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Dimensional Analysis Problem Solving
Goals:
• under standing what equals, “=“ , means;
the equal sign means “the same as”;
• identifying conversion “equivalent” statements used in Factor-Label Method Approach to Dimensional Analysis Problem Solving
Factor Label Method supplemental packet page 21
Based on the following mathematical principles:1. Multiplying any quantity by 1 does not change its value:
4 cents x 1 = 4 cents 3 cm x 1 = 3 cm
2. Dividing any quantity by itself is equal to 1.
3. Any two quantities that are equal to one another, when made into afraction give 1.
4 = 4 ∴ 4/4 = 1
1 = 12 foot inches ∴
4
4 _________ = 1 Z cm
Z cm _________ = 1= 1 _________ 3 apples
3 apples= 1= 1 _________ 12 inches
1 foot12 inches
1 foot _________ and
The basic idea is that multiplying a quantity times a fraction(or several fractions) that equal one does not change the value of the quantity but may change the units that express the quantity.
Writing metric equivalent statements:1. Always make the metric prefix equal to the numerical value of thefundamental unit:
2. The above equivalent statements can lead to either of two conversionfactors:
1 mg = 10–3 g 1 kg = 103 g 10–3 g
1 mg _________ _________ 1 mg
10–3 goror103 g
1 kg _________ _________ 1 kg
103 g
3. Which conversion factor shall we use? The one that cancels theunwanted labels (units) and gives the desired label.
Example : Convert 50 grams to milligrams: x kg = 50 g
NOT
5 x 10–2 kgx kg = 50 g 103 g
1 kg _________ x =
1 kg
g 25 x 104=xx kg = 50 g
103 g
1 kg _________
x kg = 420 g 103 g
1 kg _________ x =?
How many kilograms are the same as 420 grams?
In other words, convert 420 grams to kilograms
1. Begin by writing down conversion factors and their ratios.
or103 g
1 kg
_________ _________ 1 kg
103 g1 kg = 103 g
4.2 x 10–1 kg
2. Choose conversion factors that cancels the units we want to discard and leaves the units we want in the result
x kg = 420 g ?
x Mg = 0.000719 g 10
6 g
1 Mg _______ x?
How many Megagrams are the same as 0.000719 grams?
In other words, convert 0.000719 grams to Megagrams.
1. Begin by writing down conversion factors and their ratios.
_________ or10
6 g
1 Mg
_________ 1 Mg
106
g1 Mg = 10
6 g
= 7.19 • 10 –10
Mg
2. Choose conversion factors that cancels the units we want to discard and leaves the units we want in the result
Mg = 0.000719 g ?
How many Megaliters are the same as 22.6 centiliters?
In other words, convert 22.6 centiliters to Megaliters.
1. Begin by writing down conversion factors and their ratios.
_________ or10
6 L
1 M L
_________ 1 M L
106 L
1 M L = 10 6
L
2. Choose conversion factors that cancels the units we want to discard and leaves the units we want in the result
cL = 22.6 ML ?
_________ or10
-2 L
1 cL
_________ 1 cL
10-2 L
1 c L = 10 -2
L
2.26 EE 9 cL=x _________ 1 cL
10–2 L cL = 22.6 ML x
106 L
1 ML
_________ ?
How many nanometer are the same as 2260 micrometers?
In other words, convert 2260 micrometers to nanometers.
1. Begin by writing down conversion factors and their ratios.
_________ or10
-6 m
1 m
_________ 1 m
10-6 m
1 m = 10 -6
m
2. Choose conversion factors that cancels the units we want to discard and leaves the units we want in the result
nm = 2260 m ?
_________ or10
-9 m
1 nm
_________ 1 nm
10-9 m
1 nm = 10 -9
m
=x _________ 1 nm
10–9 mnm = 2660 m x
10 – 6 m
1 m
_________ ? = 2.66 EXP 6 nm
How many deciseconds are the same as 0.75 milliseconds?
In other words, convert 0.75 milliseconds to deciseconds.
1. Begin by writing down conversion factors and their ratios.
_________ or10
-3 s
1 ms
_________ 1 ms
10-3 s
1 ms = 10 -3
s
2. Choose conversion factors that cancels the units we want to discard and leaves the units we want in the result
ds = 0.75 ms ?
_________ or10
-1 s
1 ds
_________ 1 ds
10-1 s
1 ds = 10 -1
s
= 7.5 • 10–3 ds? _________
1 ms
10 –3 s ds = 0.75 ms x
10 –1 s
1 ds _________ x
Think Metric…or Else!
2.54 0.946 454
Memorize these equivalent statements forEnglish to Metric conversions
bottom of supplement packet page 22
kg = 7.71 x 10–7 lbs 1 lbs
454 g _________ x =x _________ 1 kg
103 g?
How many kilograms are the same as 7.71 x 10-7 pounds?
In other words, convert 7.71 x 10-7 pounds to kilograms.
1. Begin by writing down conversion factors and their ratios.
_________ or454 g
1 lbs
_________ 1 lbs
454 g1 lbs = 454
g
2. Choose conversion factors that cancels the units we want to discard and leaves the units we want in the result
kg = 7.71 x 10-7 lbs ?
_________ or10
3 g
1 kg
_________ 1 kg
103 g
1 kg = 10 3
g
3.50 x 10–7 kg
How many decimeters are the same as 8.1 x 103 inches?
1 cm
10–2 m _________ x dm = 8.1 x 103 inches =x _________ 1 dm
10–1 mx _________ 2.54 cm
1 in?
In other words, convert 8.1 x 103 inches to decimeters.1. Begin by writing down conversion factors and their ratios.
_________ or2.54 cm
1 inch
_________ 1 inch
2.54 cm1 inch = 2.54
cm
2. Choose conversion factors that cancels the units we want to discard and leaves the units we want in the result
dm = 8.1 x 103 inches ?
_________ or10
-2 m
1 cm
_________ 1 cm
10-2 m
1 cm = 10 -2
m
_________ or10
3 m
1 dm
_________ 1 m
103 m
1 dm = 10 -1
m
2.1•103 dm
x g = 136 lbs 1 lbs
454 g _______ x?
How many grams are the same as 136 pounds?
In other words, convert 136 pounds to grams.
1. Begin by writing down conversion factors and their ratios.
_________ or454 g
1 lbs
_________ 1 lbs
454 g1 lbs = 454
g
= 6.17 EE 4 g
2. Choose conversion factors that cancels the units we want to discard and leaves the units we want in the result
g = 136 lbs ?
How many centiliters are the same as 6.70 x 104 quarts?
In other words, convert 6.70 x 104 quarts to centiliters.
1. Begin by writing down conversion factors and their ratios.
_________ or0.964 L
1 qt
_________ 1 lbs
0.946 L1 qt = 0.946
L
2. Choose conversion factors that cancels the units we want to discard and leaves the units we want in the result
cL = 6.70 x 104 qt ?
_________ or10
-2 L
1 cL
_________ 1 cL
10-2 cL
1 cL = 10 -2
L
6.34 EXP 6 cL? _________
1 qt
0.946 L cL = 6.70 x 104 qt x
10 –2 L
1 cL _________ x =
Charlie Brown Handout• Applying sigfigs and metric conversion
1.6093
10. km x ________ = 1.6093 km
1 mile
x ________ =1 hour
60 min x __________ =1.6093 km1 mile
6.2 miles
19 km_____1 hour
12 miles_____5 min
1 mile