Solved Examples: The Metric System of Measurements and Units

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NOTE: Unless specified otherwise:
(1.) Use only the tables provided for you.
(2.) State the measurement.
(3.) Use at least two methods as applicable.
(4.) Please do not approximate intermediate calculations.
(5.) Please do not approximate final calculations. Leave your final answer as is.
Metric to Metric Conversions
Prefix Symbol Multiplication Factor
yocto y $10^{-24}$
zepto z $10^{-21}$
atto a $10^{-18}$
femto f $10^{-15}$
pico p $10^{-12}$
nano n $10^{-9}$
micro $\mu$ $10^{-6}$
milli m $10^{-3}$
centi c $10^{-2}$
deci d $10^{-1}$
deka da $10^1$
hecto h $10^2$
kilo K $10^3$
mega M $10^6$
giga G $10^9$
tera T $10^{12}$
peta P $10^{15}$
exa E $10^{18}$
zetta Z $10^{21}$
yotta Y $10^{24}$


(1.) Convert 16.27 meters to kilometers


Measurement is Length

$ \underline{First\:\:Method:\:\:Unity\:\:Fraction\:\:Method} \\[3ex] 16.27\:m\:\:to\:\:km \\[3ex] = 16.27\:m * \dfrac{.....km}{.....m} \\[5ex] = 16.27\:m * \dfrac{1\:km}{10^3\:m} \\[5ex] = 16.27\:m * \dfrac{1\:km}{1000\:m} \\[5ex] = \dfrac{16.27}{1000}\:km \\[5ex] = 0.01627\:km \\[3ex] \underline{Second\:\:Method:\:\:Proportional\:\:Reasoning\:\:Method} \\[3ex] 1\:km = 10^3\:m \\[3ex] 1\:km = 1000\:m \\[3ex] Let\:\:p = length\:\:of\:\:16.27\:m\:\:in\:\:km \\[3ex] $
$km$ $m$
$1$ $1000$
$p$ $16.27$

$ \dfrac{p}{1} = \dfrac{16.27}{1000} \\[5ex] p = 0.01627\:km \\[3ex] \therefore 16.27\:m = 0.01627\:km $
(2.) Convert 16.27 kilometers to meters


Measurement is Length

$ \underline{First\:\:Method:\:\:Unity\:\:Fraction\:\:Method} \\[3ex] 16.27\:km\:\:to\:\:m \\[3ex] = 16.27\:km * \dfrac{.....m}{.....km} \\[5ex] = 16.27\:km * \dfrac{10^3\:m}{1\:km} \\[5ex] = 16.27\:km * \dfrac{1000\:m}{1\:km} \\[5ex] = 16.27 * 1000 \\[3ex] = 16270\:m \\[3ex] \underline{Second\:\:Method:\:\:Proportional\:\:Reasoning\:\:Method} \\[3ex] 1\:km = 10^3\:m \\[3ex] 1\:km = 1000\:m \\[3ex] Let\:\:p = length\:\:of\:\:16.27\:km\:\:in\:\:m \\[3ex] $
$km$ $m$
$1$ $1000$
$16.27$ $p$

$ \dfrac{p}{1000} = \dfrac{16.27}{1} \\[5ex] \dfrac{p}{1000} = 16.27 \\[5ex] Multiply\:\:both\:\:sides\:\:by\:\:1000 \\[3ex] 1000 * \dfrac{p}{1000} = 1000(16.27) \\[5ex] p = 16270\:m \\[3ex] \therefore 16.27\:km = 16270\:m $
(3.) Convert 25 dekagrams to decigrams


Measurement is Mass

$ \underline{First\:\:Method:\:\:Unity\:\:Fraction\:\:Method} \\[3ex] 25\:dag\:\:to\:\:dg \\[3ex] = 25\:dag * \dfrac{.....g}{.....dag} * \dfrac{.....dg}{.....g} \\[5ex] = 25\:dag * \dfrac{10\:g}{1\:dag} * \dfrac{1\:dg}{10^{-1}\:g} \\[5ex] = 25\:dag * \dfrac{10\:g}{1\:dag} * \dfrac{1\:dg}{0.1\:g} \\[5ex] = \dfrac{25 * 10}{0.1}\:dg \\[5ex] = \dfrac{250}{0.1}\:dg \\[5ex] = 2500\:dg \\[3ex] \underline{Third\:\:Method:\:\:Fast\:\:Proportional\:\:Reasoning\:\:Method} \\[3ex] 1\:dag = 10\:g \\[3ex] 1\:dg = 10^{-1}\:g \\[3ex] 1\:dg = 0.1\:g \\[3ex] $
$dag$ $g$ $dg$
$1$ $10$
$k$ $0.1$ $1$
$25$ $p$

$ Let\:\:k = length\:\:of\:\:1\:dg\:\:in\:\:dag \\[3ex] Let\:\:p = length\:\:of\:\:25\:dag\:\:in\:\:dg \\[3ex] First: \\[3ex] \dfrac{k}{1} = \dfrac{0.1}{10} \\[5ex] k = 0.01\:dag \\[3ex] Next: \\[3ex] \dfrac{p}{1} = \dfrac{25}{k} \\[5ex] p = \dfrac{25}{0.01} \\[5ex] p = 2500\:dg \\[3ex] \therefore 25\:dag = 2500\:dg $
(4.) Convert 25 decigrams to dekagrams


Measurement is Mass

$ \underline{First\:\:Method:\:\:Unity\:\:Fraction\:\:Method} \\[3ex] 25\:dg\:\:to\:\:dag \\[3ex] = 25\:dg * \dfrac{.....g}{.....dg} * \dfrac{.....dag}{.....g} \\[5ex] = 25\:dag * \dfrac{10^{-1}\:g}{1\:dg} * \dfrac{1\:dag}{10\:g} \\[5ex] = 25\:dag * \dfrac{0.1\:g}{1\:dg} * \dfrac{1\:dag}{10\:g} \\[5ex] = \dfrac{25 * 0.1}{10}\:dg \\[5ex] = \dfrac{2.5}{10}\:dag \\[5ex] = 0.25\:dag \\[3ex] \underline{Third\:\:Method:\:\:Fast\:\:Proportional\:\:Reasoning\:\:Method} \\[3ex] 1\:dg = 10^{-1}\:g \\[3ex] 1\:dg = 0.1\:g \\[3ex] 1\:dag = 10\:g \\[3ex] $
$dg$ $g$ $dag$
$1$ $0.1$ $a$
$10$ $1$
$25$ $c$

$ Let\:\:a = length\:\:of\:\:1\:dg\:\:in\:\:dag \\[3ex] Let\:\:p = length\:\:of\:\:25\:dg\:\:in\:\:dag \\[3ex] First: \\[3ex] \dfrac{a}{1} = \dfrac{0.1}{10} \\[5ex] a = 0.01\:dag \\[3ex] Next: \\[3ex] \dfrac{c}{a} = \dfrac{25}{1} \\[5ex] \dfrac{c}{0.01} = 25 \\[5ex] Multiply\:\:both\:\:sides\:\:by\:\: 0.01 \\[3ex] 0.01 * \dfrac{c}{0.01} = 0.01(25) \\[5ex] c = 0.25\:dag \\[3ex] \therefore 25\:dg = 0.25\:dag $
(5.)


(6.) Convert 4,100,000 micrometers to meters.


Measurement is Length

$ \underline{First\:\:Method:\:\:Unity\:\:Fraction\:\:Method} \\[3ex] 4100000\:\mu m\:\:to\:\:m \\[3ex] = 4100000\:\mu m * \dfrac{.....m}{.....\mu m} \\[5ex] = 4100000\:\mu m * \dfrac{10^{-6}\:m}{1\:\mu m} \\[5ex] = 4.1 * 10^6 * 10^{-6} \\[4ex] = 4.1 * 10^{6 + (-6)} \\[4ex] = 4.1 * 10^{6 - 6} \\[4ex] = 4.1 * 10^0 \\[4ex] = 4.1 * 1 \\[3ex] = 4.1\:m \\[3ex] \underline{Second\:\:Method:\:\:Proportional\:\:Reasoning\:\:Method} \\[3ex] 1\:\mu m = 10^{-6}\:m \\[4ex] $
$\mu m$ $m$
1 $10^{-6}$
4100000 what

$ \dfrac{what}{4100000} = \dfrac{10^{-6}}{1} \\[5ex] \dfrac{what}{4100000} = 10^{-6} \\[5ex] what = 4100000 * 10^{-6} \\[4ex] what = 4100000 * 0.000001 \\[3ex] what = 4.1\:m $
(7.) CSEC The scale on a map is 1:25000
(i) Anderlin and Jersey are 31.8cm apart on the map.
Determine, in km, the actual distance between Anderlin and Jersey.

(ii) The actual distance between Clifton and James Town is 2.75km
How many units apart are they on the map?


$ (i) \\[3ex] Scale = 1:25000 \\[3ex] 31.8\:cm \rightarrow ? \\[3ex] \underline{Second\:\:Method:\:\:Proportional\:\:Reasoning\:\:Method} \\[3ex] $
$Scale$ $Actual$
$1$ $25000$
$31.8$ $y$

$ \dfrac{1}{31.8} = \dfrac{25000}{y} \\[5ex] Cross\:\:Multiply \\[3ex] 1(y) = 31.8(25000) \\[3ex] y = 795000\:cm \\[3ex] \therefore 31.8\:cm = 795000\:cm \\[3ex] $ We need to convert this distance to $km$

$ \underline{First\:\:Method - Unity\:\:Fraction\:\:Method} \\[3ex] 795000\:cm\:\:to\:\:km \\[3ex] = 795000\:cm * \dfrac{0.01\:m}{1\:cm} * \dfrac{1\:km}{1000\:m} \\[5ex] = \dfrac{7950}{1000}\:km \\[5ex] = 7.95\:km \\[3ex] \underline{Second\:\:Method:\:\:Proportional\:\:Reasoning\:\:Method} \\[3ex] $
$cm$ $m$
$1$ $0.01$
$795000$ $x$

$ \dfrac{1}{795000} = \dfrac{0.01}{x} \\[5ex] Cross\:\:Multiply \\[3ex] 1(x) = 795000(0.01) \\[3ex] x = 7950\:m \\[3ex] $
$m$ $km$
$1000$ $1$
$7950$ $p$

$ \dfrac{1000}{7950} = \dfrac{1}{p} \\[5ex] Cross\:\:Multiply \\[3ex] 1000p = 7950(1) \\[3ex] 1000p = 7950 \\[3ex] p = \dfrac{7950}{1000} \\[5ex] p = 7.95\:km \\[3ex] (ii) \\[3ex] \underline{First\:\:Method - Unity\:\:Fraction\:\:Method} \\[3ex] 2.75\:km\:\:to\:\:cm \\[3ex] = 2.75\:km * \dfrac{1000\:m}{1\:km} * \dfrac{1\:cm}{0.01\:m} \\[5ex] = \dfrac{2750}{0.01}\:km \\[5ex] = 275000\:cm \\[3ex] \underline{Second\:\:Method:\:\:Proportional\:\:Reasoning\:\:Method} \\[3ex] $
$Scale$ $Actual$
$1$ $25000$
$d$ $275000$

$ \dfrac{1}{d} = \dfrac{25000}{275000} \\[5ex] \dfrac{25000}{275000} = \dfrac{5}{55} = \dfrac{1}{11} \\[5ex] \dfrac{1}{d} = \dfrac{1}{11} \\[5ex] Cross\:\:Multiply \\[3ex] d = 1(11) \\[3ex] d = 11\:cm \\[3ex] $ Clifton and James Town are $11\:cm$ apart on the map.
(8.) Convert 0.026 kilograms to grams.


Measurement is Mass

$ \underline{First\:\:Method:\:\:Unity\:\:Fraction\:\:Method} \\[3ex] 0.026kg\:\:to\:\:g \\[3ex] = 0.026kg * \dfrac{.......g}{.......kg} \\[5ex] = 0.026kg * \dfrac{10^3\:g}{1kg} \\[5ex] = 0.026 * 1000 \\[3ex] = 26g \\[5ex] \underline{Second\:\:Method:\:\:Proportional\:\:Reasoning\:\:Method} \\[3ex] 1kg = 10^3 g \\[4ex] 1kg = 1000g \\[3ex] $
$kg$ $g$
1 1000
0.026 what

$ \dfrac{what}{1000} = \dfrac{0.026}{1} \\[5ex] what = 1000 * 0.026 \\[3ex] what = 26g $
(9.) Complete the following blank spaces
(a.) A gigameter is $10^{.......}$ times as large as a meter.
(b.) A liter is $10^{.......}$ times as large as a milliliter.
(c.) A kilogram is $10^{.......}$ times as large as a nanogram.
(d.) A liter is $10^{.......}$ times as large as a picoliter.
(e.) A terameter is $10^{.......}$ times as large as a centimeter.
(f.) A gigabyte is $10^{.......}$ times as large as a kilobyte.
(g.) A terabyte is $10^{.......}$ times as large as a megabyte.
(h.) A megawatt is $10^{.......}$ times as large as a kilowatt.
(i.) A terajoule is $10^{.......}$ times as large as a picojoule.
(j.) A square meter is $10^{.......}$ times as large as a square picometer
(k.) A cubic meter is $10^{.......}$ times as large as a cubic micrometer


(a.) 1 Gm = $10^9$ m
A gigameter is $10^9$ times as large as a meter.

(b.) 1 mL = $10^{-3}$ L
Multiply both sides by $10^3$

$ 10^3 * 1 mL = 10^3 * 10^{-3} L \\[4ex] 1000 * 1 mL = 10^{3 + (-3)} L \\[4ex] 1000mL = 10^{0}L \\[4ex] 1000mL = 1L \\[3ex] 1L = 1000mL \\[2ex] $ A liter is $10^{3}$ times as large as a milliliter.

(c.) 1 kg = $10^3$g
1 ng = $10^{-9}$g

$ 1\;kg * \dfrac{.......g}{.......kg} * \dfrac{.......ng}{.......g} \\[5ex] 1\;kg * \dfrac{10^3\;g}{1\;kg} * \dfrac{1\;ng}{10^{-9}g} \\[5ex] \dfrac{10^3}{10^{-9}}ng \\[5ex] 10^{3 -(-9)}\;ng \\[4ex] 10^{3 + 9}\;ng \\[4ex] 10^{12}\;ng \\[2ex] $ A kilogram is $10^{12}$ times as large as a nanogram.

(d.) 1 pL = $10^{-12}$ L
Multiply both sides by $10^{12}$

$ 10^{12} * 1 pL = 10^{12} * 10^{{-12}} L \\[4ex] 10^{12} * 1 pL = 10^{12 + (-12)} L \\[4ex] 10^{12}pL = 10^{0}L \\[4ex] 10^{12}pL = 1L \\[3ex] 1L = 10^{12}pL \\[2ex] $ A liter is $10^{12}$ times as large as a picoliter.

(e.) 1 Tm = $10^{12}$m
1 cm = $10^{-2}$m

$ 1\;Tm * \dfrac{.......m}{.......Tm} * \dfrac{.......cm}{.......m} \\[5ex] 1\;Tm * \dfrac{10^{12}\;m}{1\;Tm} * \dfrac{1\;cm}{10^{-2}m} \\[5ex] \dfrac{10^{12}}{10^{-2}}cm \\[5ex] 10^{12 -(-2)}\;cm \\[4ex] 10^{12 + 2}\;cm \\[4ex] 10^{14}\;cm \\[2ex] $ A terameter is $10^{14}$ times as large as a centimeter.

(f.) 1 GB = $10^9$B
1 KB = $10^3$B

$ 1\;GB * \dfrac{.......B}{.......GB} * \dfrac{.......KB}{.......B} \\[5ex] 1\;GB * \dfrac{10^9\;B}{1\;GB} * \dfrac{1\;KB}{10^3\;B} \\[5ex] \dfrac{10^9}{10^3}KB \\[5ex] 10^{9 - 3}\;KB \\[4ex] 10^6\;KB \\[2ex] $ A gigabyte is $10^{6}$ times as large as a kilobyte.

(g.) 1 TB = $10^{12}$B
1 MB = $10^6$B

$ 1\;TB * \dfrac{.......B}{.......TB} * \dfrac{.......MB}{.......B} \\[5ex] 1\;TB * \dfrac{10^{12}\;B}{1\;TB} * \dfrac{1\;MB}{10^6\;B} \\[5ex] \dfrac{10^{12}}{10^6}MB \\[5ex] 10^{12 - 6}\;MB \\[4ex] 10^6\;MB \\[2ex] $ A terabyte is $10^{6}$ times as large as a megabyte.

(h.) 1 MW = $10^6$W
1 KW = $10^3$W

$ 1\;MW * \dfrac{.......W}{.......MW} * \dfrac{.......KW}{.......W} \\[5ex] 1\;MW * \dfrac{10^6\;W}{1\;MW} * \dfrac{1\;KW}{10^3\;W} \\[5ex] \dfrac{10^6}{10^3}KW \\[5ex] 10^{6 - 3}\;KW \\[4ex] 10^3\;KW \\[2ex] $ A megawatt is $10^{3}$ times as large as a kilowatt.

(i.) 1 TJ = $10^{12}$J
1 pJ = $10^{-12}$J

$ 1\;TJ * \dfrac{.......J}{.......TJ} * \dfrac{.......pJ}{.......J} \\[5ex] 1\;TJ * \dfrac{10^{12}\;J}{1\;TJ} * \dfrac{1\;pJ}{10^{-12}J} \\[5ex] \dfrac{10^{12}}{10^{-12}}pJ \\[5ex] 10^{12 -(-12)}\;pJ \\[4ex] 10^{12 + 12}\;pJ \\[4ex] 10^{24}\;pJ \\[2ex] $ A terajoule is $10^{24}$ times as large as a picojoule.

(j.) 1 pm = $10^{-12}$m
I m² = 1 m * m

$ 1\;m * m * \dfrac{.......pm}{.......m} * \dfrac{.......pm}{.......m} \\[5ex] 1\;m * m * \dfrac{1\;pm}{10^{-12}\;m} * \dfrac{1\;pm}{10^{-12}\;m} \\[5ex] \dfrac{1}{10^{-12 + (-12)}} \\[4ex] \dfrac{1}{10^{-12 - 12}} \\[4ex] \dfrac{10^0}{10^{-24}} \\[4ex] 10^{0 - (-24)}\;pm^2 \\[4ex] 10^{0 + 24}\;pm^2 \\[4ex] 10^{24}\;pm^2 \\[2ex] $ A square meter is $10^{24}$ times as large as a square picometer

(j.) 1 µm = $10^{-6}$m
I m³ = 1 m * m * m

$ 1\;m * m * m * \dfrac{.......\mu m}{.......m} * \dfrac{.......\mu m}{.......m} * \dfrac{.......\mu m}{.......m} \\[5ex] 1\;m * m * m * \dfrac{1\;\mu m}{10^{-6}\;m} * \dfrac{1\;\mu m}{10^{-6}\;m} * \dfrac{1\;\mu m}{10^{-6}\;m} \\[5ex] \dfrac{1}{10^{-6 + (-6) + (-6)}} \\[4ex] \dfrac{1}{10^{-6 - 6 - 6}} \\[4ex] \dfrac{10^0}{10^{-18}} \\[4ex] 10^{0 - (-18)}\;\mu m^3 \\[4ex] 10^{0 + 18}\;\mu m^3 \\[4ex] 10^{18}\;\mu m^3 \\[2ex] $ A cubic meter is $10^{18}$ times as large as a cubic micrometer.
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