# 5.6: Unit Conversion- Metric System

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- David Arnold
- Retired Professor (Mathematics) at College of the Redwoods

The metric system of units is the standard system of units preferred by scientists. It is based on the base ten number system and its decimal format is more friendly to users of this system. There is a common set of prefixes adopted by the metric system to indicate a power of ten to apply to the base unit.

Metric System Preferences

This is a list of standard prefixes for the metric system and their meanings.

\[ \begin{array}{c c} \text{deka = 10} & \text{deci = 1/10} \\ \text{hecto = 100} & \text{centi = 1/100} \\ \text{kilo = 1000} & \text{milli = 1/1000} \end{array}\nonumber \]

Thus, for example, a decameter is 10 meters, a hectoliter is 100 liters, and a kilogram is 1000 grams. Similarly, a decimeter is 1/10 of a meter, a centiliter is 1/100 of a liter, and a milligram is 1/1000 of a gram.

## Units of Length

The standard measure of length in the metric system is the meter.

Historically, the meter was defined by the French Academy of Sciences as the length between two marks on a platinum-iridium bar, which was designed to represent 110,000,000 of the distance from the equator to the north pole through Paris. In 1983, it was redefined by the International Bureau of Weights and Measures (BIPM) as the distance travelled by light in free space in 1299,792,458 of a second. (Wikipedia)

We can apply the standard prefixes to get the following result.

Metric Units of Length

These units of length are used in the metric system.

Unit Length |
Unit Abbreviation |
---|---|

1 kilometer = 1000 meters | km |

1 hectometer = 100 meters | hm |

1 dekameter = 10 meters | dam |

1 meter | m |

1 decimeter = \(\frac{1}{10}\) meter | dm |

1 centimeter = \(\frac{1}{100}\) meter | cm |

1 millimeter = \(\frac{1}{1000}\) meter | mm |

We can use these facts to build conversion factors as we did in Section 6.3. For example, because

1 km = 1000 m,

we can divide both sides by 1000 m to produce the conversion factor

\[1 = \frac{1 \text{ km}}{1000 \text{ m}}.\nonumber \]

This conversion factor can help change meters into kilometers.

Before using this conversion factor in an example, we repeat here the rules for multiplying and dividing by powers of ten. We will be making heavy use of these rules in this section.

Multiplying and Dividing by Powers of Ten

- Multiplying a decimal number by 10
^{n}will move the decimal point*n*places to the right. For example, 3.2567 · 10^{2}= 3.2567 · 100 = 325.67. - Dividing a decimal number by 10
^{n}will move the decimal point*n*places to the left. For example, 3.2567/10^{2}= 3.2567/100 = 0.032567.

And now the example.

Example 1

Change 2,326 meters to kilometers.

**Solution**

Multiply by the conversion factor 1 km/1000m.

\[ \begin{aligned} 2326 \text{ m} = 2326 \text{ m} \cdot \frac{1 \text{ km}}{1000 \text{ m}} ~ & \textcolor{red}{ \text{ Apply conversion factor.}} \\ = 2326 \cancel{ \text{ m}} \cdot \frac{1 \text{ km}}{1000 \cancel{ \text{ m}}} ~ & \textcolor{red}{ \text{ Cancel common units.}} \\ = \frac{2326 \cdot 1}{1000} \text{ km} ~ & \textcolor{red}{ \text{ Multiply fractions.}} \\ = 2.326 \text{ km } & \textcolor{red}{ \text{ Simplify.}} \end{aligned}\nonumber \]

In the last step, note that dividing by 1000 moves the decimal point three places to the left. Thus, 2326 meters is equal to 2.326 kilometers.

**Alternate Solution**

A second solution depends upon the fact that multiplying or dividing by a power of ten will move the decimal point right or left a number of places equal to the number of zeros present in the multiplier or divisor. Thus, as we saw above, dividing by 1000 moved the decimal point 3 places to the left. Suppose that we arrange the metric units of length in order, from largest to smallest, as shown below.

Note that we must move 3 places left to move from the meters (m) abbreviation to the kilometers (km) abbreviation. In like manner, if we write 2,326 meters as 2,326.0meters, then we can convert to kilometers by moving the decimal 3 places to the left.

Exercise

Change 1,156 meters to kilometers

**Answer**-
1.156 kilometers

Example 2

Change 537 centimeters to meters.

**Solution**

We know that

\[1 \text{ cm } = \frac{1}{100} \text{ m,}\nonumber \]

or multiplying both sides of this result by 100,

\[100 \text{ cm } = 1 \text{ m.}\nonumber \]

Dividing both sides of this last result by 100 cm, we obtain the conversion factor 1 m/100 cm.

\[\begin{aligned} 537 \text{ cm } = 537 \text{ cm } \cdot \frac{1 \text{ m}}{100 \text{ cm}} ~ & \textcolor{red}{ \text{ Apply conversion factor.}} \\ = 537 ~ \cancel{ \text{cm}} \cdot \frac{1 \text{ m}}{100 ~ \cancel{ \text{ cm}}} ~ & \textcolor{red}{ \text{ Cancel common units.}} \\ = \frac{537 \cdot 1}{100} \text{ m} ~ & \textcolor{red}{ \text{ Multiply fractions.}} \\ = 5.37 \text{ m} \end{aligned}\nonumber \]

In the last step, note that dividing by 100 moves the decimal point two places to the left. Alternately, we can set up our ordered list of units.

Note that we must move 2 places left to move from the centimeters (cm) abbreviation to the meters (m) abbreviation. In like manner, if we write 537 centimeters as 537.0 centimeters, then we can convert to meters by moving the decimal 2 places to the left.

Sometimes more than one conversion factor is needed.

Exercise

Change 276 centimeters to meters.

**Answer**-
2.76 meters

Example 3

Change 10.2 dekameters to centimeters.

**Solution**

We have two facts:

- 1 dam=10 m, which yields the conversion factor 10 m/1 dam.
- 1 cm=(1/100) m or 100 cm=1 m, which yields the conversion factor 100 cm/1 m.

\[ \begin{aligned} 10.2 \text{ dam } = 10.2 \text{ dam } \cdot \frac{10 \text{ m}}{1 \text{ dam}} \cdot \frac{100 \text{ cm}}{1 \text{ m}} ~ & \textcolor{red}{ \text{ Apply conversion factors.}} \\ = 10.2 ~ \cancel{\text{dam}} \cdot \frac{10 ~ \cancel{\text{m}}{1 ~ \cancel{\text{dam}} \cdot \frac{100 \text{ cm}}{1 ~ \cancel{\text{m}} ~ & \textcolor{red}{ \text{ Cancel common units.}} \\ = \frac{10.2 \cdot 10 \cdot 100}{1 \cdot 1} \text{ cm } & \textcolor{red}{ \text{ Multiply fractions.}} \\ = 10, 200 \text{ cm} \end{aligned}\nonumber \]

In the last step, note that multiplying by 10, then by 100, moves the decimal point three places to the right.

Alternately, we can set up our ordered list of units.

Note that we must move 3 places right to move from the dekameters (dam) abbreviation to the centimeters (cm) abbreviation. In like manner, we can convert 10.2 dekameters to centimeters by moving the decimal 3 places to the right.

Exercise

Change 13.5 dekameters to centimeters

**Answer**-
13,500 centimeters

## Units of Mass

The fundamental unit of mass in the metric system is called a gram. Originally, it was defined to be equal to one cubic centimeter of water measured at the temperature of melting ice. Now it is simply defined as 1/1000 of a kilogram, which is defined by a physical prototype preserved by the International Bureau of Weights and Measures (Wikipedia). The mass of an object is not the same as an object’s weight, but rather a resistance to motion when an external force is applied.

The same metric system prefixes apply.

Metric Units of Mass

These units of mass are used in the metric system.

Units of Mass |
Unit Abbreviation |

1 kilogram = 1000 grams | kg |

1 hectogram = 100 grams | hg |

1 dekagram = 10 grams | dag |

1 gram | g |

1 decigram = \(\frac{1}{10}\) gram | dg |

1 centigram = \(\frac{1}{100}\) gram | cg |

1 milligram = \(\frac{1}{1000}\) gram | mg |

Example 4

Convert 0.025 dekagrams to milligrams.

**Solution**

We’ll use two conversion factors:

- 1 dag=10 g, which yields the conversion factor 10 g/1 dag.
- 1 mg=(1/1000) g, which yields the conversion factor 1000 mg/1 g.

\[ \begin{aligned} 0.025 \text{ dag } = 0.025 \text{ dag } \cdot \frac{10 \text{ g}}{1 \text{ dag}} \cdot \frac{1000 \text{ mg}}{1 \text{ g}} ~ & \textcolor{red}{ \text{ Apply conversion factors.}} \\ = 0.025 ~ \cancel{\text{dag}} \cdot \frac{10 ~ \cancel{\text{g}}{1 ~ \cancel{\text{dag}} \cdot \frac{1000 \text{ mg}}{1 ~ \cancel{\text{g}} ~ & \textcolor{red}{ \text{ Cancel common units.}} \\ = \frac{0.025 \cdot 10 \cdot 1000}{1 \cdot 1} \text{ mg } & \textcolor{red}{ \text{ Multiply fractions.}} \\ = 250 \text{ mg} \end{aligned}\nonumber \]

Alternately, we can set up our ordered list of units.

Note that we must move 4 places right to move from the dekagrams (dag) abbreviation to the milligrams (mg) abbreviation. In like manner, we can convert 0.025 dekagrams to milligrams by moving the decimal 4 places to the right.

Exercise

Convert 0.05 dekagrams to milligrams.

**Answer**-
500 milligrams

## Units of Volume

The fundamental unit of volume in the metric system is called a litre. Originally, one litre was defined as the volume of one kilogram of water measured at 4^{◦} C at 760 millimeters of mercury (Wikipedia). Currently, 1 litre is defined as 1 cubic decimeter (imagine a cube with each edge 1/10 of a meter).

The same metric system prefixes apply.

Metric Units of Volume

These units of volume are used in the metric system.

Unit of Volume |
Unit Abbreviation |

1 kilolitre = 1000 litres | kL |

1 hectolitre = 100 litres | hL |

1 dekalitre = 10 litres | daL |

1 litre | L |

1 decilitre = \(\frac{1}{10}\) litre | dL |

1 centilitre = \(\frac{1}{100}\) litre | cL |

1 millilitre = \(\frac{1}{1000}\) litre | mL |

Example 5

Convert 11,725 millilitres to dekalitres.

**Solution**

We’ll use two conversion factors:

- 1 daL=10 L, which yields the conversion factor 1 daL/10 L.
- 1 mL=(1/1000) L, which yields the conversion factor 1 L/1000 mL.

\[ \begin{aligned} 11, 725 \text{ mL } = 11, 725 \text{ mL } \cdot \frac{1 \text{ L}}{1000 \text{ mL}} \cdot \frac{1 \text{ daL}}{10 \text{ L}} ~ & \textcolor{red}{ \text{ Apply conversion factors.}} \\ = 11, 725 ~ \cancel{\text{mL}} \cdot \frac{1 ~ \cancel{\text{L}}}{1000 ~ \cancel{\text{mL}}} \cdot \frac{1 \text{ daL}}{10 ~ \cancel{\text{L}}} ~ & \textcolor{red}{ \text{ Cancel common units.}} \\ = \frac{11, 725 \cdot 1 \cdot 1}{1000 \cdot 10} \text{ daL } & \textcolor{red}{ \text{ Multiply fractions.}} \\ = 1.1725 \text{ daL} \end{aligned}\nonumber \]

Alternately, we can set up our ordered list of units.

Note that we must move 4 places left to move from the millitres (mL) abbreviation to the dekalitres (daL) abbreviation. In like manner, we can convert 11,725 millilitres to dekalitres by moving the decimal 4 places to the left.

Exercise

Convert 5,763 millilitres to dekalitres.

**Answer**-
0.5763 dekalitres

## Exercises

1. What is the meaning of the metric system prefix centi?

2. What is the meaning of the metric system prefix deka?

3. What is the meaning of the metric system prefix hecto?

4. What is the meaning of the metric system prefix kilo?

5. What is the meaning of the metric system prefix deci?

6. What is the meaning of the metric system prefix milli?

7. What is the meaning of the metric system abbreviation mg?

8. What is the meaning of the metric system abbreviation g?

9. What is the meaning of the metric system abbreviation m?

10. What is the meaning of the metric system abbreviation km?

11. What is the meaning of the metric system abbreviation kL?

12. What is the meaning of the metric system abbreviation daL?

13. What is the meaning of the metric system abbreviation hm?

14. What is the meaning of the metric system abbreviation dm?

15. What is the meaning of the metric system abbreviation dam?

16. What is the meaning of the metric system abbreviation cm?

17. What is the meaning of the metric system abbreviation dL?

18. What is the meaning of the metric system abbreviation L?

19. What is the meaning of the metric system abbreviation hg?

20. What is the meaning of the metric system abbreviation kg?

21. What is the meaning of the metric system abbreviation dg?

22. What is the meaning of the metric system abbreviation dag?

23. What is the meaning of the metric system abbreviation hL?

24. What is the meaning of the metric system abbreviation cL?

25. Change 5,490 millimeters to meters.

26. Change 8,528 millimeters to meters.

27. Change 64 meters to millimeters.

28. Change 65 meters to millimeters.

29. Change 4,571 millimeters to meters.

30. Change 8,209 millimeters to meters.

31. Change 15 meters to centimeters.

32. Change 12 meters to centimeters.

33. Change 569 centimeters to meters.

34. Change 380 centimeters to meters.

35. Change 79 meters to centimeters.

36. Change 60 meters to centimeters.

37. Change 7.6 kilometers to meters.

38. Change 4.9 kilometers to meters.

39. Change 861 centimeters to meters.

40. Change 427 centimeters to meters.

41. Change 4,826 meters to kilometers.

42. Change 1,929 meters to kilometers.

43. Change 4,724 meters to kilometers.

44. Change 1,629 meters to kilometers.

45. Change 6.5 kilometers to meters.

46. Change 7.9 kilometers to meters.

47. Change 17 meters to millimeters.

48. Change 53 meters to millimeters.

49. Change 512 milligrams to centigrams.

50. Change 516 milligrams to centigrams.

51. Change 541 milligrams to centigrams.

52. Change 223 milligrams to centigrams.

53. Change 70 grams to centigrams.

54. Change 76 grams to centigrams.

55. Change 53 centigrams to milligrams.

56. Change 30 centigrams to milligrams.

57. Change 83 kilograms to grams.

58. Change 70 kilograms to grams.

59. Change 8,196 grams to kilograms.

60. Change 6,693 grams to kilograms.

61. Change 564 centigrams to grams.

62. Change 884 centigrams to grams.

63. Change 38 grams to centigrams.

64. Change 88 grams to centigrams.

65. Change 77 centigrams to milligrams.

66. Change 61 centigrams to milligrams.

67. Change 5,337 grams to kilograms.

68. Change 4,002 grams to kilograms.

69. Change 15 kilograms to grams.

70. Change 45 kilograms to grams.

71. Change 833 centigrams to grams.

72. Change 247 centigrams to grams.

73. Change 619,560 centilitres to kilolitres.

74. Change 678,962 centilitres to kilolitres.

75. Change 15.2 litres to millilitres.

76. Change 9.7 litres to millilitres.

77. Change 10,850 centilitres to litres.

78. Change 15,198 centilitres to litres.

79. Change 10.7 litres to millilitres.

80. Change 17.3 litres to millilitres.

81. Change 15,665 millilitres to litres.

82. Change 12,157 millilitres to litres.

83. Change 6.3 kilolitres to centilitres.

84. Change 8.3 kilolitres to centilitres.

85. Change 4.5 kilolitres to centilitres.

86. Change 6.2 kilolitres to centilitres.

87. Change 10.6 litres to centilitres.

88. Change 16.6 litres to centilitres.

89. Change 14,383 centilitres to litres.

90. Change 11,557 centilitres to litres.

91. Change 9.9 litres to centilitres.

92. Change 19.5 litres to centilitres.

93. Change 407,331 centilitres to kilolitres.

94. Change 827,348 centilitres to kilolitres.

95. Change 14,968 millilitres to litres.

96. Change 18,439 millilitres to litres.

## Answers

1. 1/100

3. 100

5. 1/10

7. milligram

9. meter

11. kilolitre or kiloliter

13. hectometer

15. dekameter

17. decilitre or deciliter

19. hectogram

21. decigram

23. hectolitre or hectoliter

25. 5.49 meters

27. 64,000 millimeters

29. 4.571 meters

31. 1,500 centimeters

33. 5.69 meters

35. 7,900 centimeters

37. 7,600 meters

39. 8.61 meters

41. 4.826 kilometers

43. 4.724 kilometers

45. 6,500 meters

47. 17,000 millimeters

49. 51.2 centigrams

51. 54.1 centigrams

53. 7,000 centigrams

55. 530 milligrams

57. 83,000 grams

59. 8.196 kilograms

61. 5.64 grams

63. 3,800 centigrams

65. 770 milligrams

67. 5.337 kilograms

69. 15,000 grams

71. 8.33 grams

73. 6.1956 kilolitres

75. 15,200 millilitres

77. 108.5 litres

79. 10,700 millilitres

81. 15.665 litres

83. 630,000 centilitres

85. 450,000 centilitres

87. 1,060 centilitres

89. 143.83 litres

91. 990 centilitres

93. 4.07331 kilolitres

95. 14.968 litres