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 foot = 12 inches ∴

44 _________ = 1

Z cmZ cm _________ = 1= 1 _________ 3 apples

3 apples= 1= 1 _________ 12 inches

1 foot12 inches1 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 goror 103 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 g1 kg _________ x =

1 kg g 25 x 104=xx kg = 50 g

103 g1 kg

_________

Lets work out the dimensional analysis problems together supplement packet page 23

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 106 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.

_________ or106 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.

_________ or106 L

1 M L _________ 1 M L

106 L1 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 L1 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 m1 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 m1 nm = 10 -9 m

=x _________ 1 nm

10–9 mnm = 2660 x

10 – 6

1 _________ ? = 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 s1 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 s1 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

Lets work out the dimensional analysis problems together supplement packet page 23

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 ?

_________ or103 g

1 kg _________ 1 kg

103 g1 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 m1 cm = 10 -2 m

_________ or103 m

1 dm _________ 1 m

103 m1 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 cL1 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 km1 mile

x ________ =1 hour60 min x __________ =1.6093 km

1 mile

6.2 miles

19 km_____1 hour

12 miles_____5 min1 mile

Sally and Charlie

10 mg = 1 cg 10 dg = 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

10 • 10 -3 g = 1 •10 -2 g 10 • 10 -1 g = 1 g