Lesson 4-4
Working with Dimensional Analysis
The term dimensional analysis sounds more like something you might find in rocket science than medicine. However, dimensional analysis is actually a simple, but powerful idea. It is simple because it always takes the same form:  a series of two or more ratios that are multiplied together. It is powerful because it virtually eliminates the need for recalling large numbers of measurement conversions and formulas.  Master the basic idea of dimensional analysis, and you no longer have to ask, " Do I multiply or divide to make this conversion or calculate this dosage?"

Dimensional analysis is NOT rocket science.

Relative Amounts and Ratios

Recall that a ratio expresses the relative value of two amounts. "Two out of three, " for example,  can be expressed as the ratio 2/3. "One in a hundred" can be expressed as 1/100.

Now consider the fact that there are 16 fl. oz in a pint. That can also be expressed as a ratio:  . That is saying "Sixteen fluid ounces per one pint." It's a ratio that happens to include labels as well as numerical values.

You know that there are 100 cm in a meter. This can be expressed in two ways:

  says there are 100 centimeters in a meter.

says there is 1 meter per 100 centimeters.

Both are correct. Both say the same thing. The ratios are equal.

The success of dimensional analysis depends your knowing:
  • A ratio and its inverse (flipped-over version) are equal
  • Which of the two ratios are to be used solving a given problem.

Examples 1:

Express each of the following conversions as two ratios.

Known Conversion Ratios
There are 12 inches in one foot
12 in
1 ft
1 ft
12 in
There are 1000 ml in a L
1000 ml
1 L
1 L
1000 ml
16 oz = 1 lb
16 oz
1 lb
1 lb
16 oz
You can substitute 3 teaspoons of medicine for one tablespoon.
3 tsp
1 tbs
1 tbs
3 tsp

 

You know it is not unusual to have medications mixed with sterile water. The concentration of the medication can then be expressed as mg of medication per ml of fluid. So if there is 10 mg of medication in 100 ml mixture, the concentration can be expressed as:

Ten milligrams per milliliter
or
One hundred milliliters per milligram
 

Expressing Relative Amounts as a Pair of Ratios

Work with this drill routine until you can consistently make the conversions correctly.

Click the Answer button to see the correct answer to the question.
Click the Next button to see the next question.

Setting Up Ratios to "Cancel" Labels

When you first studied rations and proportions, you found that you can "cancel" equal terms in order to simplify the multiplication of fractions or ratios.

Example

Given:

5

x

3  
8 5

Cancel "crosswise" terms that are equal

5 x 3 = 1 x 3
8 5 8 1

Complete the multiplication

1 x 3 = 3
8 1 8

For the purposes of dimensional analysis, you can also "cancel" labels:

 

 

 

Explanation

In the first step, you are simply setting up a product of two ratios you should know:  there are 100 mm in 1 cm, and there are 10 cm in 1 m.

100 mm

x

10 cm  
1 cm 1 m

In the second step, notice that cm appears in the denominator of one term and in the numerator of the other term. This means you can "cancel" the cm labels not the numerical values, but the labels, themselves.

100 mm

x

10 cm  
1 cm 1 m

The third step simply shows the product of the two ratios, having cancelled out the common labels (cm, in this example)

100 mm

x

10  
1 1 m

And finally, you complete the multiplication of the numerical terms, sliding the remaining labels over into the final ratio.

100 mm x 10 = 1000 mm
1 1 m 1 m

The final result says that there are 1000 mm in one meter. And of course that is  true.

Example 2:

Use a product of ratios (dimensional analysis) to convert 575 milligrams to grams.

The given amount is 575 mg. Expressing this as a ratio:

You should know that the conversion ratio for milligrams and grams is

Set up those two ratios for multiplication:

Strike out labels that can be cancelled:

Complete the multiplication:

Answer: 575 mg is equal to 0.575 g

 

"Cancelling Labels" and Completing the Multipication

The ratios have been set up for you in this drill. Your job is to complete the task by cancelling labels are appropriate and doing the multiplication.

Click the Answer button to see the correct answer to the question.
Click the Next button to see the next question.

 

Important

The whole trick to dimensional analysis is to set up the ratios in such a way that all the labels you don't  want in the final result are "cancelled." 

Example 3

Convert 1500 mg to grams.

The given ratio is and you want the result to be in grams. So you need a conversion ratio for converting between milligrams and grams.

The conversion ratios you can choose from are and . These ratios are equal. They are both used for converting between units of milligrams and grams. The big question is this: which one do you use in our example for converting 1500 mg to grams?

Let's try the analysis using the conversion ratio, .

The trouble with using this form of the conversion ratio is that none of the labels cancel.  So It must be the wrong choice.

So try the other version, ,and see what happens:

Now the mg labels cancel, leaving the answer in grams.

1500 mg is equal to 1.5 g.

Example 4

Convert 0.2 pounds to ounces.

The given ratio is:

The conversion ratios for pounds and ounces are and

Which one should you use for converting the given ratio to units of ounces?

Try doing the analysis with the first conversion ratio:

You can see that the pound (lb) labels will cancel, leaving on the oz label for the answer. So that is the correct choice:

Try the other conversion ratio, , and you will see that nothing can be cancelled.

Ans: 0.2 lb = 3.2 oz.

 

Choosing Conversion Ratios

Select the ratio that allows you to cancel all unwanted labels, leaving only the label(s) required for the final answer.

Click the Answer button to see the correct answer to the question.
Click the Next button to see the next question.

The Structure of Dimensional Analysis

The most convincing reason for using dimensional analysis is simplicity. Every conversion problem and every dosage calculation look exactly the same way products of ratios. And there are just three basic kinds of ratios:

The Given Ratio is the one that shows the information given by the problem. If you are required to convert 122 inches to feet, for example, the given amount is 126 in, and given ratio is .

The Conversion Ratio is the one that contains the necessary conversion numbers and labels. When converting between inches and feet, the correct conversion ratio is .

The Result Ratio is the product of the given ratio and conversion ratio:  . (The Result Ratio should be simplified to a single term when you have finished the work. In this example,  should be simplified as 10.5 ft).

There can be only one Given Ratio and One Result Ratio, but there can be more than one Conversion Ratio. This happens when there is more than one unit conversion involved in the problem.

Here is an example of a dimensional analysis that uses two conversion ratios to complete the job. Don't panic over the  complicated-looking equation. Just identify the parts and note that each one says something you probably understand rather well. In this example, you convert a speed expressed in meters per second into a speed expressed in kilometers per hour.

Such problems usually get people all anxious about which to multiply or divide my which. Dimensional analysis, however, orders everything.

Example 5

Convert 2000 meters per second into kilometers per hour.

The Given Ratio is

One of the conversion ratios has to deal with seconds and hours: or

The second conversion ratio works with meters and kilometers:  or

The dimensional analysis look like this:

Note how the conversion ratios are selected so that the meters and seconds cancel, leaving just the desired kilometer and hour.

0405

Lesson 4-5
Mastering Length Conversions

 

Length Conversions

 

 

 

Units of Measure for Length
Metric US Standard
millimeter
centimeter
meter
kilometer
mm
cm
m
km
inch
foot
yard
mile
in
ft
yd
mi

 

Most Common Conversion Factors for Length

Metric US Standard System Conversion
1000 m per 1 km

100 cm per 1 m

1000 mm per 1 m

12 in  per 1 ft

3 ft per 1 yd

5280 ft per 1 mi

2.54 cm per 1 in

1.61 km per 1 mi

 

Length Conversion Map

Metric Length Conversions

Metric length conversions are relatively easy because the conversion factors are all multiples of 10.  You don't need the aid of a calculator because all the calculations are simply multiplying or dividing by multiples of 10.

100 mm per 1 cm
or
1 cm per 100 mm
100 cm per 1 m
or
1 m per 100 cm
1000 m per 1 km
or
1 km per 1000 m

 

Example 1: Converting meters to centimeters

Convert 0.12 m to cm.

Given Length:  0.12 m
Conversion Factor: 
100 cm/m
 
Set up the ratios:
Cancel the labels:
Complete the math:
Result: 120 cm  

Example 2: Converting centimeters to meters

Convert 1500 cm to m.

Given Length:  1500 cm
Conversion Factor: 
1 m/100 cm
 
Set up the ratios:
Cancel the labels:
Complete the math:
Result: 1.5 m  

 

 
Emoticon - Brilliant Idea Get the idea? Systematic dimensional analysis (conversion ratios) can save a lot of grief!

You can use the same easily remembered conversion factor 100 cm per 1 m to convert meters to centimeters or vice versa.

You don't have to deal with that old problem of trying to remember whether to multiply or to divide. Simply set up the conversion ratio so it cancels the label for the given amount.

Example 3: Converting meters to kilometers

Convert 16500 m to km

Given Length:  16500 m
Conversion Factor:  1 km/
1000 m
 
Set up the ratios:
Cancel the labels:
Complete the math:
Result: 16.5 km  

Example 4: Converting centimeters to millimeters

Convert 0.175 cm to mm

Given Length:  0.175 cm
Conversion Factor:  100 mm/
1 cm
 
Set up the ratios:
Cancel the labels:
Complete the math:
Result: 17.5 mm  

 

Metric-to-Metric Conversion Drill

Work with this drill routine until you can consistently make the conversions correctly.

Click the Answer button to see the correct answer to the question.
Click the Next button to see the next question.

US Standard Length Conversions

US Standard conversions are a bit more difficult than metric conversions, mainly because the conversion factors are not simple multiples of 10:

So a calculator can be helpful at times for making conversions that include any US Standard lengths.

NOTE

Leave answers in decimal form. All but the US building trades use decimal forms of US Standard lengths. For example, cite 1.25 ft instead of 1 ft 3 in., and 1.5 inches instead of 1 ".

Example 5: Converting Inches to Feet

Convert 44 in to ft.

Given Length:  44 in
Conversion Factor: 
1 ft/12 in
 
Set up the ratios:
Cancel the labels:
Complete the math:
Result: 3.67 ft  

Example 6: Converting Yards to Feet

Convert 1.25 yd in to ft.

Given Length:  1.25 yd
Conversion Factor: 
3 ft/1 yd
 
Set up the ratios:
Cancel the labels:
Complete the math:
Result: 3.75 ft  

US Standard Conversion Drill

Work with this drill routine until you can consistently make the conversions correctly.

Click the Answer button to see the correct answer to the question.
Click the Next button to see the next question.

Converting Between Metric and US Standard Lengths

Here is the only conversion ratio you must be able to recall. You can use dimensional analysis with two conversion factors to work with any other combination of metric and US terms.

Example 7: Converting Centimeters to Inches

Convert 43 centimeters to inches

Given Length:  43 cm
Conversion Factor:  2.54 cm/in
 
Set up the ratios:
Cancel the labels:
Complete the math:
Result: 16.93 in  

Example 8: Converting feet to centimeters

How many centimeters are in 1 foot?

If you don't happen to recall the number of centimeters in a foot, you can use a 2-step conversion: 

  1. Convert feet to inches using the ratio you know:  2.54 cm/ in
  2. Convert that result to feet using another ratio you know: 12 in/ft

Given Length:  1 ft
Conversion Factors:  12 in/ft, 2.54 cm/in
 
Set up the ratios:
Cancel the labels:
Complete the math:
Result: 30.5 cm  

Example 9: Converting miles to kilometers

How many miles are in 4 kilometers?

Make this a special challenge by assuming you do not recall the number of kilometers in one mile. All you can recall is: 2.54 cm/1 in, 12 in/1 mi, and 5280 ft/mi.

Given Length:  4 km
Conversion Factors: 
     1000 m/km, 100 cm/m, 1 in/2.54 cm, 1 ft/12 in, 1 mi/5280 ft
 
Set up the ratios:
Cancel the labels:
Complete the math:
Result: 2.49 mi  

 

Example 10: Converting millimeters to inches

Convert 150 mm to inches.

Given Length:  150 mm
Conversion Factors:  2.54 cm/in, 10 mm/cm
 
Set up Given Length and Conversion Factor 1:
Attach Conversion Factor 2:
Cancel the labels:
Complete the math:
Result: 0.59 in  

Ans: 150 mm is 0.59 in

Metric/US Standard Conversion Drill

Work with this drill routine until you can consistently make the conversions correctly.

Click the Answer button to see the correct answer to the question.
Click the Next button to see the next question.

 

0406

Lesson 4-6
Mastering Volume Conversions

 

Volume is the measure of the amount of space an object or substance occupies. A grain of sand has a minuscule volume compared with the Rock of Gibraltar; nevertheless, the lowly grain of sand has a measurable volume. Solids, liquids, and gasses that are confined to some sort of container, all occupy space and can be measured in terms of their volume.
 

In the healthcare profession--especially when talking about medications--the term volume refers to liquid measurements.

 

Units of Measure for Liquid Volume
Metric US Standard Apothecary
microliter
milliliter
liter
mcm
ml
L
fluid ounce
pint
quart
gallon
fl. oz
pt
qt
gal
grain
gr

 

NOTES:

You will occasionally see liquid volume expressed in units of cubic centimeters (cc). This isn't a cause for concern, however, because 1 cc is the same as 1 ml.

As a health professional, you won't work with microliters (mcm) very often. Science and technology professions prefer the label mL.

The US Standard ounce is properly called a fluid ounce (fl. oz) to distinguish it from the weight/mass version where 16 ounces (oz) equals one pound (lb).

 

Most Common Conversion Factors for Liquid Volume

Metric US Standard System Conversion
1,000,000 mcm per L

1000 ml per L

16 fl.oz per pt

2 pt per  qt

4 qt per gal

1 qt = 0.946 L

Volume Conversion Map

 

Metric Volume Conversions

Metric volume conversions are relatively easy because the conversion factors are all multiples of 10.  In fact, there is only one purely metric volume conversion:  milliliters to liters and vice-versa.

1000 ml per 1 L
or
1 L per 1000 ml

 

Note: The cubic centimeter (cc) measurement is still used today, but you should easily remember that 1 cc is equivalent to 1 ml.

 

Example 1: Converting Milliliters to Liters

Convert 1250 ml to L

Given Volume:  1250 ml
Conversion Factor:  1 L per 1000 ml
 
Set up the ratios:

1250 ml

x

1 L  
1 1000 ml
Cancel the labels:

1250

x

1 L  
1 1000
Complete the math:

1250

x

1 L = 1.25 L
1 1000
Result: 1.25 L  

Example 2: Converting Liters to Milliliters

Convert 0.055 L to ml

0.055 L

x

1000 ml  
1 1 L

0.055

x

1000 ml  
1 1

0.055

x

1000 ml  = 55 ml
1 1

0.055 L = 55 ml

Note

Notice how you don't have to figure out whether your are supposed to multiply or divide. You simply take the proper conversion factor ( 1000 ml per 1 L, in these examples) and arrange it so the unwanted labels cancel out.  In Example 1, you want to get rid of the ml label, so you flip over the conversion ratio so that the ml label is on the bottom. In Example 2, you want to get rid of the L label, so you flip over the same conversion factor so that L labels cancel.

Examples 3: Converting Between Liters and Cubic Centimeters

3-1. Convert 0.15 L to cc.

0.15

x

1000 cc = 150 cc
1 1

3-2  1280 cc = _____ L

1280

x

1 L

= 1.28 L
1 1000

Metric Volume Drill

Work with this drill routine until you can consistently make the conversions correctly.

Click the Answer button to see the correct answer to the question.

Click the Next button to see the next question.

US Volume Conversions

The US Standard volume measurements most commonly encountered in the healthcare professions are:

16 fl.oz per pt
or
1 pt per 16 fl.oz
2 pt per 1 qt
or
1 qt per 2 pt
4 qt per 1 gal
or
1 gal per 4 qt

Example 4: Converting Fluid Ounces to Pints

Convert 18 fl. oz to pt

Given Volume:  18 fl. oz
Conversion Factor:  1 pt per 16 fl. oz
 
Set up the ratios:

18 fl. oz

x

1 pt  
1 16 fl. oz
Cancel the labels:

18

x

1 pt  
1 16
Complete the math:

18 

x

1 pt = 1.13 pt
1 16
Result: 1.13 pt  

Examples 5: Converting US Volume Measurements

5-1. Convert 0.1 pint to units of fluid ounces.

0.1

x

16 fl. oz = 1.6 fl. oz
1 1

5-2. Convert 10 pt to qt.

10

x

1 qt = 5 qt
1 2

5-3. Convert 0.1 gallon to quarts.

0.1

x

4 qt = 0.4 qt
1 1

Sometimes you have to use multiple conversion ratios, rather that looking up conversion factors you haven't memorized.

5-4. Convert 1600 fluid ounces to gallons.

US Standard Volume Drill

Work with this drill routine until you can consistently make the conversions correctly.

Click the Answer button to see the correct answer to the question.
Click the Next button to see the next question.

 

Metric/US Volume Conversions

It is possible to make up all sort of metric-to-US and US-to-metric volume conversions, but there is just one that stands out from the rest: converting between cc (or ml) and fluid ounces. In fact, if you memorize this conversion ratio you will be able to use multiple conversion ratios to work out any other metric/US volume conversion without having to memorize or look up their ratios.

1 fl. oz per 29.6 ml
or
29.6 ml per fl. oz

Note

Recall that 1 cc = 1 ml, so you can also put the ratio this way:

1 fl. oz per 29.6 cc
or
29.6 cc per fl. oz

Example 6: Converting Fluid Ounces to Cubic Centimeters

Convert 3/4 fl. oz to cc

Given Volume:  0.75 fl. oz
Conversion Factor:  29.6 cc per fl. oz
 
Set up the ratios:

0.75 fl. oz

x

29.6 cc  
1 1 fl. oz
Cancel the labels:

0.75

x

29.6 cc  
1 1
Complete the math:

0.75

x

29.6 cc = 22.2 cc
1 1
Result: 22.2 cc  

Examples 7: Converting Between Metric and US Volume Measurements

7-1. 100 ml = _____ fl. oz

100

x

1 fl. oz = 3.4 fl. oz
1 29.6

 

7-2. 1 L = _____ fl. oz

The conversion ratio for liters and fluid ounces is not ordinarily provided on conversion charts. So it is necessary to convert the liter to another unit that is normally associated with fluid ounces ... and that would be ml.  You should know by now that 1 L = 1000 ml, so substitute that ratio:

So instead of trying to work with liters

1

x

1 fl. oz =
1 ???

work with the equivalent number of ml--a conversion ratio that is more commonly known:

1000

x

1 fl. oz = 33.8 fl. oz
1 29.6

Answer: 1 L = 33.8 fl. oz

 

Fluid Ounces and Cubic Centimeter Drill

Work with this drill routine until you can consistently make the conversions correctly.

Click the Answer button to see the correct answer to the question.
Click the Next button to see the next question.

Household  Volume Measurements

The household volume measurements you are likely to encounter as a health worker are cup (c) , teaspoon (tsp or t), and tablespoon (tbs or T). Here are the conversion ratios:
5 ml per 1 tsp
or
1 tsp per 5 m
l
3 tsp per 1 tbs
or
1 tbs per 3 tsp
8 fl. oz per c
or
1 c per 8 fl. oz

Also remember that units of ml and cc are identical.

Examples 8: Converting Household Volume Measurements

8-1.  The instructions on a bottle of cough medicine call for taking 2 tsp every six hours. How much is each dose in ml?

8-2. How many cubic centimeters in one cup?

Recall that 1 fl. oz = 29.6 cc

8-3. 1 tbs = _____ ml

Household Measurements Drill

Work with this drill routine until you can consistently make the conversions correctly.

Click the Answer button to see the correct answer to the question.
Click the Next button to see the next question.

Topic Summary: Volume Conversions

Volume Conversions Drill

Work with this drill routine until you can consistently make the conversions correctly.

Click the Answer button to see the correct answer to the question.
Click the Next button to see the next question.

0407

Lesson 4-7
Mastering Mass/Weight Conversions

 

Metric US Standard Apothecary
microgram
milligram
gram
kilogram
mcg
mg
g
kg
ounce
pound
ton
oz
lb
T
grain
gr

 

Note

The label for micrograms (mcg) is used primarily in medicine and pharmacy. It is written as mg in most technical trades and science professions.

 

 

 

Mass (Weight) Conversion Map

 

 

Metric Mass Conversions

 

 

1000 mcg per 1 mg
or
1 mg per 1000 mcg
1000 mg per 1 g
or
1 g per 1000 mg
1000 g per 1 kg
or
1 kg per 1000 g

Example 1: Converting Micrograms to Milligrams

Convert 3500 micrograms to milligrams.

Given Mass:  3500 mcg
Conversion Factor:  1000 mcg per 1 mg
 
Set up the ratios:
Cancel the labels:
Complete the math:
Result: 3.5 mg  

 

Examples 2: Converting Metric Mass

2-1. Convert 0.02 gm to milligrams.

2-2. Express 5 mg in micrograms.

Volume Conversions Drill

Work with this drill routine until you can consistently make the conversions correctly.

Click the Answer button to see the correct answer to the question.
Click the Next button to see the next question.

 

US Standard Mass (Weight) Conversions

 

16 oz per 1 lb
or
1 lb per 16 oz
2000 lb per 1 T
or
1 T per 2000 lb

Example 3. Converting Pounds to Ounces

Convert 35 oz to pounds

Given Mass:  35 oz
Conversion Factor:  1 lb per 16 oz
 
Set up the ratios:
Cancel the labels:
Complete the math:
Result: 2.19 lb  

Mass Conversions Drill

Work with this drill routine until you can consistently make the conversions correctly.

Click the Answer button to see the correct answer to the question.
Click the Next button to see the next question.

 

 

Mixed-Standard Mass Conversions

1 gr per 60 mg
or
60 mg per 1 gr

Examples x: Converting Grains

 

 

 

How many grains in one ounce?

 

 

 

 

0408

Lesson 4-8
Mastering Time and Rate Conversions

 

 

 

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