex]F= \frac{9}{5}C+32[/latex]

Substitute 20 for C.

[latex]F= \frac{9}{5} (20)+32[/latex]

Multiply.

[latex]F=36+32[/latex]

Add.

[latex]F=68[/latex]

So 20°C is equivalent to 68°F.

Try It

Convert the Celsius temperature to degrees Fahrenheit:

The temperature in Helsinki, Finland was $15 °C .$

Show Solution

[latex]59°F[/latex]

Try It

Convert the Celsius temperatures to degrees Fahrenheit:

The temperature in Sydney, Australia was $10 °C .$

Show Solution

[latex]50°F[/latex]

Access the online resource below for additional instruction and practice with the Conversion of Temperature from Celsius to Fahrenheit:

Mathispower4u. Convert Temperature from Celsius to Fahrenheit. YouTube. https://www.youtube.com/watch?v=sBXeRYW9ibw

Mathispower4u. Convert Temperature from Fahrenheit to Celsius. YouTube. https://www.youtube.com/watch?v=hZiP3GF_tzM

Key Concepts

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

Table 1 U.S. System Units

Prefix	Symbol	Value	Example Name	Example Symbol	Example Value	Example Description

kilo	k	10³	Kilometer	km	10³ m	About 6/10 mile
hecto	h	10²	Hectoliter	hL	10² L	26 gallons
deka	da	10¹	Dekagram	dag	10¹ g	Teaspoon of butter
		10⁰ (=1)
deci	d	10^–1	Deciliter	dL	10^–1 L	Less than half a soda
centi	c	10^–2	Centimeter	cm	10^–2 m	Fingertip thickness
milli	m	10^–3	Millimeter	mm	10^–3 m	Flea at its shoulder

Table 2 Metric Prefixes and Their Symbols

Length	Volume	Mass
1 m = 1.0936 yd	1 L = 1.0567 qt	1 kg = 2.2046 lb
1 in. = 2.54 cm (exact)	1 gal = 3.7854 L	1 lb = 453.59 g
1 km = 0.62137 mi	1 ft³ = 28.317 L	1 (avoirdupois) oz = 28.349 g
1 mi = 1609.3 m	1 tbsp = 14.787 mL	1 (troy) oz = 31.103 g

Table 3 Common Equivalences

Convert Between Units

Step 1. Identify the starting unit and the target unit.
Step 2. Find a unit equivalence between the two units. (Sometimes using an intermediate unit is helpful.)
Step 3. Determine the conversion factor that will eliminate the starting unit and leave you with the target unit.
Step 4: Multiply the measurement to be converted by 1; write 1 as a fraction relating the units given and the units needed.
Step 5: Multiply.
Step 6: Simplify the fraction, performing the indicated operations and removing the common units.

Section Exercises

Make Unit Conversions in the U.S. System

1. A floor tile is $2$ feet wide. Convert the width to inches.

Show Solution

24 inches

2. A ribbon is $18$ inches long. Convert the length to feet.

3. Carson is $45$ inches tall. Convert his height to feet.

Show Solution

3.75 feet

4. Jon is $6$ feet $4$ inches tall. Convert his height to inches.

5. Faye is $4$ feet $10$ inches tall. Convert her height to inches.

Show Solution

58 inches

6. A football field is $160$ feet wide. Convert the width to yards.

7. On a baseball diamond, the distance from home plate to first base is $30$ yards. Convert the distance to feet.

Show Solution

90 feet

8. Ulises lives $1.5$ miles from school. Convert the distance to feet.

9. Denver, Colorado, is $5,183$ feet above sea level. Convert the height to miles.

Show Solution

0.98 miles

10. A killer whale weighs $4.6$ tons. Convert the weight to pounds.

11. Blue whales can weigh as much as $150$ tons. Convert the weight to pounds.

Show Solution

300,000 pounds

12. An empty bus weighs $35,000$ pounds. Convert the weight to tons.

13. At take-off, an airplane weighs $220,000$ pounds. Convert the weight to tons.

Show Solution

110 tons

14. The voyage of the Mayflower took $2$ months and $5$ days. Convert the time to days (30 days = 1 month).

15. Lynn’s cruise lasted $6$ days and $18$ hours. Convert the time to hours.

Show Solution

162 hours

16. Rocco waited $1 \frac{1}{2}$ hours for his appointment. Convert the time to seconds.

17. Misty’s surgery lasted $2 \frac{1}{4}$ hours. Convert the time to seconds.

Show Solution

8100 seconds

18. How many teaspoons are in a pint?

19. How many tablespoons are in a gallon?

Show Solution

256 tablespoons

20. JJ’s cat, Posy, weighs $14$ pounds. Convert her weight to ounces.

21. April’s dog, Beans, weighs $8$ pounds. Convert his weight to ounces.

Show Solution

128 ounces

22. Baby Preston weighed $7$ pounds $3$ ounces at birth. Convert his weight to ounces.

23. Baby Audrey weighed $6$ pounds $15$ ounces at birth. Convert her weight to ounces.

Show Solution

111 ounces

24. Crista will serve $20$ cups of juice at her son’s party. Convert the volume to gallons.

25. Lance needs $500$ cups of water for the runners in a race. Convert the volume to gallons.

Show Solution

31.25 inches

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

In the following exercises, solve and write your answer in mixed units.

26. Eli caught three fish. The weights of the fish were $2$ pounds $4$ ounces, $1$ pound $11$ ounces, and $4$ pounds $14$ ounces. What was the total weight of the three fish?

27. Judy bought $1$ pound $6$ ounces of almonds, $2$ pounds $3$ ounces of walnuts, and $8$ ounces of cashews. What was the total weight of the nuts?

Show Solution

4 lbs., 1 oz.

28. One day Anya kept track of the number of minutes she spent driving. She recorded trips of $45, 10, 8, 65, 20, and 35 minutes.$ How much time (in hours and minutes) did Anya spend driving?

29. Last year Eric went on $6$ business trips. The number of days of each was $5, 2, 8, 12, 6, and 3.$ How much time (in weeks and days) did Eric spend on business trips last year?

Show Solution

5 weeks and 1 day

30. Renee attached a $6-foot-6-inch$ cord to her computer’s $3-foot-8-inch$ power cord. What was the total length of the cords?

31. Fawzi’s SUV is $6$ feet $4$ inches tall. If he puts a $2-foot-10-inch$ box on top of his SUV, what is the total height of the SUV and the box?

Show Solution

9 ft., 2 in.

32. Leilani wants to make $8$ placemats. For each placemat she needs $18$ inches of fabric. How many yards of fabric will she need for the $8$ placemats?

33. Mireille needs to cut $24$ inches of ribbon for each of the $12$ girls in her dance class. How many yards of ribbon will she need altogether?

Show Solution

8 yards

Make Unit Conversions in the Metric System

34. Ghalib ran $5$ kilometers. Convert the length to meters.

35. Kitaka hiked $8$ kilometers. Convert the length to meters.

Show Solution

8000 meters

36. Estrella is $1.55$ meters tall. Convert her height to centimeters.

37. The width of the wading pool is $2.45$ meters. Convert the width to centimeters.

Show Solution

245 centimeters

38. Mount Whitney is $3,072$ meters tall. Convert the height to kilometers.

39. The depth of the Mariana Trench is $10,911$ meters. Convert the depth to kilometers.

Show Solution

10.911 kilometers

40. June’s multivitamin contains $1,500$ milligrams of calcium. Convert this to grams.

41. A typical ruby-throated hummingbird weights $3$ grams. Convert this to milligrams.

Show Solution

3000 milligrams

42. One stick of butter contains $91.6$ grams of fat. Convert this to milligrams.

43. One serving of gourmet ice cream has $25$ grams of fat. Convert this to milligrams.

Show Solution

25,000 milligrams

44. The maximum mass of an airmail letter is $2$ kilograms. Convert this to grams.

45. Dimitri’s daughter weighed $3.8$ kilograms at birth. Convert this to grams.

Show Solution

3800 grams

46. A bottle of wine contained $750$ milliliters. Convert this to liters.

47. A bottle of medicine contained $300$ milliliters. Convert this to liters.

Show Solution

0.3 liters

48. If an aspirin tablet contains 325 mg aspirin, how many grams of aspirin does it contain?

49. Many medical laboratory tests are run using 5.0 microliter blood serum. What is this volume in milliliters if there are 1 million microliters in one liter?

Solution

0.005 mL

50. 8.160 m = ________ cm

51. 3779 mg = ________ g

Solution

3.779 g

52. 781 mL = ________ L

53. 4.18 kg = ________ g

Solution

4180 g

54. 27.8 m = ________ km

55. 0.13 mL = ________ L

Solution

0.00013 L

56. 1738 km = ________ m

57. 1.9 g = ________ kg

Solution

0.0019 kg

Use Mixed Units of Measurement in the Metric System

In the following exercises, solve and write your answer in mixed units.

58. Matthias is $1.8$ meters tall. His son is $89$ centimeters tall. How much taller, in centimeters, is Matthias than his son?

59. Stavros is $1.6$ meters tall. His sister is $95$ centimeters tall. How much taller, in centimeters, is Stavros than his sister?

Show Solution

65 centimeters

60. A typical dove weighs $345$ grams. A typical duck weighs $1.2$ kilograms. What is the difference, in grams, of the weights of a duck and a dove?

61. Concetta had a $2-kilogram$ bag of flour. She used $180$ grams of flour to make biscotti. How many kilograms of flour are left in the bag?

Show Solution

1.82 kilograms

62. Harry mailed $5$ packages that weighed $420$ grams each. What was the total weight of the packages in kilograms?

63. One glass of orange juice provides $560$ milligrams of potassium. Linda drinks one glass of orange juice every morning. How many grams of potassium does Linda get from her orange juice in $30$ days?

Show Solution

16.8 grams

64. Jonas drinks $200$ milliliters of water $8$ times a day. How many liters of water does Jonas drink in a day?

65. One serving of whole grain sandwich bread provides $6$ grams of protein. How many milligrams of protein are provided by $7$ servings of whole grain sandwich bread?

Show Solution

42,0000 milligrams

Convert Between U.S. and Metric Systems

In the following exercises, make the unit conversions. Round to the nearest tenth.

66. Bill is $75$ inches tall. Convert his height to centimeters.

67. Frankie is $42$ inches tall. Convert his height to centimeters.

Show Solution

106.7 centimeters

68. Marcus passed a football $24$ yards. Convert the pass length to meters.

69. Connie bought $9$ yards of fabric to make drapes. Convert the fabric length to meters.

Show Solution

8.2 meters

70. Each American throws out an average of $1,650$ pounds of garbage per year. Convert this weight to kilograms (2.20 pounds = 1 kilogram).

71. An average American will throw away $90,000$ pounds of trash over his or her lifetime. Convert this weight to kilograms (2.20 pounds = 1 kilogram).

Show Solution

40,900 kilograms

72. A $5K$ run is $5$ kilometers long. Convert this length to miles.

73. Kathryn is $1.6$ meters tall. Convert her height to feet.

Show Solution

5.2 feet

74. Dawn’s suitcase weighed $20$ kilograms. Convert the weight to pounds.

75. Jackson’s backpack weighs $15$ kilograms. Convert the weight to pounds.

Show Solution

33 pounds

76. Ozzie put $14$ gallons of gas in his truck. Convert the volume to liters.

77. Bernard bought $8$ gallons of paint. Convert the volume to liters.

Show Solution

30.2 liters

78. How many milliliters of a soft drink are contained in a 12.0-oz can?

79. A barrel of oil is exactly 42 gal. How many liters of oil are in a barrel?

Solution

159 L

80. The diameter of a red blood cell is about 0.0003 in. What is its diameter in centimeters?

81. Is a 197-lb weight lifter light enough to compete in a class limited to those weighing 90 kg or less?

Solution

No, the weight lifter is about 198.4 lbs.

82. A very good weight lifter lifted 192 kg in a move called the clean and jerk. What was the mass of the weight lifted in pounds?

83. Gasoline is sold by the liter in many countries. How many liters are required to fill a 12.0-gal gas tank?

Solution

45.4 L

84. Milk is sold by the liter in many countries. What is the volume of a half-gallon of milk in liters?

85. A long ton is defined as exactly 2240 lb. What is this mass in kilograms?

Solution

1016 kg

86. Make the conversion indicated in each of the following:

(a) the men’s world record long jump, 29 ft 4¼ in., to meters

(b) the greatest depth of the ocean, about 6.5 mi, to kilometers

(c) the volume of 1 gill (exactly 4 oz) to milliliters

(d) the mass of a 3525-lb car to kilograms

(e) the mass of a 2.3-oz egg to grams

87. Make the conversion indicated in each of the following:

(a) the length of a soccer field, 120 m, to feet

(b) the height of Mt. Kilimanjaro, at 19,565 ft, the highest mountain in Africa, to kilometers

(c) the mass of a pencil, 0.0085 kg, to ounces

(d) the mass of a bushel of rye, 32.0 lb, to kilograms

(e) the mass of a 5.00-grain aspirin tablet to milligrams (1 grain = 0.00229 oz)

Solution

(a) 394 ft, (b) 5.96 km, (c) 0.3 oz, (d) 14.5 kg, (e) 0.01145 oz

88. Soccer fields vary in size. A large soccer field is 115 m long and 85 m wide. What are its dimensions in feet and inches? (Assume that 1 meter equals 3.281 feet.)

89. What is the height in meters of a person who is 6 ft 1 in. tall? (Assume that 1 meter equals 39.37 in.)

Solution

1.85 m

90. Mount Everest, at 29,028 feet, is the tallest mountain on the Earth. What is its height in kilometers? (Assume that 1 kilometer equals 3,281 feet.)

91. An algebra student is 159 cm tall and weighs 45.8 kg. What is her height in inches and weight in pounds?

Convert between Fahrenheit and Celsius

In the following exercises, convert the Fahrenheit temperature to degrees Celsius. Round to the nearest tenth.

92. 86°F

93. 77°F

Show Solution

25°C

94. 104°F

95. 14°F

Show Solution

-10°C

96. 72°F

97. 4°F

Show Solution

-15.6°C

98. 0°F

99. 120°F

Show Solution

48.9°C

In the following exercises, convert the Celsius temperatures to degrees Fahrenheit. Round to the nearest tenth.

100. 5°C

101. 25°C

Show Solution

77°F

102. −10°C

103. −15°C

Show Solution

5°F

104. 22°C

105. 8°C

Show Solution

46.4°F

106. 43°C

107. $16 °C$

Show Solution

60.8°F

Everyday Math

108. Nutrition Julian drinks one can of soda every day. Each can of soda contains $40$ grams of sugar. How many kilograms of sugar does Julian get from soda in $1$ year?

109. Reflectors The reflectors in each lane-marking stripe on a highway are spaced $16$ yards apart. How many reflectors are needed for a one-mile-long stretch of highway?

Show Solution

110 reflectors

110. Soccer Soccer is played with a round ball having a circumference between 27 and 28 in. and a weight between 14 and 16 oz. What are these specifications in units of centimeters and grams?

111. Basketball A women’s basketball has a circumference between 28.5 and 29.0 inches and a maximum weight of 20 ounces. What are these specifications in units of centimeters and grams?

Solution

The circumference is between 72.4 and 73.7 centimeters, and the weight is between 397 and 454 grams.

112. To prepare for a laboratory period, a student lab assistant needs 125 g of a compound. A bottle containing 1/4 lb is available. Did the student have enough of the compound?

Writing Exercises

113. Some people think that $65 °$ to $75 °$ Fahrenheit is the ideal temperature range. (a) What is your ideal temperature range? Why do you think so? (b) Convert your ideal temperatures from Fahrenheit to Celsius.

114. (a) Did you grow up using the U.S. customary or the metric system of measurement? (b) Describe two examples in your life when you had to convert between systems of measurement. (c) Which system do you think is easier to use? Explain.

Glossary

conversion factor

A conversion factor (or unit conversion factor) is a ratio of two equivalent quantities expressed with different measurement units.

metric system

The metric system is a decimal system of measurement which is based on a fundamental set of units including the meter, the second, and the kilogram.

rate

A rate is a quantity which is computed from measurements in two or more different units.

rate conversion

Rate conversions are unit conversions involving several different quantities.

unit

Units are standardized values used to express measurements of physical quantities.

unit equivalence

A unit equivalence tells you how many of one unit are equivalent to exactly one of another unit.

Media Attributions

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Make Unit Conversions in the U.S. System

How To

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

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Make Unit Conversions in the Metric System

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Use Mixed Units of Measurement in the Metric System

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Convert Between U.S. and Metric Systems of Measurement

Try It

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Convert Between Fahrenheit and Celsius Temperatures

Try It

Try It

Try It

Try It

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 U.S. and Metric Systems

Convert between Fahrenheit and Celsius

Everyday Math

Writing Exercises

Glossary

Media Attributions

License

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