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2.1 Linear measures: the metric system of measurement

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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses the Metric System of measurement. By the end of the module students should be more familiar with some of the advantages of the base ten number system, know the prefixes of the metric measures, be familiar with the metric system of measurement and be able to convert from one unit of measure in the metric system to another unit of measure

Section overview

  • The Advantages of the Base Ten Number System
  • Prefixes
  • Conversion from One Unit to Another Unit
  • Conversion Table

The advantages of the base ten number system

The metric system of measurement takes advantage of our base ten number sys­tem. The advantage of the metric system over the United States system is that in the metric system it is possible to convert from one unit of measure to another simply by multiplying or dividing the given number by a power of 10. This means we can make a conversion simply by moving the decimal point to the right or the left.

Prefixes

Common units of measure in the metric system are the meter (for length), the liter (for volume), and the gram (for mass). To each of the units can be attached a prefix. The metric prefixes along with their meaning are listed below.

    Metric prefixes

  • thousand
  • tenth
  • hundred
  • hundredth
  • ten
  • thousandth

For example, if length is being measured,
1 kilometer is equivalent to 1000 meters.
1 centimeter is equivalent to one hundredth of a meter.
1 millimeter is equivalent to one thousandth of a meter.

Conversion from one unit to another unit

Let's note three characteristics of the metric system that occur in the metric table of measurements.

  1. In each category, the prefixes are the same.
  2. We can move from a larger to a smaller unit of measure by moving the decimal point to the right .
  3. We can move from a smaller to a larger unit of measure by moving the decimal point to the left .

The following table provides a summary of the relationship between the basic unit of measure (meter, gram, liter) and each prefix, and how many places the decimal point is moved and in what direction.

kilo hecto deka unit deci centi milli

Basic Unit to Prefix Move the Decimal Point
unit to deka 1 to 10 1 place to the left
unit to hector 1 to 100 2 places to the left
unit to kilo 1 to 1,000 3 places to the left
unit to deci 1 to 0.1 1 place to the right
unit to centi 1 to 0.01 2 places to the right
unit to milli 1 to 0.001 3 places to the right

Conversion table

Listed below, in the unit conversion table, are some of the common metric units of measure.

Unit Conversion Table
Length 1   kilometer  ( km ) = 1,000 meters  ( m ) size 12{1" kilometer " \( "km" \) ="1,000 meters " \( m \) } {} 1, 000 × 1 m size 12{1,"000" times 1" m"} {}
1   hectometer  ( hm ) = 100 meters size 12{1" hectometer " \( "hm" \) ="100 meters "} {} 100 × 1 m size 12{"100" times 1" m"} {}
1   dekameter  ( dam ) = 10 meters size 12{1" dekameter " \( "dam" \) ="10 meters"} {} 10 × 1 m size 12{"10" times 1" m"} {}
1 meter (m) 1 × 1 m size 12{1 times 1" m"} {}
1   decimeter  ( dm ) = 1 10   meter size 12{1" decimeter " \( "dm" \) = { {1} over {"10"} } " meter"} {} . 1 × 1 m size 12{ "." 1 times 1" m"} {}
1   centimeter  ( cm ) = 1 100   meter size 12{1" centimeter " \( "cm" \) = { {1} over {"100"} } " meter"} {} . 01 × 1 m size 12{ "." "01" times 1" m"} {}
1   millimeter  ( mm ) = 1 1,000   meter size 12{1" millimeter " \( "mm" \) = { {1} over {"1,000"} } " meter"} {} . 001 × 1 m size 12{ "." "001" times 1" m"} {}
Mass 1   kilogram   ( kg ) = 1,000 grams   ( g ) size 12{"1 kilogram " \( "kg" \) ="1,000 grams " \( g \) } {} 1, 000 × 1 g size 12{1,"000" times 1" g"} {}
1   hectogram  ( hg ) = 100 grams size 12{"1 hectogram " \( "hg" \) ="100 grams"} {} 100 × 1 g size 12{"100" times 1" g"} {}
1   dekagram   ( dag ) = 10 grams size 12{"1 dekagram " \( "dag" \) ="10 grams"} {} 10 × 1 g size 12{"10" times 1" g"} {}
1   gram   (g) 1 × 1 g size 12{1 times 1" g"} {}
1   decigram   ( dg ) = 1 10   gram size 12{"1 decigram " \( "dg" \) = { {1} over {"10"} } " gram"} {} . 1 × 1 g size 12{ "." 1 times 1" g"} {}
1   centigram   ( cg ) = 1 100   gram size 12{"1 centigram " \( "cg" \) = { {1} over {"100"} } " gram"} {} . 01 × 1 g size 12{ "." "01" times 1" g"} {}
1   milligram   ( mg ) = 1 1,000   gram size 12{"1 milligram " \( "mg" \) = { {1} over {"1,000"} } " gram"} {} . 001 × 1 g size 12{ "." "001" times 1" g"} {}
Volume 1   kiloliter   ( kL ) = 1, 000   liters   ( L ) size 12{"1 kiloliter " \( "kL" \) =1,"000"" liters " \( L \) } {} 1, 000 × 1 L size 12{1,"000" times 1" L"} {}
1   hectoliter   ( hL ) = 100   liters size 12{"1 hectoliter " \( "hL" \) ="100"" liters"} {} 100 × 1 L size 12{"100" times 1" L"} {}
1   dekaliter   ( daL ) = 10   liters size 12{"1 dekaliter " \( "daL" \) ="10"" liters"} {} 10 × 1 L size 12{"10" times 1" L"} {}
1 liter (L) 1 × 1 L size 12{1 times 1" L"} {}
1   deciliter   ( dL ) = 1 10   liter size 12{"1 deciliter " \( "dL" \) = { {1} over {"10"} } " liter"} {} . 1 × 1 L size 12{ "." 1 times 1" L"} {}
1   centiliter   ( cL ) = 1 100   liter size 12{"1 centiliter " \( "cL" \) = { {1} over {"100"} } " liter"} {} . 01 × 1 L size 12{ "." "01" times 1" L"} {}
1   milliliter   ( mL ) = 1 1, 000   liter size 12{"1 milliliter " \( "mL" \) = { {1} over {1,"000"} } " liter"} {} . 001 × 1 L size 12{ "." "001" times 1" L"} {}
Time Same as the United States system

Distinction between mass and weight

There is a distinction between mass and weight. The weight of a body is related to gravity whereas the mass of a body is not. For example, your weight on the earth is different than it is on the moon, but your mass is the same in both places. Mass is a measure of a body's resistance to motion. The more massive a body, the more resistant it is to motion. Also, more massive bodies weigh more than less massive bodies.

Converting metric units

To convert from one metric unit to another metric unit:

  1. Determine the location of the original number on the metric scale (pictured in each of the following examples).
  2. Move the decimal point of the original number in the same direction and same number of places as is necessary to move to the metric unit you wish to go to.

We can also convert from one metric unit to another using unit fractions. Both methods are shown in [link] of [link] .

Sample set a

Convert 3 kilograms to grams.

  1. 3 kg can be written as 3.0 kg. Then,

    A line with hash marks dividing the line into seven segments. The segments are labeled, from left to right, kg, hg, dag, g, dg, cg, and mg. Below kg, hg, dagga, and g are arrows pointing from each segment to the neighboring segment on the right. These arrows are labeled 1, 2, and 3, indicating the number of places to the right.

    3.0 kg is equal to 3000g. An arrow is drawn under the three zeros in 3000, counting three decimal places to the right.

    Thus, 3kg = 3,000 g size 12{"3kg"="3,000 g"} {} .

  2. We can also use unit fractions to make this conversion.

    Since we are converting to grams, and 1, 000     g = 1   kg size 12{1,"000"g="1kg"} {} , we choose the unit fraction 1,000   g 1   kg size 12{ { {"1,000 g"} over {"1 kg"} } } {} since grams is in the numerator.

    3   kg = 3 kg ⋅ 1,000   g 1 kg = 3   kg ⋅ 1,000   g 1   kg = 3 ⋅ 1,000   g = 3,000   g

Convert 67.2 hectoliters to milliliters.

A line with hash marks dividing the line into seven segments. The segments are labeled, from left to right, kL, hL, dal, L, dL, cL, and mL. Below hL, dal, L, dL, cL, and mL are arrows pointing from each segment to the neighboring segment on the right. These arrows are labeled 1 through 5 indicating the number of places to the right.

62.7 hL is equal to 6720000 mL. An arrow is drawn under the rightmost five digits in 6720000, counting five decimal places to the right.

Thus, 67 . 2 hL = 6,720,000 mL size 12{"67" "." "2 hL"="6,720,000 mL"} {} .

Convert 100.07 centimeters to meters.

A line with hash marks dividing the line into seven segments. The segments are labeled, from left to right, km, hm, dam, m, dm, cm, mm. Below cm, dm, and m are arrows pointing from each segment to the neighboring segment on the left. These arrows are labeled 1 and 2, indicating the number of places to the left.

100.07 cm equals 1.0007 m. Arrows under the two leftmost zeros are labeled 1 and 2, pointing to the left, indicating the number of decimal places moved.

Thus, 100 . 07 cm = 1 . 0007 m size 12{"100" "." "07 cm "=" 1" "." "0007 m"} {} .

Convert 0.16 milligrams to grams.

A line with hash marks dividing the line into seven segments. The segments are labeled, from left to right, kg, hg, dg, g, dg, cg, and mg. Below g, dg, cg, and mg are arrows pointing from each segment to the neighboring segment on the left. These arrows are labeled 1, 2, and 3, indicating the number of places to the left.

0.16mg equals 0.00016g. Underneath the rightmost three zeros are arrows pointing to the left, labeled 1, 2, and 3, indicating the movement of the decimal point.

Thus, 0 . 16   mg = 0 . 00016 g . size 12{0 "." "16"" mg "=" 0" "." "00016 g" "." } {}

Practice set a

Convert 411 kilograms to grams.

411,000 g

Convert 5.626 liters to centiliters.

562.6 cL

Convert 80 milliliters to kiloliters.

0.00008 kL

Convert 150 milligrams to centigrams.

15 cg

Convert 2.5 centimeters to meters.

0.025 m

Exercises

Make each conversion.

87 m to cm

8,700 cm

905 L to mL

16,005 mg to g

16.005 g

48.66 L to dL

11.161 kL to L

11,161 L

521.85 cm to mm

1.26 dag to dg

126 dg

99.04 dam to cm

0.51 kL to daL

5.1 daL

0.17 kL to daL

0.05 m to dm

0.5 dm

0.001 km to mm

8.106 hg to cg

81,060 cg

17.0186 kL to mL

3 cm to m

0.03 m

9 mm to m

4 g to mg

4,000 mg

2 L to kL

6 kg to mg

6,000,000 mg

7 daL to mL

Exercises for review

( [link] ) Find the value of 5 8 − 1 3 + 3 4 size 12{ { {5} over {8} } - { {1} over {3} } + { {3} over {4} } } {} .

25 24 = 1 1 24 size 12{ { {"25"} over {"24"} } =1 { {1} over {"24"} } } {}

( [link] ) Solve the proportion: 9 x = 27 60 size 12{ { {9} over {x} } = { {"27"} over {"60"} } } {} .

( [link] ) Use the method of rounding to estimate the sum: 8, 226 + 4, 118 size 12{8,"226"+4,"118"} {} .

12,300 (12,344)

( [link] ) Use the clustering method to estimate the sum: 87 + 121 + 118 + 91 + 92 size 12{"87"+"121"+"118"+"91"+"92"} {} .

( [link] ) Convert 3 in. to yd.

0 . 08 3 ¯ size 12{0 "." "08" {overline {3}} } {} yard

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Source:  OpenStax, Contemporary math applications. OpenStax CNX. Dec 15, 2014 Download for free at http://legacy.cnx.org/content/col11559/1.6
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