3.1 Conversions Within and Between the Metric and Imperial Systems

When we measure something, we figure out the size or amount of it. When expressing the size or amount of something, we need to include a unit of measurement to indicate what is being measured. There are two widely used systems of measurement commonly used around the world: the metric system and the imperial system. Most countries use the metric system, and Canada uses the metric system as its official measurement system. With that being said, the imperial system is also often used in Canada.

The metric system uses units that are related by powers of 10. The base unit for measuring length is the metre (m), the base unit for measuring weight is the gram (g), and the base unit for measuring volume is the litre (L). A prefix added to the base unit determines its value. We will be exploring the following prefixes in this book: kilo-, hecto-, deca-, deci-, centi-, and milli-.

The imperial system of measurement uses inches, feet, yards, and miles to measure length; ounces, pounds, and tons to measure weight; and fluid ounces, cups, pints, quarts, and gallons to measure volume.

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

We can convert units within the imperial system, within the metric system, and between the imperial and metric systems. The equivalencies of measurements are shown in the tables below. The common abbreviations for each measurement are shown in parentheses.

Imperial System

Table 3.1.1
Length Weight Volume
1 foot (ft) = 12 inches (in)

1 yard (yd) = 3 feet (ft)

1 mile (mi) = 5 280 feet (ft)

 1 pound (lb) = 16 ounces (oz)

1 ton (t) = 2 000 pounds (lb)

 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)

Metric System

Table 3.1.2
Length Weight Volume
1 kilometre (km) = 1 000 m

1 hectometre (hm) = 100 m

1 decametre (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 decagram (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 decalitre (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 centimetre

1 metre = 1 000 millimetres

 1 gram = 100 centigrams 

1 gram = 1 000 milligrams

 1 litre = 100 centilitres 

1 litre = 1 000 millilitres

Imperial System and Metric System

Table 3.1.3
Metric to Imperial Imperial to Metric
Length

1 kilometre ≈ 0.6214 miles

1 metre ≈ 1.0936 yards

1 metre ≈ 3.2808 feet

1 centimetre ≈ 0.3937 inches

 1 mile ≈ 1.6093 kilometres

1 yard ≈ 0.9144 metres

1 foot ≈  0.3048 metres

1 inch ≈ 2.54 centimetres
Weight 1 kilogram ≈  2.2046 pounds

1 gram ≈  0.0353 ounces

 1 pound ≈ 0.4536 kilograms

1 ounce ≈  28.3495 grams
Volume 1 litre ≈  0.2200 gallons

1 litre ≈  0.8799 quarts

 1 gallon ≈  4.5461 litres

1 quart ≈  1.1365 litres

To convert between units, we can use proportional reasoning or unit analysis. We will go over both of these processes.

Proportional Reasoning 

Table 3.1.4
Steps for Converting Units Using Proportional Reasoning
Step 1: Identify the relation between the units given and the units needed.

Step 2: Write a proportion.

Step 3: Cross-multiply.
Example 1

Convert 2.3 metres into centimetres using proportional reasoning.

Table 3.1.5
Steps Solution
Identify the relationship between metres and centimetres. 1 m = 100 cm
Write a proportion. \frac{1 m}{2.3 m}=\frac{100 cm}{x cm}
Cross-multiply. 1 \bullet \text{x}=2.3 \bullet 100
x = 230
2.3 metres = 230 centimetres
Practice 1

Convert the following amounts to the indicated measurements using proportional reasoning. Round to the nearest thousandth where necessary.

a) 18 yd =                    ft

 

 

 

 b) 528 cm =                    m

 

 

 
c) 12 oz =                    g

 

 

 

 

 

 d) 26 cm =                    ft

 

 

 

 

 

Unit Analysis 

Table 3.1.6
Steps for Converting Units Using Unit Analysis
Step 1: Multiply the measurement to be converted by 1.

Step 2: Write 1 as a fraction relating the units given and the units needed.

Step 3: Multiply.

Step 4: Simplify the fraction.

Step 5: Simplify.
Example 2

Convert 3.2 tons into pounds using unit analysis.

Table 3.1.7
Steps Solution
Multiply the measurement to be converted by 1. \text{3.2 tons} \bullet 1
Write 1 as a fraction relating tons and pounds. \text{3.2 tons} \bullet \dfrac{\text{2 000 pounds}}{\text{1 ton}}
Simplify. \frac{{3.2 tons} \bullet \text{2 000 pounds}}{\text{1 ton}}
Multiply. 3.2 tons = 6 400 pounds
Practice 2

Convert the following amounts to the indicated measurements using unit analysis. Round to the nearest thousandth where necessary.

a) 15 lb =                    kg

 

 

 

 b) 30 in =                    ft

 

 

 
c) 18 km =                    ft

 

 

 

 

 d) 18 kL =                    mL

 

 

 

 

Metric Unit Chart

As you become familiar with the metric system, you may see a pattern that can make converting within the metric system quicker. Because the system is based on multiples of 10, the calculations involve multiplying by multiples of 10. We can 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. The chart below can be used to figure out how many places the decimal should be moved and in which direction. The unit will change depending on what type of measure you are converting. For example, if you are converting a length, you would be working with metres as the unit.

Table 3.1.8
K H D U D C M
kilo-

 

 hecto-

 

 deca-

 

 unit
– metre
– gram
– litre		 deci-

 

 centi-

 

 milli-

 
Example 3

Convert 350 L into kilolitres using the chart provided. 

Table 3.1.9
Steps Solution
Count the number of spaces between litres and kilolitres on the chart. Notice which direction you are moving when you count from litres to kilolitres. To move from U (litres) to K (kilolitres), we move three spaces to the left. This means we need to move the decimal in 350 L three spaces to the left.
Move the decimal three units to the left. 350 L = 0.35 kL
Practice 3

Convert the following amounts to the indicated measurements using the chart provided.

a) 1.58 m =                    cm

 

 b) 16 hg =                    cg

 
c) 16 mm =                    dam

 

 d) 6.3 L =                    mL

 

Conversion Word Problems

Practice 4

Convert the amounts to the indicated measurement in each of the following word problems.

a) The depth of the Mariana Trench is 10 911 metres. Convert the depth to kilometres.

 

 

 

 

 

 b) On average, a Canadian will throw away 163 000 pounds of trash over their lifetime. If a garbage truck can hold 21 000 kg of garbage, how many trucks would be needed to carry all of the garbage over your lifetime?

 

 

 

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

 

 

 

 

 d) Kaya is 4 feet 10 inches tall. Convert her height to inches.

 

 

 

 

 

Homework

  1. Convert the following amounts to the indicated measurements. Round to the nearest hundredth where necessary.

    a) 150 cm =                      m

     

     

     

     

    		 b) 0.8 hg =                      kg

     

     

     

     

    		 c) 3.5 L =                      mL

     

     

     

     
    d) 10 in =                      cm

     

     

     

     

    		 e) 4 lb =                      kg

     

     

     

     

    		 f) 5 km =                      mi

     

     

     

     
    g) 5 L =                      qt

     

     

     

     

    		 h) 10 lb =                      g

     

     

     

     

    		 i) 3 ft =                      cm

     

     

     

     
    j) 12 ft =                      in

     

     

     

     

    		 k) 200 m =                      yd

     

     

     

     

    		 l) 6 km =                      mi

     

     

     

     
    m) 50 mi =                      km

     

     

     

     

    		 n) 6 daL =                      L

     

     

     

     

    		 o) 150 g =                      oz

     

     

     

     
    p) 3 oz =                      g

     

     

     

     

    		 q) 3 pt =                      L

     

     

     

     

    		 r) 500 cm =                      m

     

     

     

     
    s) 2.5 kg =                       lb

     

     

     

     

    		 t) 1 000 mm =                  m

     

     

     

     

    		 u) 500 g =                      oz

     

     

     

     
    v) 0.78 qt =                       pt

     

     

     

     

    		 w) 0.5 kL =                       L

     

     

     

     

    		 x) 72 cup =                       fl oz

     

     

     

     
    y) 2 lb =                       kg

     

     

     

     

    		 z) 0.125 gal =                       cup

     

     

     

     

    		 aa) 0.381 m =                       cm

     

     

     

     
    bb) 500 L =                      hL

     

     

     

     

     

    		 cc) 300 dm =                  m

     

     

     

     

     

    		 dd) 500 m =                      yd

     

     

     

     

     

  2. Convert to the indicated measurement in the each of the following word problems. Round to the nearest tenth where necessary.

    a) A park bench is 6 feet long. Convert the length to inches.

     

     

     

    		 b) Ghalib ran 5 kilometres. Convert the length to metres.

     

     

     
    c) Jonas drinks 200 millilitres of water 8 times a day. How many litres of water does Jonas drink in a day?

     

     

     

    		 d) Ulises lives 1.5 miles from school. Convert the distance to feet.

     

     

     
    e) Mount Whitney is 3 072 metres tall. Convert the height to kilometres.

     

     

     

     

    		 f) An empty bus weighs 35 000 pounds. Convert the weight to tons.

     

     

     

     
    g) JJ’s cat, Posy, weighs 14 pounds. Convert her weight to ounces.

     

     

     

    		 h) Ali will serve 20 cups of juice at his son’s party. Convert the volume to gallons.

     

     

     
    i) Paul is 6 feet 4 inches tall. Convert his height to inches.

     

     

     

    		 j) Baby Isla weighed 7 pounds 3 ounces at birth. Convert her weight to ounces.

     

     

     
    k) Vera put 14 gallons of gas in his truck. Convert the volume to litres.

     

     

    		 l) June’s multivitamin contains 1 500 milligrams of calcium. Convert this to grams.

     

     

     
    m) Dawn’s suitcase weighed 20 kilograms. Convert the weight to pounds.

     

     

     

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

     

     

     
    o) Luna is 4 feet 10 inches tall. Convert her height to inches.

     

     

     

     

    		 p) According to research, Canadians regrettably produce more garbage per capita than any other country on Earth, at 2 172.6 pounds per person annually. Convert the amount of waste to kilograms.

     

     

     

  3. Answer the following questions.

    a) 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?

     

     

     

     

    		 b) Giulia attached an extension cord measuring 6 feet 6 inches to her computer’s power cord, which measures 3 feet 8 inches. What was the total length of the cords?

     

     

     

     
    c) 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?

     

     

     

     

    		 d) Matthias is 1.8 metres tall. His son is 89 centimetres tall. How much taller is Matthias than his son?

     

     

     

     
    e) 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?

     

     

     

     

    		 f) Alejandro mailed five packages that weighed 420 grams each. What was the total weight of the packages in kilograms?

     

     

     

     

Answers

1.

a) 1.5 m b) 0.08 kg c) 3 500 mL d) 25.4 cm e) 1.81 kg
f) 3.11 mi g) 4.40 qt h) 4 536 g i) 91.44 cm j) 144 in
k) 218.72 yd l) 3.73 mi m) 80.47 km n) 60 L o) 5.30 oz
p) 85.05 g q) 1.70 L r) 5 m s) 5.51 lb t) 1 m
u) 17.65 oz v) 1.56 pt w) 500 L x) 576 fl oz y) 0.91 kg
z) 2 cups aa) 38.1 cm bb) 5 hL cc) 30 m dd) 546.80 yd

2.

a) 72 in b) 5 000 m c) 1.6 L d) 7 920 ft
e) 3.1 km f) 17.5 t g) 224 oz h) 1.3 gal
i) 76 in j) 115 oz k) 63.6 L l) 1.5 g
m) 44.09 lb n) 1 mi o) 58 in p) 985.49 kg

3.

a) 8 lb 13 oz b) 10 ft 2 in c) 4 yd
d) 91 cm e) 855 g f) 2.1 kg

Attribution

Unless otherwise indicated, material on this page has been adapted from the following resources:

Flinn, C., & Overgaard, M. (2021). Math for trades: Volume 2. BCcampus. https://collection.bccampus.ca/textbooks/math-for-trades-volume-2-bccampus-238/, licensed under CC BY 4.0

Mazur, I. (2021). Introductory algebra. BCcampus. https://collection.bccampus.ca/textbooks/intermediate-algebra-bccampus-412/, licensed under CC BY 4.0

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