4.1 Systems of Measurement

Learning Objectives

By the end of this section, you will be able to:

  • Make unit conversions in the imperial system
  • Use mixed units of measurement in the imperial system
  • Make unit conversions in the metric system
  • Use mixed units of measurement in the metric system
  • Convert between the imperial and the metric systems of measurement
  • Convert between Fahrenheit and Celsius temperatures

Make Unit Conversions in the Imperial System

There are two systems of measurement commonly used around the world. Most countries use the metric system. Canada uses the metric system, and the United States use the imperial system of measurement. However, people in Canada often use imperial measurements as well. We will look at the imperial system first.

The imperial 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 imperial system and the metric system measure time in seconds, minutes, and hours.

The equivalencies of measurements are shown in the table below. The table also shows, in parentheses, the common abbreviations for each measurement.

Imperial System of Measurement
\begin{array}{ccc}\mathbf{\text{Length}}\hfill & & \begin{array}{ccc}1\phantom{\rule{0.2em}{0ex}}\text{foot}\phantom{\rule{0.2em}{0ex}}\text{(ft.)}\hfill & =\hfill & 12\phantom{\rule{0.2em}{0ex}}\text{inches}\phantom{\rule{0.2em}{0ex}}\text{(in.)}\hfill \\ 1\phantom{\rule{0.2em}{0ex}}\text{yard}\phantom{\rule{0.2em}{0ex}}\text{(yd.)}\hfill & =\hfill & 3\phantom{\rule{0.2em}{0ex}}\text{feet}\phantom{\rule{0.2em}{0ex}}\text{(ft.)}\hfill \\ 1\phantom{\rule{0.2em}{0ex}}\text{mile}\phantom{\rule{0.2em}{0ex}}\text{(mi.)}\hfill & =\hfill & 5,280\phantom{\rule{0.2em}{0ex}}\text{feet}\phantom{\rule{0.2em}{0ex}}\text{(ft.)}\hfill \end{array}\hfill \end{array} \begin{array}{ccc}\mathbf{\text{Volume}}\hfill & & \begin{array}{ccc}3\phantom{\rule{0.2em}{0ex}}\text{teaspoons}\phantom{\rule{0.2em}{0ex}}\text{(t)}\hfill & =\hfill & 1\phantom{\rule{0.2em}{0ex}}\text{tablespoon}\phantom{\rule{0.2em}{0ex}}\text{(T)}\hfill \\ \text{16 tablespoons}\phantom{\rule{0.2em}{0ex}}\text{(T)}\hfill & =\hfill & \text{1 cup}\phantom{\rule{0.2em}{0ex}}\text{(C)}\hfill \\ \text{1 cup}\phantom{\rule{0.2em}{0ex}}\text{(C)}\hfill & =\hfill & \text{8 fluid ounces}\phantom{\rule{0.2em}{0ex}}\text{(fl. oz.)}\hfill \\ \text{1 pint}\phantom{\rule{0.2em}{0ex}}\text{(pt.)}\hfill & =\hfill & \text{2 cups}\phantom{\rule{0.2em}{0ex}}\text{(C)}\hfill \\ \text{1 quart}\phantom{\rule{0.2em}{0ex}}\text{(qt.)}\hfill & =\hfill & \text{2 pints}\phantom{\rule{0.2em}{0ex}}\text{(pt.)}\hfill \\ \text{1 gallon}\phantom{\rule{0.2em}{0ex}}\text{(gal)}\hfill & =\hfill & \text{4 quarts}\phantom{\rule{0.2em}{0ex}}\text{(qt.)}\hfill \end{array}\hfill \end{array}
\begin{array}{ccc}\mathbf{\text{Weight}}\hfill & & \begin{array}{ccc}\text{1 pound}\phantom{\rule{0.2em}{0ex}}\text{(lb.)}\hfill & =\hfill & \text{16 ounces}\phantom{\rule{0.2em}{0ex}}\text{(oz.)}\hfill \\ \text{1 ton}\hfill & =\hfill & \text{2000 pounds}\phantom{\rule{0.2em}{0ex}}\text{(lb.)}\hfill \end{array}\hfill \end{array} \begin{array}{ccc}\mathbf{\text{Time}}\hfill & & \phantom{\rule{1em}{0ex}}\begin{array}{ccc}\text{1 minute}\phantom{\rule{0.2em}{0ex}}\text{(min)}\hfill & =\hfill & \text{60 seconds}\phantom{\rule{0.2em}{0ex}}\text{(sec)}\hfill \\ \text{1 hour}\phantom{\rule{0.2em}{0ex}}\text{(hr)}\hfill & =\hfill & \text{60 minutes}\phantom{\rule{0.2em}{0ex}}\text{(min)}\hfill \\ \text{1 day}\hfill & =\hfill & \text{24 hours}\phantom{\rule{0.2em}{0ex}}\text{(hr)}\hfill \\ \text{1 week}\phantom{\rule{0.2em}{0ex}}\text{(wk)}\hfill & =\hfill & \text{7 days}\hfill \\ \text{1 year}\phantom{\rule{0.2em}{0ex}}\text{(yr)}\hfill & =\hfill & \text{365 days}\hfill \end{array}\hfill \end{array}

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.

Identity Property of Multiplication

For any real number a:

a\cdot1=a 1\cdot a=a

1 is the multiplicative identity.

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 \dfrac{1\text{ foot}}{12\text{ inches}}. When we multiply by this fraction we do not change the value, but just change the units.

But \dfrac{12\text{ inches}}{1\text{ foot}} also equals 1. How do we decide whether to multiply by \dfrac{1\text{ foot}}{12\text{ inches}} or \dfrac{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?

Two expressions are given: 66 inches times the fraction (1 foot) over (12 inches), and 66 inches times the fraction (12 inches) over (1 foot). This second expression is crossed out. Below this, it is stated that “The first form works since 66 inches times the fraction (1 foot) over (12 inches), with inches crossed off in both instances.

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

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

Solution
Step 1. Multiply the measurement to be converted by 1; write  1  as a fraction  relating the units given and  the units needed. Multiply 66 inches by 1 , writing  1 as a fraction relating inches and feet.  We need inches in the denominator so that the inches will divide out! \begin{array}{l} 66 \text { inches } \cdot 1 \\ 66 \text { inches } \cdot \frac{1 \text { foot }}{12 \text { inches }} \end{array}
Step 2. Multiply. Think of 66 inches as \frac{66 \text { inches }} { 1} \dfrac{66 \text { inches } \cdot 1 \text { foot }}{12 \text { inches }}
Step 3. Simplify the fraction. Notice: inches divide out. \dfrac{66 {\cancel{\text { inches }}} \cdot 1 \text { foot }}{12 {\cancel{\text { inches }}}}\\ =\dfrac{66\text{ feet}}{12}
Step 4. Simplify. Divide 66  by  12. 5.5 feet

TRY IT 1.1

Lexie is 30 inches tall. Convert her height to feet.

Answer

2.5 feet

TRY IT 1.2

Rene bought a hose that is 18 yards long. Convert the length to feet.

Answer

54 feet

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.
  4. Simplify.

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

A female orca in the Salish Sea 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 \dfrac{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 \dfrac{\text{2,000 pounds}}{\text{1 ton}}
Simplify.

    {\large \[ \displaystyle \frac{3.2 {\color{red}{\cancel{ \color{black} \text{ tons}}}} \cdot 2,000 \, \text{pounds}}{1 {\color{red}{\cancel{ \color{black}\text{ ton}}}}} \] }
Multiply. \text{6,400 pounds}
The female orca weighs almost 6,400 pounds.

TRY IT 2.1

Arnold’s SUV weighs about 4.3 tons. Convert the weight to pounds.

Answer

8,600 pounds

TRY IT 2.2

The Carnival Destiny cruise ship weighs 51,000 tons. Convert the weight to pounds.

Answer

102,000,000 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 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}}. \dfrac{\text{9 wk}}{1}\cdot \dfrac{\text{7 days}}{\text{1 wk}}\cdot \dfrac{\text{24 hr}}{\text{1 day}}\cdot \dfrac{\text{60 min}}{\text{1 hr}}
Divide out the common units. \dfrac{\text{9 \cancel{ wk}}}{1}\cdot \dfrac{\text{7\cancel{ days}}}{\text{1\cancel{ wk}}}\cdot \dfrac{\text{24 \cancel{hr}}}{\text{1\cancel{ day}}}\cdot \dfrac{\text{60 min}}{\text{1 \cancel{hr}}}
Multiply. \dfrac{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!).

TRY IT 3.1

The distance between the earth and the moon is about 250,000 miles. Convert this length to yards.

Answer

440,000,000 yards

TRY IT 3.2

The astronauts of Expedition 28 on the International Space Station spend 15 weeks in space. Convert the time to minutes.

Answer

151,200 minutes

EXAMPLE 4

How many ounces are in 1 gallon?

Solution

We will convert gallons to ounces by multiplying by several conversion factors. Refer to the table on Imperial Systems of Measurement.

1 gallon
Multiply the measurement to be converted by 1. \dfrac{\text{1 gallon}}{1}\cdot \dfrac{\text{4 quarts}}{\text{1 gallon}}\cdot \dfrac{\text{2 pints}}{\text{1 quart}}\cdot \dfrac{\text{2 cups}}{\text{1 pint}}\cdot \dfrac{\text{8 ounces}}{\text{1 cup}}
Use conversion factors to get to the right unit.
Simplify.		 \dfrac{\text{1\cancel{ gallon}}}{1}\cdot \dfrac{\text{4\cancel{ quarts}}}{\text{1\cancel{ gallon}}}\cdot \dfrac{\text{2 \cancel{pints}}}{\text{1 \cancel{quart}}}\cdot \dfrac{\text{2 \cancel{cups}}}{\text{1 \cancel{pint}}}\cdot \dfrac{\text{8 ounces}}{\text{1 \cancel{cup}}}
Multiply. \dfrac{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.

TRY IT 4.1

How many cups are in 1 gallon?

Answer

16 cups

TRY IT 4.2

How many teaspoons are in 1 cup?

Answer

48 teaspoons

Use Mixed Units of Measurement in the Imperial 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 5

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. \begin{array}{rr} \quad 14 \text { ounces } \\ 1 \text { pound } \quad 2 \text { ounces } \\ +1 \text { pound } \quad 6 \text { ounces } \\ \hline 2 \text { pounds } 22 \text { ounces } \end{array}
Convert 22 ounces to pounds and ounces. 1 pound, 6 ounces
Add the pounds and ounces. 2 pounds + 1 pound + 6 ounces
Answer Seymour bought 3 pounds 6 ounces of steak.

TRY IT 5.1

Laura gave birth to triplets weighing 3 pounds 6 ounces, 3 pounds 5 ounces, and 2 pounds 13 ounces. What was the total birth weight of the three babies?

Answer

9 lbs. 8 oz

TRY IT 5.2

Stan cut two pieces of crown molding for his family room that were 8 feet 7 inches and 12 feet 11 inches. What was the total length of the molding?

Answer

21 ft. 6 in.

EXAMPLE 6

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. \begin{aligned} &\begin{array}{lr} 6 \text { feet } & 4 \text { inches } \\ \times & 4 \\ \hline 24 \text { feet } & 16 \text { inches } \end{array} \end{aligned}
Convert the 16 inches to feet.
Add the feet.		 24 \text{ feet} + 1\text{ foot} 4 \text{ inches} =25\text{ feet} 4\text{ inches}
Anthony bought 25 feet and 4 inches of wood.

TRY IT 6.1

Henri wants to triple his vegan spaghetti sauce recipe that uses 1 pound 8 ounces of black beans. How many pounds of black beans will he need?

Answer

4 lbs. 8 oz.

TRY IT 6.2

Joellen wants to double a solution of 5 gallons 3 quarts. How many gallons of solution will she have in all?

Answer

11 gallons 2 qt.

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 metre. One kilometre is 1,000 metres; the prefix kilo means thousand. One centimetre is \dfrac{1}{100} of a metre, just like one cent is \dfrac{1}{100} of one dollar.

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

Metric System of Measurement
Length Mass Capacity
1 kilometre (km) = 1,000 m

1 hectometre (hm) = 100 m

1 dekametre (dam) = 10 m

1 metre (m) = 1 m

1 decimetre (dm) = 0.1 m

1 centimetre (cm) = 0.01 m

1 millimetre (mm) = 0.001 m

 1 kilogram (kg) = 1,000 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 kilolitre (kL) = 1,000 L

1 hectolitre (hL) = 100 L

1 dekalitre (daL) = 10 L

1 litre (L) = 1 L

1 decilitre (dL) = 0.1 L

1 centilitre (cL) = 0.01 L

1 millilitre (mL) = 0.001 L
1 metre = 100 centimetres

1 metre = 1,000 millimetres

 1 gram = 100 centigrams

1 gram = 1,000 milligrams

 1 litre = 100 centilitres

1 litre = 1,000 millilitres

To make conversions in the metric system, we will use the same technique we did in the Imperial 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 kilometres. The metric system is commonly used in Canada when talking about the length of a race.

EXAMPLE 7

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

Solution

We will convert kilometres to metres using the identity property of multiplication.

Nick ran 10 kilometres
Multiply the measurement to be converted by 1. 10 kilometres × 1
Write 1 as a fraction relating kilometres and metres. 10\text{ kilometres}\times\dfrac{1,000\text{ metres}}{1\text{ kilometres}}
Simplify. 10\text{ \cancel{kilometres}}\times\dfrac{1,000\text{ metres}}{1\text{ \cancel{kilometres}}}
Multiply. 10,000 metres
Nick ran 10,000 metres.

TRY IT 7.1

Sandy completed her first 5K race! How many metres did she run?

Answer

5,000 metres

TRY IT 7.2

Herman bought a rug 2.5 metres in length. How many centimetres is the length?

Answer

250 centimetres

EXAMPLE 8

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

Solution

We will convert grams into kilograms.

Eleanor’s baby weighs 3200 {\color{red} \text { grams}}
Multiply the measurement to be converted by 1. 3200 {\color{red} \text { grams}} \cdot 1
Write 1 as a function relating kilograms and grams. 3,200 {\color{red} {\text{ grams }}} \cdot \dfrac{1 \text{kg}}{1,000 {\color{red} {\text{ grams }}}}}
Simplify.  3,200 {\color{red} {\cancel {\text{ grams }}}} \cdot \dfrac{1 \text{kg}}{1,000 {\color{red}{\cancel {\text{ grams }}}}}}
Multiply. \dfrac{\text{3,200 kilograms}}{1,000}
Divide. 3.2 kilograms
The baby weighed 3.2 kilograms.

TRY IT 8.1

Kari’s newborn baby weighed 2,800 grams. How many kilograms did the baby weigh?

Answer

2.8 kilograms

TRY IT 8.2

Anderson received a package that was marked 4,500 grams. How many kilograms did this package weigh?

Answer

4.5 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 8, we changed 3,200 grams to kilograms by multiplying by \dfrac{1}{1000} (or 0.001). This is the same as moving the decimal three places to the left.

We have the statement 3200 g times the fraction 1 kg over 1000 g, with the g’s crossed out. Below this, we have 3.2. We also have the statement 3200 times 1/1000, with an arrow drawn from the right of the final 0 in 3200 to the space between the 0’s, to the space between the 2 and the 0, and then to the space between the 3 and the 2. Below this, we have 3.2.
Figure.1

EXAMPLE 9

Convert a) 350 L to kilolitres b) 4.1 L to millilitres.

Solution
  1. We will convert litres to kilolitres. In the Metric System of Measurement table, we see that 1\phantom{\rule{0.2em}{0ex}}\text{kilolitre}=\text{1,000 litres.}
    Given amount \text{350 L}
    Multiply by 1, writing 1 as a fraction relating litres to kilolitres. \text{350 L}\cdot \dfrac{\text{1 kL}}{\text{1,000 L}}
    Simplify. 350\cancel{\text{L}}\cdot \dfrac{\text{1 kL}}{1,000\cancel{\text{L}}}
    Move the decimal 3 units to the left. \text{0.35 kL}
  2. We will convert litres to millilitres. From Metric System of Measurement table we see that \text{1 litre}=\text{1,000 millilitres.}
    Given amount 4.1 \text{L}
    Multiply by 1, writing 1 as a fraction relating litres to millilitres. 4.1 \text{L} \cdot \dfrac{1,000 \text{mL}}{1 \text{L}}
    Simplify. 4.1 {\color{red}{\cancel{\text{L}}}} \cdot \dfrac{1,000 \text{mL}}{1 {\color{red}{\cancel{\text{L}}}}}
    Move the decimal 3 units to the right. . = 4,100 \text{mL}

TRY IT 9.1

Convert: a) 725 L to kilolitres b) 6.3 L to millilitres

Answer

a) 0.725 kilolitres b) 6,300 millilitres

TRY IT 9.2

Convert: a) 350 hL to litres b) 4.1 L to centilitres

Answer

a) 35,000 litres b) 410 centilitres

Use Mixed Units of Measurement in the Imperial System

Performing arithmetic operations on measurements with mixed units of measures in the imperial system requires the same care we used in the Canadian system. Make sure to add or subtract like units.

EXAMPLE 10

Ryland is 1.6 metres tall. His younger brother is 85 centimetres tall. How much taller is Ryland than his younger brother?

Solution

We can convert both measurements to either centimetres or metres. Since metres is the larger unit, we will subtract the lengths in metres. We convert 85 centimetres to metres by moving the decimal 2 places to the left.

Write the 85 centimetres as metres. \begin{array}{r}1.60\text{m}\\ \underline{-0.85\text{m}} \\ 0.75\text{m}\hfill \end{array}

Ryland is 0.75 m taller than his brother.

TRY IT 10.1

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

Answer

83 centimetres

TRY IT 10.2

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

Answer

1.04 metres

EXAMPLE 11

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

Solution

We will find the amount of olive oil in millileters then convert to litres.

What do we need to do? Triple 150 mL
Translate to algebra. 3 \cdot 150 \text{ mL}
Multiply. 450 mL
Convert to litres. 450\cdot \dfrac{0.001\text{ L}}{1\text{ mL}}
Simplify. 0.45 L
Dena needs 0.45 litres of olive oil.

TRY IT 11.1

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

Answer

2 litres

TRY IT 11.2

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?

Answer

2.4 kilograms

Convert Between the Imperial and the Metric Systems of Measurement

Many measurements in Canada are made in metric units. Our soda may come in 2-litre 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.

The table below shows some of the most common conversions.

Conversion Factors Between Imperial and Metric Systems
Length Mass Capacity
\begin{array}{c}\begin{array}{ccc}\text{1 in.}\hfill & =\hfill & \text{2.54 cm}\hfill \\ \text{1 ft.}\hfill & =\hfill & \text{0.305 m}\hfill \\ \text{1 yd.}\hfill & =\hfill & \text{0.914 m}\hfill \\ \text{1 mi.}\hfill & =\hfill & \text{1.609 km}\hfill \\ \text{1 m}\hfill & =\hfill & \text{3.28 ft.}\hfill \end{array}\hfill \end{array} \begin{array}{c}\begin{array}{ccc}\text{1 lb.}\hfill & =\hfill & \text{0.45 kg}\hfill \\ \text{1 oz.}\hfill & =\hfill & \text{28 g}\hfill \\ \text{1 kg}\hfill & =\hfill & \text{2.2 lb.}\hfill \end{array}\hfill \end{array} \begin{array}{c}\begin{array}{ccc}\text{1 qt.}\hfill & =\hfill & \text{0.95 L}\hfill \\ \text{1 fl. oz.}\hfill & =\hfill & \text{30 mL}\hfill \\ \text{1 L}\hfill & =\hfill & \text{1.06 qt.}\hfill \end{array}\hfill \end{array}

(Figure.2) shows how inches and centimetres are related on a ruler.

A ruler with inches and centimetres.
Figure.2

(Figure.3) shows the ounce and millilitre markings on a measuring cup.

A measuring cup showing millilitre s and ounces.
Figure.3

(Figure.4) shows how pounds and kilograms marked on a bathroom scale.

We are given an image of a bathroom scale showing pounds.
Figure.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 ounces are in the bottle? Round to the nearest tenth of an ounce.

Solution
Given amount 500 mL
Multiply by a unit conversion factor relating mL and ounces. 500{\color{red}{\cancel{\text{ millilitres}}}}\cdot \dfrac{1\text{ ounce}}{30{\color{red}{\cancel{\text{ millilitres}}}}}
Simplify. \dfrac{\text{500 ounce}}{30}
Divide. 16.7 ounces.
The water bottle has 16.7 ounces.

TRY IT 12.1

How many quarts of soda are in a 2-L bottle? Round your answer to nearest tenth.

Answer

2.1 quarts

TRY IT 12.2

How many litres are in 4 quarts of milk?

Answer

3.8 litres

EXAMPLE 13

Soleil was on a road trip and saw a sign that said the next rest stop was in 100 kilometres. How many miles until the next rest stop? Round your answer to nearest mile.

Solution
Distance to the next stop 100 kilometres
Multiply by a unit conversion factor relating km and mi. 100\text{ kilometres}\cdot \dfrac{1\text{ mile}}{1.609\text{ kilometre}}
Simplify. \dfrac{\text{100 miles}}{1.609}
Divide. 62 miles.
Soleil will travel 62 miles.

TRY IT 13.1

The height of Mount Kilimanjaro is 5,895 metres. Convert the height to feet. Round your answer to nearest feet.

Answer

19,336 feet or 19,341 feet

TRY IT 13.2

The flight distance from Toronto to Vancouver is 3,364 kilometres. Convert the distance to miles. Round your answer to nearest mile.

Answer

2,090 miles or 2,091 miles

Convert between Fahrenheit and Celsius Temperatures

Have you ever been in a foreign country and heard the weather forecast? If the forecast is for 71°F what does that mean?

The Canadian and imperial systems use different scales to measure temperature. The Canadian system uses degrees Celsius, written °C. The imperial system uses degrees Fahrenheit, written°F. (Figure.5) shows the relationship between the two systems.

The diagram shows normal body temperature, along with the freezing and boiling temperatures of water in degrees Fahrenheit and degrees Celsius.
Two thermometres are shown, one in Celsius (°C) and another in Fahrenheit (°F). They are marked “Water boils” at 100°C and 212°F. They are marked “Normal body temperature” at 37°C and 98.6°F. They are marked “Water freezes” at 0°C and 32°F.
Figure.5

Temperature Conversion

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

C=\dfrac{5}{9}\left(F-32\right).

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

F=\dfrac{9}{5}C+32.

EXAMPLE 14

Convert 50° Fahrenheit into degrees Celsius.

Solution

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

C=\frac{5}{9}(F-32)
Substitute \color{red}{50} for F. C=\frac{5}{9}({\color{red}{50}}-32)
Simplify in parentheses. C=\frac{5}{9}(18)
Multiply. C=10
So we found that 50°F is equivalent to 10°C.

TRY IT 14.1

Convert the Fahrenheit temperature to degrees Celsius: 59° Fahrenheit.

Answer

15°C

TRY IT 14.2

Convert the Fahrenheit temperature to degrees Celsius: 41° Fahrenheit.

Answer

5°C

EXAMPLE 15

While visiting Paris, Woody saw the temperature was 20° Celsius. Convert the temperature into degrees Fahrenheit.

Solution

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

F=\frac{9}{5} C+32
Substitute  {\color{red}{ 20}}  for C. F=\frac{9}{5}({\color{red}20})+32
Multiply. F=36+32
Add. F=68
So we found that 20°C is equivalent to 68°F.

TRY IT 15.1

Convert the Celsius temperature to degrees Fahrenheit: the temperature in Helsinki, Finland, was 15° Celsius.

Answer

59°F

TRY IT 15.2

Convert the Celsius temperature to degrees Fahrenheit: the temperature in Sydney, Australia, was 10° Celsius.

Answer

50° F

Key Concepts

  • Metric System of Measurement
    • Length
      \begin{array}{lcl}\text{1 kilometre (km)} & = & \text{1,000 m} \\ \text{1 hectometre (hm)} & = & \text{100 m} \\ \text{1 dekametre (dam)} & = & \text{10 m} \\ \text{1 metre (m)} & = & \text{1 m} \\ \text{1 decimetre (dm)} & = & \text{0.1 m} \\ \text{1 centimetre (cm)} & = & \text{0.01 m}\hfill \\ \text{1 millimetre (mm)} & = & \text{0.001 m} \\ \text{1 metre} & = & \text{100 centimetres} \\ \text{1 metre} & = & \text{1,000 millimetres} \end{array}
    • Mass
      \begin{array}{lcl}\text{1 kilogram (kg)} & = & \text{1,000 g} \\ \text{1 hectogram (hg)} & = & \text{100 g} \\ \text{1 dekagram (dag)} & = & \text{10 g} \\ \text{1 gram (g)} & = & \text{1 g} \\ \text{1 decigram (dg)} & = & \text{0.1 g} \\ \text{1 centigram (cg)} & = & \text{0.01 g} \\ \text{1 milligram (mg)} & = & \text{0.001 g} \\ \text{1 gram} & = & \text{100 centigrams} \\ \text{1 gram} & = & \text{1,000 milligrams} \end{array}
    • Capacity
      \begin{array}{lcl}\text{1 kilolitre (kL)} & = & \text{1,000 L} \\ \text{1 hectolitre (hL)} & = & \text{100 L} \\ \text{1 dekalitre (daL)} & = & \text{10 L} \\ \text{1 litre (L)} & = & \text{1 L} \\ \text{1 decilitre (dL)} & = & \text{0.1 L} \\ \text{1 centilitre (cL)} & = & \text{0.01 L} \\ \text{1 millilitre (mL)} & = & \text{0.001 L} \\ \text{1 litre} & = & \text{100 centilitres} \\ \text{1 litre} & = & \text{1,000 millilitres}\end{array}
  • Temperature Conversion
    • To convert from Fahrenheit temperature, F, to Celsius temperature, C, use the formula \text{C}=\dfrac{5}{9}\left(\text{F}-32\right)
    • To convert from Celsius temperature, C, to Fahrenheit temperature, F, use the formula \text{F}=\dfrac{9}{5}\text{C}+32

Practice Makes Perfect

Make Unit Conversions in the Imperial System

In the following exercises, convert the units.

1. A park bench is 6 feet long. Convert the length to inches. 2. A floor tile is 2 feet wide. Convert the width to inches.
3. A ribbon is 18 inches long. Convert the length to feet. 4. Carson is 45 inches tall. Convert his height to feet.
5. A football field is 160 feet wide. Convert the width to yards. 6. On a baseball diamond, the distance from home plate to first base is 30 yards. Convert the distance to feet.
7. Ulises lives 1.5 miles from school. Convert the distance to feet. 8. Denver, Colorado, is 5,183 feet above sea level. Convert the height to miles.
9. A killer whale weighs 4.6 tons. Convert the weight to pounds. 10. Blue whales can weigh as much as 150 tons. Convert the weight to pounds.
11. An empty bus weighs 35,000 pounds. Convert the weight to tons. 12. At take-off, an airplane weighs 220,000 pounds. Convert the weight to tons.
13. Rocco waited 1\dfrac{1}{2} hours for his appointment. Convert the time to seconds. 14. Misty’s surgery lasted 2\dfrac{1}{4} hours. Convert the time to seconds.
15. How many teaspoons are in a pint? 16. How many tablespoons are in a gallon?
17. JJ’s cat, Posy, weighs 14 pounds. Convert her weight to ounces. 18. April’s dog, Beans, weighs 8 pounds. Convert his weight to ounces.
19. Crista will serve 20 cups of juice at her son’s party. Convert the volume to gallons. 20. Lance needs 50 cups of water for the runners in a race. Convert the volume to gallons.
21. Jon is 6 feet 4 inches tall. Convert his height to inches. 22. Faye is 4 feet 10 inches tall. Convert her height to inches.
23. The voyage of the Mayflower took 2 months and 5 days. Convert the time to days. 24. Lynn’s cruise lasted 6 days and 18 hours. Convert the time to hours.
25. Baby Preston weighed 7 pounds 3 ounces at birth. Convert his weight to ounces. 26. Baby Audrey weighted 6 pounds 15 ounces at birth. Convert her weight to ounces.

Use Mixed Units of Measurement in the Imperial System

In the following exercises, solve.

27. 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? 28. Judy bought 1 pound 6 ounces of almonds, 2 pounds 3 ounces of walnuts, and 8 ounces of cashews. How many pounds of nuts did Judy buy?
29. One day Anya kept track of the number of minutes she spent driving. She recorded 45, 10, 8, 65, 20, and 35. How many hours did Anya spend driving? 30. Last year Eric went on 6 business trips. The number of days of each was 5, 2, 8, 12, 6, and 3. How many weeks did Eric spend on business trips last year?
31. Renee attached a 6 feet 6 inch extension cord to her computer’s 3 feet 8 inch power cord. What was the total length of the cords? 32. Fawzi’s SUV is 6 feet 4 inches tall. If he puts a 2 feet 10 inch box on top of his SUV, what is the total height of the SUV and the box?
33. 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? 34. 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?

Make Unit Conversions in the Metric System

In the following exercises, convert the units.

35. Ghalib ran 5 kilometres. Convert the length to metres. 36. Kitaka hiked 8 kilometres. Convert the length to metres.
37. Estrella is 1.55 metres tall. Convert her height to centimetres. 38. The width of the wading pool is 2.45 metres. Convert the width to centimetres.
39. Mount Whitney is 3,072 metres tall. Convert the height to kilometres. 40. The depth of the Mariana Trench is 10,911 metres. Convert the depth to kilometres.
41. June’s multivitamin contains 1,500 milligrams of calcium. Convert this to grams. 42. A typical ruby-throated hummingbird weights 3 grams. Convert this to milligrams.
43. One stick of butter contains 91.6 grams of fat. Convert this to milligrams. 44. One serving of gourmet ice cream has 25 grams of fat. Convert this to milligrams.
45. The maximum mass of an airmail letter is 2 kilograms. Convert this to grams. 46. Dimitri’s daughter weighed 3.8 kilograms at birth. Convert this to grams.
47. A bottle of wine contained 750 millilitre s. Convert this to litres. 48. A bottle of medicine contained 300 millilitre s. Convert this to litres.

Use Mixed Units of Measurement in the Metric System

In the following exercises, solve.

49. Matthias is 1.8 metres tall. His son is 89 centimetres tall. How much taller is Matthias than his son? 50. Stavros is 1.6 metres tall. His sister is 95 centimetres tall. How much taller is Stavros than his sister?
51. 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? 52. 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?
53. Harry mailed 5 packages that weighed 420 grams each. What was the total weight of the packages in kilograms? 54. 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?
55. Jonas drinks 200 millilitre s of water 8 times a day. How many litres of water does Jonas drink in a day? 56. 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?

Convert Between the Imperial and the Metric Systems of Measurement

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

57. Bill is 75 inches tall. Convert his height to centimetres. 58. Frankie is 42 inches tall. Convert his height to centimetres.
59. Marcus passed a football 24 yards. Convert the pass length to metres 60. Connie bought 9 yards of fabric to make drapes. Convert the fabric length to metres.
61. According to research conducted by the CRC, Canadians regrettably produce more garbage per capita than any other country on earth, at 2,172.6 pounds per person annually. Convert the waste to kilograms. 62. An average Canadian will throw away 163,000 pounds of trash over his or her lifetime. Convert this weight to kilograms.
63. A 5K run is 5 kilometres long. Convert this length to miles. 64. Kathryn is 1.6 metres tall. Convert her height to feet.
65. Dawn’s suitcase weighed 20 kilograms. Convert the weight to pounds. 66. Jackson’s backpack weighed 15 kilograms. Convert the weight to pounds.
67. Ozzie put 14 gallons of gas in his truck. Convert the volume to litres. 68. Bernard bought 8 gallons of paint. Convert the volume to litres.

Convert between Fahrenheit and Celsius Temperatures

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

69. 86° Fahrenheit 70. 77° Fahrenheit
71. 104° Fahrenheit 72. 14° Fahrenheit
73. 72° Fahrenheit 74. 4° Fahrenheit
75. 0° Fahrenheit 76. 120° Fahrenheit

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

77. 5° Celsius 78. 25° Celsius
79. -10° Celsius 80. -15° Celsius
81. 22° Celsius 82. 8° Celsius
83. 43° Celsius 84. 16° Celsius

Everyday Math

85. 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? 86. 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 lane-marking stripe?

Writing Exercises

87. 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.

 88.

a) Did you grow up using the Canadian. or the Imperial system of measurement?

b) Describe two examples in your life when you had to convert between the two systems of measurement.

Answers

1. 72 inches 3. 1.5 feet 5. 53\dfrac{1}{3} yards
7. 7,920 feet 9. 9,200 pounds 11. 17\dfrac{1}{2} tons
13. 5,400 s 15. 96 teaspoons 17. 224 ounces
19. 1\dfrac{1}{4} gallons 21. 76 in. 23. 65 days
25. 115 ounces 27. 8 lbs. 13 oz. 29. 3.05 hours
31. 10 ft. 2 in. 33. 4 yards 35. 5,000 metres
37. 155 centimetres 39. 3.072 kilometres 41. 1.5 grams
43. 91,600 milligrams 45. 2,000 grams 47. 0.75 litres
49. 91 centimetres 51. Typically, a duck weighs 855 g  more than a dove.
53. 2.1 kilograms 55. 1.6 litres 57. 190.5 centimetres
59. 21.9 metres 61. 985.5 kilograms 63. 3.1 miles
65. 44 pounds 67. 53.2 litres 69. 30°C
71. 40°C 73. 22.2°C 75. -17.8°C
77. 41°F 79. 14°F 81. 71.6°F
83. 109.4°F 85. 14.6 kilograms 87. Answers may vary.

Attributions

This chapter has been adapted from “Systems of Measurement” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

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Intermediate Algebra I Copyright © 2021 by Pooja Gupta is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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