6.1: Systems of Measurement
Chapter 6: Learning Outcomes
By the end of this chapter, you should be able to:
- Use the common metric units for temperature, length, area, volume/capacity, and mass.
- Use the common imperial for temperature, length, area, volume/capacity, and force.
- Convert between and within metric and imperial units using tables and/or calculators.
- Use proportional reasoning for conversions.
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 for measuring, while the United States uses the imperial system. However, people in Canada often use imperial measurements as well. We will look at the imperial system first.
The imperial system of measurement uses the following units:
- Length — inch, foot, yard, and mile
- Weight — pound and ton
- Capacity — cup, pint, quart, and gallon
The imperial and metric systems both measure time in seconds, minutes, and hours.
The equivalencies of measurements are shown in the lists below. The lists also show, in parentheses, the common abbreviations for each measurement.
- 1 foot (ft.) = 12 inches (in.)
- 1 yard (yd.) = 3 feet (ft.)
- 1 mile (mi.) = 5280 feet (ft.)
Weight:
- 1 pound (lb.) = 16 ounces (oz.)
- 1 ton = 2000 pounds (lb.)
Volume:
- 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.)
Time:
- 1 minute (min) = 60 seconds (sec)
- 1 hour (hr) = 60 minutes (min)
- 1 day = 24 hours (hr)
- 1 week (wk) = 7 days
- 1 year (yr) = 365 days
In many real-life applications, we need to convert between units of measurement, such as feet and yards, minutes and seconds, and quarts and gallons. We will use the identity property of multiplication to do these conversions. We will restate the identity property of multiplication here for easy reference.
Identity property of multiplication:
For any real number a
[latex]a\cdot1=a[/latex]
[latex]1\cdot a=a[/latex]
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 [latex]\dfrac{1\text{ foot}}{12\text{ inches}}[/latex]. When we multiply by this fraction, we do not change the value; instead, we just change the units.
However, [latex]\dfrac{12\text{ inches}}{1\text{ foot}}[/latex] also equals 1. How do we decide whether to multiply by [latex]\dfrac{1\text{ foot}}{12\text{ inches}}[/latex] or [latex]\dfrac{12\text{ inches}}{1\text{ foot}}[/latex]?
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?
[latex]66 \text{ inches} \cdot \dfrac{1 \text{ foot}}{12 \text{ inches}}[/latex] or [latex]\require{cancel}\cancel{66 \text{ inches} \cdot \dfrac{12 \text{ inches}}{1 \text{ foot}}}[/latex]
The first form works because [latex]\require{cancel}66 \cancel{\text{ inches}} \cdot \dfrac{1 \text{ foot}}{12 \cancel{\text{ inches}}}[/latex]
The inches divide out and leave only feet. The second form does not have any units that will divide out, so it will not help us.
Example A
MaryAnne is 66 inches tall. Convert her height into feet.
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.
[latex]66 \text{ inches} \times 1[/latex]
[latex]66 \text{ inches} \times \dfrac{1 \text{ foot}}{12 \text{inches}}[/latex]
Step 2: Multiply.
Think of 66 inches as [latex]\dfrac{66 \text{ inches}}{1}[/latex]
[latex]\dfrac{66 \text{ inches} \times1 \text{ foot}}{12 \text{ inches}}[/latex]
Step 3: Simplify the fraction.
[latex]\require{cancel}\dfrac{66 \cancel{\text{ inches}} \times1 \text{ foot}}{12 \cancel{\text{inches}}}[/latex]
Notice how the inches divide out.
[latex]\dfrac{66 \text{ feet}}{12}[/latex]
Step 4: Simplify.
Divide 66 by 12.
5.5 feet
Exercise 1
Lexie is 30 inches tall. Convert her height to feet.
Exercise 1 Answer
2.5 feet
Exercise 2
Rene bought a hose that is 18 yards long. Convert the length to feet.
Exercise 2 Answer
54 feet
To make unit conversions:
- Multiply the measurement to be converted by 1; write 1 as a fraction relating the units given and the units needed.
- Multiply.
- Simplify the fraction.
- 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 B
A female orca in the Salish Sea weighs almost 3.2 tons. Convert her weight to pounds.
We will convert 3.2 tons into pounds. We will use the identity property of multiplication, writing 1 as the fraction [latex]\dfrac{2000\text{ pounds}}{1\text{ ton}}[/latex].
Step 1: Multiply the measurement to be converted, by 1.
[latex]\text{3.2 tons}\cdot 1[/latex]
Step 2: Write 1 as a fraction relating tons and pounds.
[latex]\text{3.2 tons}\cdot \dfrac{\text{2,000 pounds}}{\text{1 ton}}[/latex]
Step 3: Simplify.
[latex]\require{cancel}\dfrac{3.2 \cancel{\text{ tons}} \cdot 2\,000 \text{ pounds}}{1\cancel{\text{ ton}}}[/latex]
Step 4: Multiply.
6400 pounds
The female orca weighs almost 6400 pounds.
Exercise 3
Arnold’s SUV weighs about 4.3 tons. Convert the weight to pounds.
Exercise 3 Answer
8600 pounds
Exercise 4
The Carnival Destiny cruise ship weighs 51000 tons. Convert the weight to pounds.
Exercise 4 Answer
102000000 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 C
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.
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.
Step 1: Write 1 as [latex]\dfrac{\text{7 days}}{\text{1 week}}, \dfrac{\text{24 hours}}{\text{1 day}}, \text{ and } \dfrac{\text{60 minutes}}{\text{1 hour}}[/latex].
[latex]\dfrac{9\text{ 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}}[/latex]
Step 2: Divide out the common units.
[latex]\require{cancel}\dfrac{9\cancel{\text{ wk}}}{1}\cdot \dfrac{7\cancel{\text{ days}}}{1\cancel{\text{ wk}}}\cdot \dfrac{24\cancel{\text{ hr}}}{1\cancel{\text{ day}}}\cdot \dfrac{60\text{ min}}{1\cancel{\text{ hr}}}[/latex]
Step 3: Multiply.
[latex]\dfrac{9\cdot 7\cdot 24\cdot \text{60 min}}{1\cdot 1\cdot 1\cdot 1}[/latex]
Step 4: Multiply again.
90720 min
Juliet and her boyfriend will be apart for 90720 minutes (although it may seem like an eternity!).
Exercise 5
The distance between the earth and the moon is about 250000 miles. Convert this length to yards.
Exercise 5 Answer
440000000 yards
Exercise 6
The astronauts of Expedition 28 on the International Space Station spent 15 weeks in space. Convert the time to minutes.
Exercise 6 Answer
151200 minutes
Example D
How many ounces are in 1 gallon?
We will convert gallons to ounces by multiplying by several conversion factors. Refer to the Imperial Systems of Measurement listed above.
Step 1: Multiply the measurement to be converted by 1.
[latex]\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}}[/latex]
Step 2: Use conversion factors to get to the right unit. Simplify.
[latex]\require{cancel}\dfrac{1\cancel{\text{ gallon}}}{1}\cdot \dfrac{4\cancel{\text{ quarts}}}{1\cancel{\text{ gallon}}}\cdot \dfrac{2\cancel{\text{ pints}}}{1\cancel{\text{ quart}}}\cdot \dfrac{2\cancel{\text{ cups}}}{1\cancel{\text{ pint}}}\cdot \dfrac{\text{8 ounces}}{1\cancel{\text{ cup}}}[/latex]
Step 3: Multiply.
[latex]\dfrac{1\cdot 4\cdot 2\cdot 2\cdot \text{8 ounces}}{1\cdot 1\cdot 1\cdot 1\cdot 1}[/latex]
Step 4: Simplify.
128 ounces
There are 128 ounces in a gallon.
Exercise 7
How many cups are in 1 gallon?
Exercise 7 Answer
16 cups
Exercise 8
How many teaspoons are in 1 cup?
Exercise 8 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 E
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?
We will add the weights of the steaks to find the total weight of the steaks.
Step 1: Add the ounces. Then, add the pounds.
[latex]\begin{array}[t]{rlr}&&14\text{ ounces} \\ &1\text{ pound} &2\text{ ounces} \\ +&1\text{ pound}& 6\text{ ounces} \\ \hline &2\text{ pounds} &22\text{ ounces} \\ \end{array}[/latex]
Step 2: Convert 22 ounces to pounds and ounces.
[latex]1 \text{ pound}, 6 \text{ ounces}[/latex]
Step 3: Add the pounds and ounces.
[latex]2\text{ pound} + 1\text{ pound} + 6\text{ ounces}[/latex]
[latex] 3 \text{ pounds}, 6 \text{ ounces}[/latex]
Seymour bought 3 pounds, 6 ounces of steak.
Exercise 9
Laura gave birth to triplets weighing 3 pounds 3 ounces, 3 pounds 3 ounces, and 2 pounds 9 ounces. What was the total birth weight of the three babies?
Exercise 9 Answer
9 lbs. 8 oz
Exercise 10
Stan cut two pieces of crown moulding for his family room that were 8 feet 7 inches and 12 feet 11 inches. What was the total length of the moulding?
Exercise 10 Answer
21 ft. 6 in.
Example F
Anthony bought four planks of wood that were each 6 feet 4 inches long. What is the total length of the wood he purchased?
We will multiply the length of one plank to find the total length.
Step 1: Multiply the inches and then the feet.
[latex]\begin{array}[t]{rlr}&6 \text{ feet} &4\text{ inches} \\ \times &&4 \\ \hline &24 \text{ feet} &16\text{ inches} \\\end{array}[/latex]
Step 2: Convert the 16 inches to feet.
[latex]16 \text{ inches} = 1 \text{ foot}, 4 \text{ inches}[/latex]
Step 3: Add the feet.
[latex]24 \text{ feet} + 1\text{ foot } 4\text{ inches}[/latex]
[latex]25 \text{ feet}, 4 \text{ inches}[/latex]
Anthony bought 25 feet and 4 inches of wood.
Exercise 11
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?
Exercise 11 Answer
4 lbs. 8 oz.
Exercise 12
Joellen wants to double a solution of 5 gallons 3 quarts. How many gallons of solution will she have in all?
Exercise 12 Answer
11 gallons 2 qt.
Make Unit Conversions in the Metric System
In the metric system, units are related by powers of 10. The root words of their names reflect this relation. For example, the basic unit for measuring length is a metre. One kilometre is 1000 metres; the prefix kilo means thousand. One centimetre is [latex]\tfrac{1}{100}[/latex] of a metre, just like one cent is [latex]\tfrac{1}{100}[/latex] of one dollar.
The equivalencies of measurements in the metric system are shown in the lists below. The common abbreviations for each measurement are given in parentheses.
- 1 kilometre (km) = 1000 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 metre = 100 centimetres
- 1 metre = 1000 millimetres
Mass:
- 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 gram = 100 centigrams
- 1 gram = 1000 milligrams
Capacity:
- 1 kilolitre (kL) = 1000 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 litre = 100 centilitres
- 1 litre = 1000 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 G
Nick ran a 10K race. How many metres did he run?
We will convert kilometres to metres using the identity property of multiplication.
Step 1: Multiply the measurement to be converted by 1.
[latex]10\text{ kilometres} \times 1[/latex]
Step 2: Write 1 as a fraction relating kilometres and metres.
[latex]10\text{ kilometres}\times\dfrac{1,000\text{ metres}}{1\text{ kilometres}}[/latex]
Step 3: Simplify.
[latex]10\cancel{\text{ kilometres}}\times\dfrac{1,000\text{ metres}}{1\cancel{\text{ kilometres}}}[/latex]
Step 4: Multiply.
[latex]10\,000 \text{ metres}[/latex]
Nick ran 10,000 metres.
Exercise 13
Sandy completed her first 5K race! How many metres did she run?
Exercise 13 Answer
5000 metres
Exercise 14
Herman bought a rug 2.5 metres in length. How many centimetres is the length?
Exercise 14 Answer
250 centimetres
Example H
Eleanor’s newborn baby weighed 3200 grams. How many kilograms did the baby weigh?
We will convert grams into kilograms.
Step 1: Multiply the measurement to be converted by 1.
[latex]3\,200\text{ grams}\cdot1[/latex]
Step 2: Write 1 as a function relating kilograms and grams.
[latex]3\,200\text{ grams}\cdot\dfrac{1\text{ kg}}{1\,000\text{ grams}}[/latex]
Step 3: Simplify.
[latex]3\,200\cancel{\text{ grams}}\cdot\dfrac{1\text{ kg}}{1\,000\cancel{\text{ grams}}}[/latex]
Step 4: Multiply.
[latex]\dfrac{3\,200\text{ kilograms}}{1\,000}[/latex]
Step 5: Divide.
[latex]3.2 \text{ kilograms}[/latex]
The baby weighed 3.2 kilograms.
Exercise 15
Kari’s newborn baby weighed 2800 grams. How many kilograms did the baby weigh?
Exercise 15 Answer
2.8 kilograms
Exercise 16
Anderson received a package that was marked 4500 grams. How many kilograms did this package weigh?
Exercise 16 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 1000, 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 3200 grams to kilograms by multiplying by [latex]\dfrac{1}{1\,000}[/latex] (or 0.001). This is the same as moving the decimal three places to the left (see Figure 6.1.1).
Example I
Convert:
- 350 L to kilolitres
- 4.1 L to millilitres
a. 350 L to kilolitres
We will convert litres to kilolitres. In the Metric System of Measurement listed above, we see that 1 kL = 1000 litres.
Step 1: Multiply by 1, writing 1 as a fraction relating litres to kilolitres.
[latex]350\text{ L}\cdot\dfrac{1\text{ kL}}{1\,000\text{ L}}[/latex]
Step 2: Simplify.
[latex]\require{cancel}350\cancel{\text{L}}\cdot \dfrac{\text{1 kL}}{1,000\cancel{\text{L}}}[/latex]
Step 3: Move the decimal 3 units to the left.
[latex]0.35 \text{ kL}[/latex]
b. 4.1 L to millilitres
We will convert litres to millilitres. In the Metric System of Measurement listed above, we see that 1 litre = 1000 millilitres.
Step 1: Multiply by 1, writing 1 as a fraction relating litres to millilitres.
[latex]4.1\text{ L}\cdot \dfrac{1\,000\text{ mL}}{1\text{ L}}[/latex]
Step 2: Simplify.
[latex]\require{cancel}4.1\cancel{\text{ L}}\cdot \dfrac{1\,000\text{ mL}}{1\cancel{\text{ L}}}[/latex]
Step 3: Move the decimal 3 units to the right.
[latex]4\,100 \text{ mL}[/latex]
Exercise 17
Convert:
- 725 L to kilolitres
- 6.3 L to millilitres
Exercise 17 Answers
- 7250 kilolitres
- 6300 millilitres
Exercise 18
Convert:
- 350 hL to litres
- 4.1 L to centilitres
Exercise 18 Answers
- 35000 litres
- 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 J
Ryland is 1.6 metres tall. His younger brother is 85 centimetres tall. How much taller is Ryland than his younger brother?
Step 1: Convert both measurements to either centimetres or metres. Since metres is the larger unit, we will subtract the lengths in metres.
Step 2: Convert 85 centimetres to metres by moving the decimal 2 places to the left to get 0.85 m.
[latex]\begin{array}[t]{rr}&1.60\text{ m}\\ -&0.85\text{ m} \\ \hline &0.75\text{ m} \\\end{array}[/latex]
Ryland is 0.75 m taller than his brother.
Exercise 19
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.
Exercise 19 Answer
83 centimetres
Exercise 20
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.
Exercise 20 Answer
1.04 metres
Example K
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?
We will find the amount of olive oil in millilitres and then convert it to litres.
Step 1: Translate to algebra.
[latex]3 \cdot 150 \text{ mL}[/latex]
Step 2: Multiply.
[latex]450 \text{ mL}[/latex]
Step 3: Convert to litres.
[latex]450\text{ mL}\cdot \dfrac{0.001\text{ L}}{1\text{ mL}}[/latex]
Step 4: Simplify.
[latex]450\cancel{\text{ mL}}\cdot \dfrac{0.001\text{ L}}{1\cancel{\text{ mL}}}[/latex]
[latex]0.45 \text{ L}[/latex]
Dena needs 0.45 litres of olive oil.
Exercise 21
A recipe for Alfredo sauce calls for 250 millilitres 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?
Exercise 21 Answer
2 litres
Exercise 22
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?
Exercise 22 Answer
2.4 kilograms
Convert Between the Imperial & 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 lists below shows some of the most common conversions.
Length:
- 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.
Mass:
- 1 lb. = 0.45 kg
- 1 oz. = 28 g
- 1 kg = 2.2 lb.
Capacity:
- 1 qt. = 0.95 L
- 1 fl. oz. = 30 mL
- 1 L = 1.06 qt.
Figure 6.1.2 shows how inches and centimetres are related on a ruler.
Figure 6.1.3 shows the ounce and millilitre markings on a measuring cup.
Figure 6.1.4 shows how pounds and kilograms marked on a bathroom scale.
We make conversions between the systems just as we do within the systems — by multiplying by unit conversion factors.
Example L
Lee’s water bottle holds 500 mL of water. How many ounces are in the bottle? Round to the nearest tenth of an ounce.
Step 1: Multiply by a unit conversion factor relating to mL and ounces.
[latex]500\text{ millilitres}\cdot \dfrac{1\text{ ounce}}{30\text{ millilitres}}[/latex]
Step 2: Simplify.
[latex]\dfrac{50\text{ ounce}}{30}[/latex]
Step 3: Divide.
[latex]16.7 \text{ ounces}[/latex]
The water bottle has 16.7 ounces.
Exercise 23
How many quarts of soda are in a 2-L bottle?
Exercise 23 Answer
2.12 quarts
Exercise 24
How many litres are in 4 quarts of milk?
Exercise 24 Answer
3.8 litres
Example M
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?
Step 1: Multiply by a unit conversion factor relating km and mi.
[latex]100\text{ kilometres}\cdot \dfrac{1\text{ mile}}{1.61\text{ kilometre}}[/latex]
Step 2: Simplify.
[latex]\dfrac{\text{100 miles}}{1.61}[/latex]
Step 3: Divide.
[latex]62 \text{ ounces}[/latex]
Soleil will travel 62 miles.
Exercise 25
The height of Mount Kilimanjaro is 5895 metres. Convert the height to feet.
Exercise 25 Answer
19335.6 feet
Exercise 26
The flight distance from Toronto to Vancouver is 3364 kilometres. Convert the distance to miles.
Exercise 26 Answer
2090 miles
Convert Between Fahrenheit & Celsius Temperatures
Have you ever been in a foreign country and heard the weather forecast? If the forecast is for [latex]71°F[/latex], what does that mean?
The Canadian and imperial systems use different scales to measure temperature. The Canadian system uses degrees Celsius, written as °C. The imperial system uses degrees Fahrenheit, written as °F. Figure 6.1.5 shows the relationship between the two systems.
Figure 6.1.5 shows normal body temperature, along with the freezing and boiling temperatures of water in degrees Fahrenheit and degrees Celsius.
Temperature conversions:
To convert from Fahrenheit temperature, F, to Celsius temperature, C, use the formula:
[latex]C=\dfrac{5}{9}\left(F-32\right)[/latex].
To convert from Celsius temperature, C, to Fahrenheit temperature, F, use the formula:
[latex]F=\dfrac{9}{5}C+32[/latex].
Example N
Convert 50° Fahrenheit into degrees Celsius.
We will substitute 50°F into the formula to find C.
Step 1: Substitute 50 for F.
[latex]C = \dfrac{5}{9}(50-32)[/latex]
Step 2: Simplify in parentheses.
[latex]C = \dfrac{5}{9}(18)[/latex]
Step 3: Multiply.
[latex]C = 10[/latex]
50°F is equivalent to 10°C.
Exercise 27
Convert the Fahrenheit temperature to degrees Celsius:
59° Fahrenheit.
Exercise 27 Answer
15°C
Exercise 28
Convert the Fahrenheit temperature to degrees Celsius:
41° Fahrenheit.
Exercise 28 Answer
5°C
Example O
While visiting Paris, Woody saw the temperature was 20° Celsius. Convert the temperature to degrees Fahrenheit.
We will substitute 20°C into the formula to find F.
Step 1: Substitute 20 for C.
[latex]F = \dfrac{9}{5}(20) + 32[/latex]
Step 2: Multiply.
[latex]F = 36 + 32[/latex]
Step 3: Add.
[latex]F = 68[/latex]
20°C is equivalent to 68°F.
Exercise 29
Convert the Celsius temperature to degrees Fahrenheit:
The temperature in Helsinki, Finland, was 15° Celsius.
Exercise 29 Answer
59°F
Exercise 30
Convert the Celsius temperature to degrees Fahrenheit:
The temperature in Sydney, Australia, was 10° Celsius.
Exercise 30 Answer
50° F
Key Concepts
Metric System of Measurement
Length
- 1 kilometre (km) = 1000 m
- 1 hectometre (hm) = 100 m
- 1 dekatmetre (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 metre = 100 centimetres
- 1 metre = 1000 millimetres
Mass
- 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 gram = 100 centigrams
- 1 gram = 1000 milligrams
Capacity
- 1 kilolitre (kL) = 1000 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 litre = 100 centilitres
- 1 litre = 1000 millilitres
Temperature Conversion
- To convert from Fahrenheit temperature, F, to Celsius temperature, C, use the formula [latex]\text{C}=\dfrac{5}{9}\left(\text{F}-32\right)[/latex]
- To convert from Celsius temperature, C, to Fahrenheit temperature, F, use the formula [latex]\text{F}=\dfrac{9}{5}\text{C}+32[/latex]
6.1: Practice Questions
1. Unit Conversion in the Imperial System
In the following exercises, convert between imperial units
-
- A park bench is 6 feet long. Convert the length to inches.
- A floor tile is 2 feet wide. Convert the width to inches.
- A ribbon is 18 inches long. Convert the length to feet.
- Carson is 45 inches tall. Convert his height to feet.
- A football field is 160 feet wide. Convert the width to yards.
- On a baseball diamond, the distance from home plate to first base is 30 yards. Convert the distance to feet.
- Ulises lives 1.5 miles from school. Convert the distance to feet.
- Denver, Colorado, is 5183 feet above sea level. Convert the height to miles.
- A killer whale weighs 4.6 tons. Convert the weight to pounds.
- Blue whales can weigh as much as 150 tons. Convert the weight to pounds.
- An empty bus weighs 35000 pounds. Convert the weight to tons.
- At take-off, an airplane weighs 220000 pounds. Convert the weight to tons.
- Rocco waited [latex]1\dfrac{1}{2}[/latex] hours for his appointment. Convert the time to seconds.
- Misty’s surgery lasted [latex]2\dfrac{1}{4}[/latex] hours. Convert the time to seconds.
- How many teaspoons are in a pint?
- How many tablespoons are in a gallon?
- JJ’s cat, Posy, weighs 14 pounds. Convert her weight to ounces.
- April’s dog, Beans, weighs 8 pounds. Convert his weight to ounces.
- Crista will serve 20 cups of juice at her son’s party. Convert the volume to gallons.
- Lance needs 50 cups of water for the runners in a race. Convert the volume to gallons.
- Jon is 6 feet 4 inches tall. Convert his height to inches.
- Faye is 4 feet 10 inches tall. Convert her height to inches.
- The voyage of the Mayflower took 2 months and 5 days. Convert the time to days.
- Lynn’s cruise lasted 6 days and 18 hours. Convert the time to hours.
- Baby Preston weighed 7 pounds 3 ounces at birth. Convert his weight to ounces.
- Baby Audrey weighed 6 pounds 15 ounces at birth. Convert her weight to ounces.
2. Mixed Units of Measurement in the Imperial System
Solve the following exercises using mixed imperial units.
-
- 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?
- 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?
- 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?
- 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?
- Renee attached a 6-foot, 6-inch extension cord to her computer’s 3-foot, 8-inch power cord. What was the total length of the cords?
- 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?
- 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?
- 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?
3. Unit Conversions in the Metric System
In the following exercises, convert between metric units.
-
- Ghalib ran 5 kilometres. Convert the length to metres.
- Kitaka hiked 8 kilometres. Convert the length to metres.
- Estrella is 1.55 metres tall. Convert her height to centimetres.
- The width of the wading pool is 2.45 metres. Convert the width to centimetres.
- Mount Whitney is 3072 metres tall. Convert the height to kilometres.
- The depth of the Mariana Trench is 10911 metres. Convert the depth to kilometres.
- June’s multivitamin contains 1500 milligrams of calcium. Convert this to grams.
- A typical ruby-throated hummingbird weighs 3 grams. Convert this to milligrams.
- One stick of butter contains 91.6 grams of fat. Convert this to milligrams.
- One serving of gourmet ice cream has 25 grams of fat. Convert this to milligrams.
- The maximum mass of an airmail letter is 2 kilograms. Convert this to grams.
- Dimitri’s daughter weighed 3.8 kilograms at birth. Convert this to grams.
- A bottle of wine contained 750 millilitres. Convert this to litres.
- A bottle of medicine contained 300 millilitres. Convert this to litres.
4. Mixed Units of Measurement in the Metric System
Solve the following exercises using mixed metric units.
-
- Matthias is 1.8 metres tall. His son is 89 centimetres tall. How much taller is Matthias than his son?
- Stavros is 1.6 metres tall. His sister is 95 centimetres tall. How much taller is Stavros than his sister?
- A typical dove weighs 345 grams. A typical duck weighs 1.2 kilograms. What is the difference, in grams, between the weights of a duck and a dove?
- 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?
- Harry mailed 5 packages that weighed 420 grams each. What was the total weight of the packages in kilograms?
- 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?
- Jonas drinks 200 millilitres of water 8 times a day. How many litres of water does Jonas drink in a day?
- 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?
5. Converting Between Imperial & Metric Systems of Measurement
In the following exercises, convert between imperial and metric units. Round to the nearest tenth.
-
- Bill is 75 inches tall. Convert his height to centimetres.
- Frankie is 42 inches tall. Convert his height to centimetres.
- Marcus passed a football 24 yards. Convert the pass length to metres.
- Connie bought 9 yards of fabric to make drapes. Convert the fabric length to metres.
- According to research conducted by the CRC, Canadians regrettably produce more garbage per capita than any other country on earth, at 2172.6 pounds per person annually. Convert the waste to kilograms.
- An average Canadian will throw away 163000 pounds of trash over their lifetime. Convert this weight to kilograms.
- A 5K run is 5 kilometres long. Convert this length to miles.
- Kathryn is 1.6 metres tall. Convert her height to feet.
- Dawn’s suitcase weighed 20 kilograms. Convert the weight to pounds.
- Jackson’s backpack weighed 15 kilograms. Convert the weight to pounds.
- Ozzie put 14 gallons of gas in his truck. Convert the volume to litres.
- Bernard bought 8 gallons of paint. Convert the volume to litres.
6. Converting Between Fahrenheit & Celsius Temperatures
In the following exercises, convert from Fahrenheit to degrees Celsius. Round to the nearest tenth.
-
- 86° Fahrenheit
- 77° Fahrenheit
- 104° Fahrenheit
- 14° Fahrenheit
- 72° Fahrenheit
- 4° Fahrenheit
- 0° Fahrenheit
- 120° Fahrenheit
In the following exercises, convert from Celsius to degrees Fahrenheit. Round to the nearest tenth.
-
- 5° Celsius
- 25° Celsius
- -10° Celsius
- -15° Celsius
- 22° Celsius
- 8° Celsius
- 43° Celsius
- 16° Celsius
7. Everyday Math
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- 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?
- 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?
8. Writing Exercises
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- Some people think that 22° to 28° Celsius is the ideal temperature range.
- What is your ideal temperature range? Why do you think so?
- Did you grow up using the Canadian Metric or the Imperial system of measurement?
- Describe two examples in your life when you had to convert between the two systems of measurement.
- Some people think that 22° to 28° Celsius is the ideal temperature range.
6.1: Practice Answers
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- Unit conversions in the imperial system.
- 72 inches
- 1.5 feet
- [latex]53\dfrac{1}{3}[/latex] yards
- 7920 feet
- 9200 pounds
- [latex]17\dfrac{1}{2}[/latex] tons
- 5400 s
- 96 teaspoons
- 224 ounces
- [latex]1\dfrac{1}{4}[/latex] gallons
- 76 in.
- 65 days
- 115 ounces
- Mixed units of measurement in the imperial system.
- 8 lbs. 13 oz.
- 3.05 hours
- 10 ft. 2 in.
- 4 yards
- Unit conversions in the metric system.
- 5000 metres
- 155 centimetres
- 3.072 kilometres
- 1.5 grams
- 91600 milligrams
- 2000 grams
- 0.75 litres
- Mixed units of measurement in the metric system.
- 91 centimetres
- 855 grams
- 2.1 kilograms
- 1.6 litres
- Coverting between imperial and metric systems of measurement.
- 190.5 centimetres
- 21.9 metres
- 977.7 kilograms
- 3.1 miles
- 44 pounds
- 53.2 litres
- Converting between Fahrenheit and Celsius temperatures.
- 30°C
- 40°C
- 22.2°C
- -17.8°C
- 41°F
- 14°F
- 71.6°F
- 109.4°F
- Everyday math.
- 14.6 kilograms
- Writing exercises.
- Answers will vary.
- Unit conversions in the imperial system.
Attributions
All figures in this chapter are from 3.1 Systems of Measurement in Introductory Algebra by Izabela Mazur, via BCcampus.
This chapter has been adapted from 3.1 Systems of Measurement in Introductory Algebra (BCcampus) by Izabela Mazur (2021), which is under a CC BY 4.0 license.
The original chapter was adapted from 1.10 Systems of Measurement in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith (2020), which is under a CC BY 4.0 license. Adapted by Izabela Mazur.