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
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.
Based on the following mat hem at ical principles:
1.
Mult iplying any quant it y by 1 does not change its value:
4 cent s x
2.
1 = 4 cents
3 cm x 1 = 3 cm
Dividing any quant it y by it self is equal t o 1.
4
_________
4
= 1
 4/4 = 1
= 1
3 apples
3.
Any t wo quantities t hat are equal
fract ion give 1.
4=4
3 appl es
_________
Z cm
_________
Z cm
= 1
t o one another, when made into a
1 foot = 12 inches

1 f oot
_________
12 inches
=
1 and
12 inches
_________
1 f oot
=
1
Writing metric equivalent statements:
1.
Always make the metric prefix equal t o the
fundam ent al unit:
2.
T he above
factors:
–3
1 m g = 10
g
3
= 10
g
1 kg
num erical value of the
equivalent st at ement s can lead t o either of two conversion
3
1 kg
_________
or
3
1 mg
_________
10 g
_________
10 g
1 kg
10
–3
g
3.
Which conversion factor shall we use? T he one t hat
unwanted labels (unit s) and gives t he desired label.
Exam ple : Convert 50 grams to milligrams:
x kg = 50 g
1 kg
x _________
3
10 g
=
5 x 10
–2
kg
NO T
–3
or
10 g
_________
1 mg
cancels the
x kg = 50 g
x kg = 50 g x
10 3 g
_________
=
1 kg
2
4
5 x 10
g
1 kg
Lets work out the dimensional analysis problems together
supplement packet page 23
?x kg = 420 g
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.
3
1 kg = 10 g
103 g
_________
1 kg
or
1 kg
_________
3
10 g
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
1
kg
_________
103 g
= 4.2 x 10–1 kg
? Mg = 0.000719 g
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.
6
6
1 Mg = 10 g
10 g
_________
1 Mg
or
1 Mg
_________
6
10 g
2. Choose conversion factors that cancels the units we want to discard
and leaves the units we want in the result
1
Mg
?x Mg = 0.000719 g x _______
6
10 g
–10
= 7.19 • 10
Mg
? cL = 22.6 ML
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.
6
6
1 M L = 10 L
10 L
_________
1ML
or
1ML
_________
6
10 L
or
1 cL
_________
-2
10 L
-2
-2
1 c L = 10 L
10 L
_________
1 cL
2. Choose conversion factors that cancels the units we want to discard
and leaves the units we want in the result
6
1 cL
10
L
_________
_________
x
= 2.26 EE 9 cL
? cL = 22.6 ML x
–2
1 ML
10 L
? nm = 2260 m m
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.
-6
-6
1 m m = 10 m
10 m
_________
1mm
or
1mm
_________
-6
10 m
or
1 nm
_________
-9
10 m
-9
1 nm = 10
-9
m
10 m
_________
1 nm
2. Choose conversion factors that cancels the units we want to discard
and leaves the units we want in the result
? nm = 2660 mm x
–6
10
m
_________
x
1 mm
1 nm
_________
10
–9
m
= EXP 6 nm
= 2.66
? ds = 0.75 ms
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.
-3
-3
1 ms = 10 s
10 s
_________
1 ms
or
1 ms
_________
-3
10 s
or
1 ds
_________
-1
10 s
-1
1 ds = 10
-1
s
10 s
_________
1 ds
2. Choose conversion factors that cancels the units we want to discard
and leaves the units we want in the result
–3
10 s
? ds = 0.75 ms x _________
1 ms
x
1 ds
_________
–3 ds
=
7.5
•
10
10 –1 s
bottom of supplement packet page 22
Think Metric…or Else!
2.54
0.946
454
Memorize these equivalent statements for
English to Metric conversions
Lets work out the dimensional analysis problems together
supplement packet page 23
? kg = 7.71 x 10-7 lbs
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.
1 lbs = 454 g
454 g
_________
1 lbs
or
1 lbs
_________
454 g
or
1 kg
_________
3
10 g
3
1 kg = 10
3
g
10 g
_________
1 kg
2. Choose conversion factors that cancels the units we want to discard
and leaves the units we want in the result
1 kg
454 g
_________
_________
x
= 3.50 x 10–7 kg
? kg = 7.71 x 10 lbs x
103 g
1 lbs
–7
? dm = 8.1 x 103 inches
How many decimeters are the same as 8.1 x 103 inches?
In other words, convert 8.1 x 103 inches to decimeters.
1. Begin by writing down conversion factors and their ratios.
1 inch = 2.54cm
2.54 cm
_________
1 inch
or
1 inch
_________
2.54 cm
or
1 cm
_________
-2
10 m
or
1m
_________
3
10 m
-2
-2
1 cm = 10 m
10 m
_________
1 cm
3
-1
1 dm = 10 m
10 m
_________
1 dm
2. Choose conversion factors that cancels the units we want to discard
and leaves the units we want in the result
–2
2.54
cm
1 dm
10
m _________
3
_________
_________
? dm = 8.1 x 10 i nches x
x
x
= 2.1•103 dm
–1
1 in
1 cm
10 m
? g = 136 lbs
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.
1 lbs = 454 g
454 g
_________
1 lbs
or
1 lbs
_________
454 g
2. Choose conversion factors that cancels the units we want to discard
and leaves the units we want in the result
?x g = 136 lbs
454
g
x _______
1 lbs
= 6.17 EE 4 g
? cL = 6.70 x 104 qt
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.
1 qt = 0.946 L
0.964 L
_________
1 qt
or
1 lbs
_________
0.946 L
or
1 cL
_________
-2
10 cL
-2
1 cL = 10
-2
L
10 L
_________
1 cL
2. Choose conversion factors that cancels the units we want to discard
and leaves the units we want in the result
1 cL
0.946 L
_________
_________
x
= 6.34 EXP 6 cL
? cL = 6.70 x 10 qt x
–2
10
L
1 qt
4
Charlie Brown Handout
• Applying sigfigs and metric conversion
1.6093
1 mile
10. km x ________
= 6.2 miles
1.6093 km
1_____
mile x ________
60 min
5 min
1 hour
miles x __________
= 12
_____
1.6093 km
1 mile
1 hour
= 19 km
Sally and Charlie
10 mg = 1 cg
10 • 10 -3 g = 1 •10 -2 g
10 dg = 1 g
10 • 10 -1 g = 1 g
m = 10 -3 substitute m for 10 -3 d = 10 -1 substitute d for 10 -1
c = 10 -2 substitute c for 10 -2