 | 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:
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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. |
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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
5x 3 = 1 x 3 8 58 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 cm1 cm1 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. |
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| 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,
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. |
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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:
- Given Ratio
- Conversion Ratio
- Result Ratio
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. |
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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
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| Metric | US Standard | ||
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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.
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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 |
 | Get the idea? Systematic dimensional analysis (conversion ratios) can save a lot of grief!
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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. |
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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:
- 12 in per 1 ft
- 3 ft per 1 yd
- 5280 ft per 1 mi
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. |
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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.
- 2.54 cm per 1 in
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:
- Convert feet to inches using the ratio you know: 2.54 cm/ in
- 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. |
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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.
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| Metric | US Standard | Apothecary | |||
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| 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.
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| 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: |
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| Cancel the labels: |
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| Complete the math: |
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| 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.
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US Volume Conversions
The US Standard volume measurements most commonly encountered in the healthcare professions are:
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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: |
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| Cancel the labels: |
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| Complete the math: |
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| 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. |
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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.
| Note Recall that 1 cc = 1 ml, so you can also put the ratio this way:
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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: |
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| Cancel the labels: |
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| Complete the math: |
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| 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. |
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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:
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. |
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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. |
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0407
| | Lesson 4-7 Mastering Mass/Weight Conversions |
| Metric | US Standard | Apothecary | |||
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| 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
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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. |
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US Standard Mass (Weight) Conversions
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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. |
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Mixed-Standard Mass Conversions
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Examples x: Converting Grains
How many grains in one ounce?
0408
| | Lesson 4-8 Mastering Time and Rate Conversions |
999
