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Clinical Calculations: Module 2: Dimensional Analysis

Dimensional Analysis

Module 2 - The Dimensional Analysis Method

 

What’s in this module?

In this module you’ll learn the methodology of dimensional analysis.  This calculation method will allow you to perform even the most complex calculations without an error.  We’ll start with some very basic calculations, so you can see how easy the methodology is. 

Even if you already know how to do clinical calculations using another method, you should take the time to learn to use dimensional analysis.  All the sample solutions in this text will use dimensional analysis.  And, most important, the most difficult problems will be much easier if you use dimensional analysis.

Once you learn how to set up a dimensional analysis equation, there is no need to memorize equations.  No math beyond the basic level is required.


Summary of problem types in this module

We’ll use basic problems to illustrate the dimensional analysis methodology.  The key is to set up the equation so the labels for each quantity can cancel out.


Equivalents to know

In this module we’ll use some equivalents that should already be familiar to you.  These equivalents are included in the household system of measurement in the United States.

Please note that ounce (oz) can be used as both a weight measurement and a volume measurement.

Equivalents can be used with either term in the numerator or denominator since the terms are equal.

Weight:

16 oz = 1 lb

Volume:

3 tsp = 1 Tbsp

2 Tbsp = 1 oz

8 oz = 1 cup

16 oz = 1 pt

2 pt = 1 qt

1 qt = 32 oz


Rounding rules to know

Only these general rounding rules for decimals will apply to this module.

  • If the answer is less than one (1), take the math out three (3) places past the decimal point (the thousandth position) and round to two (2) places past the decimal point (the hundredth position).
  • If the answer is greater than one (1), take the math out two (2) places past the decimal point (the hundredth position) and round to one (1) place past the decimal point (the tenth position).
  • Do not include trailing zeros. (Ex: 12.0 ml would simply be expressed as 12 ml and 0.40 mg would be expressed as 0.4 mg)
  • Always use a leading zero for numbers less than one. (Ex: .25ml should be expressed as 0.25ml)


Starting factors and answer units

The starting factor (SF) is the amount you start with  - the quantity and the units you know.  It is the quantity and unit of measurement to be converted.

The answer unit (AU) is the equivalent quantity expressed in the units that you have available.  The AU is the unit of measurement you have in an amount equivalent to the SF (the amount you know).


How is dimensional analysis done?

Step 1 – Identify the SF and AU for your problem.  The SF will be the first term in your equation.  The AU will be the unit of measurement label for your answer.

Step 2 – Identify the equivalents needed to work from the SF to the AU.

Step 3 – Set up your equation so that measurement labels to be canceled out are in subsequent numerator and denominator positions to allow their cancelation.

Step 4 – Work the equation by first cancelling out unwanted measurement labels.  Next reduce terms of the equation as much as possible.  Multiply remaining terms in the numerator and denominator to form your answer ratio.  Reduce the answer ratio to lowest terms.

Step 5 – If necessary, convert to decimals or whole numbers and apply the appropriate rounding rules.

Step 6 – Always double-check your answer.  Make certain that your answer makes sense as the solution to the problem.  This will be an extra step in preventing medication errors.


Example Problem 1

You want to give your client 8 oz of juice, but you have available 1 pt.  How much of the pint do you want to use?

The SF is the 8 oz of juice you want.  The AU is pt.  You will solve the problem to find out how much of the pint to use.

Here’s the problem set up in the dimensional analysis format:

SF = 8 oz

AU = pt

Equivalent:

16 oz = 1 pt

Equation:

            Set up the equation as 8 ounces over 1 times 1 pint over 16 ounces which equals 8 pints over 16 which reduces to a half pint

Note that the ounce (oz) labels cancel each other.  That leaves the answer unit you want (pt) and a fraction to reduce.

To be eliminated from the equation, or canceled out, a unit of measurement must be in both the numerator and the denominator.

Stated using decimals, the answer would be 0.5 pt.  The rounding rule for numbers less than one applies to this answer.

 
Example Problem 2

In this example a client is required to drink a fluid over a 12-hour period in preparation for a gastrointestinal exam. The fluid fills a gallon container (4 qt).  How many ounces must the client drink?

Here’s the problem set up in the dimensional analysis format:

SF = 4 qt

AU = oz

Equivalent:

1 qt = 32 oz

Equation:

            Set up the equation as 4 quarts over 1 times 32 ounces over 1 quart. When you work out the problem, it equals 128 ounces      

Note that the qt measurement labels cancel each other.  Your answer is a whole number.   

 

Example Problem 3                    

In this example a nurse is working on a labor and delivery unit.  A new baby weighs 115 ounces.  The parents want to know the baby’s weight in pounds.  How many pounds does the baby weigh?  State your answer in pounds.  Use the rounding rule for numbers greater than one.

Here’s the problem set up in the dimensional analysis format:

SF = 115 oz

AU = lb

Equivalent:

16 oz = 1 lb

Equation:

Set up the equation as 115 ounces over 1 times 1 pound over 16 ounces. Since the unit ounces is in both the numerator and denominator, it can be cancelled (crossed out) leaving you with pounds, which is your answer unit. When you work out the problem, you get 115 pounds over 16 ounces. Convert the fraction to a decimal by dividing 115 by 16. That gives you 7.18 pounds which rounds to 7.2 pounds.

 

What if the parents want the weight in pounds and ounces?

Convert the extra 0.2 pounds to ounces:

SF = 0.2 lb

AU = oz

Equivalent:

16 oz = 1 lb

Equation:

Set up the equation as 0.2 pounds over 1 times 16 ounces over 1 pound. The unit pounds can be crossed out, which leaves you with ounces. Since ounces is the answer unit, the equation is set up correctly. When worked out you get an answer of 3.2 ounces. This amount is added to the 7 pounds.

 

Your final answer is 7 lb 3.2 oz

 

Example Problem 4

A pediatric nurse needs to give a child 2 teaspoons of a liquid medication.  The medication container holds 1 ounce of medication.  How much of the container should the nurse give to the child?

Here’s the problem set up in the dimensional analysis format:

SF = 2 tsp

AU = oz

Equivalents:

3 tsp = 1 Tbsp

2 Tbsp = 1 oz

Note that two equivalents are used for this problem.

Equation for the dose in oz:

Set up the equation as 2 teaspoons over 1 times 1 tablespoon over 3 teaspoons times 1 ounce over 2 tablespoons. Since teaspoon is in the numerator and denominator, cross through it. Since tablespoon is in both the numerator and denominator, cross it out. That leaves you with ounce, which is your answer unit. Solve the equation to get 2 ounces over 6, which reduces to one third ounce.

 

The nurse will use 1/3 of the container containing 1 oz.

 

Note that all quantity measurement labels are set up in the numerator and the denominator in order to cancel out the unwanted units of measurement.  This is the key to dimensional analysis no matter how many equivalents you need to use in your equation.

Ready to try the basics of dimensional analysis?  The practice problems will start with identifying SF and AU, then graduate to setting up equations.

Practice Problems

Module 2 Practice Problems

 

What do I do with these problems?

  1. First identify the SF and AU for each problem.  Make certain that you understand this step before you attempt tthe other steps in solving each problem.
  2. Identify the equivalent or equivalents needed to solve the problem.
  3. Set up your equation making certain that you can cancel out unwanted measurement labels.
  4. Solve your equation.
  5. Make certain that you apply the relevant rounding rules.
  6. Make certain that your answer seems reasonable for the problem.

Solutions to all problems appear at the end of this module.  Before you leave this module, make certain that you can successfully complete all the steps listed above.

The purpose of this module is to make you familiar with the dimensional analysis methodology.  Most of the practice problems are easy mathematically, but your purpose is to be able to use the methodology correctly.

 

Practice Problems

1. A client is 5.4 feet tall. How many inches tall is the client?  (12 in = 1 ft)

 

2. An injectable mediation is to be given over 90 seconds.  How many minutes is 90 seconds?  (60 sec = 1 min)

 

3. One tablespoon of cough medicine is to be given to a client.  How many teaspoons should be given?

 

4. A severe wound measures 0.45 feet. How many inches long is the wound?

 

5. A container of gelatin on a client's food tray measures 0.5 pints.  How many ounces area contained in the container?

 

6. How many ounces are in 3 qts?

 

7. How many pints are in 5 qts?

 

8. A client is exercising daily by walking two miles.  How many feet does the client walk daily?  (1 mile = 5280 ft)

 

9. A nurse opens a roll of cloth tape containing 12 yards.  How many feet are in the roll of tape?  (1 yard = 3 ft)

 

10. A client needs to lie flat for 1.5 hours after a medical procedure is performed.  For how many minutes must the client lie flat?  (60 min = 1 hr)

 

11. A child weighs 820 ounces.  How many pounds does the child weigh?

 

12. The dosage strength of a medication is listed as 100 units in one ounce.  How many ounces would the nurse give to the client to administer 75 units?

 

13. A client is being fed liquid nourishment by a tube through the throat to the stomach.  The rate of feeding is 30 ouncs per hour.  How many ml of liquid will the client be fed in one hour?

 

14. A nurse is to give one tablespoon of medication to a client.  How many ounces will the nurse give?

 

15. A nurse has a package of elastic bandage that measures 3 feet.  How many inches of elastic bandage are contained in the package?

 

16. A package of gauze for a dressing weighs 6 ounces.  How many packages of gauze will be contained in a box containing 5 pounds of gauze?  (Round to a whole number)

 

17. A fluid is to be transferred from a one-quart bottle to smaller containers each containing one cup.  How many of the one cup containers will be used?

 

18. A dose of medication can be given every 15 min.  How many doses of the medication can be given in four hours?

 

19. In the first four to six months when a baby isn't eating any solids, here's a simple rule: Offer 2.5 ounces of infant formula per pound of body weight each day.  A newborn baby weighs 6.5 lbs.  How many ounces of formula should be offered every day?

 

20. The skin is the largest organ of the body, with a total area of about 20 square feet.  A client has a burn that measures 24 square inches.  What percentage of the client's body is burned?  (1 square foot = 144 square inches)  Find your answer to four decimal places, convert to a percentage, then use the rounding rule for numbers > 1 or numbers < 1 as appropriate.

Answers to Practice Problems

Answers to Module 2 Practice Problems

 

1. A client is 5.4 feet tall.  How many inches tall is the client?  (12 in = 1 ft)

First identify the SF and AU for the problem.  Make certain that you understand this step before you go further.

SF = 5.4 ft

AU = in

Identify the equivalents you need to solve the problem.

Equivalent:

1 ft = 12 in

Next, set up your equation making certain that you can cancel out unwanted measurement labels.

Equation:

Set up the equation as 5.4 feet over 1 times 12 inches over 1 foot. Then, since foot is both in the numerator and the denominator, you can cross out the unit foot from both. That leaves you with inches, which is your answer unit. When you work out the math of 5.4 over 1 times 12 over 1 you get a final answer of 64.8 inches.

When you solve your equation, make certain that you apply the appropriate rounding rules.  In this case, use the rule for numbers >1.

Also make certain that your answer seems reasonable for the problem.

 

2. An injectable medication is to be given over 90 seconds.  How many minutes is 90 seconds?  (60 sec = 1 min)

SF = 90 sec

AU = min

Equivalent:

1 min = 60 sec

Equation:

Set up the equation as 90 seconds over 1 times 1 minute over 60 seconds. Since the unit seconds is in both the numerator and the denominator, you can cross out both. That leaves you with the unit minutes, which is your answer unit. Work out the problem 90 over 1 times 1 over 60 to get a final answer of 1.5 minutes.

 

3. One tablespoon of cough medicine is to be given to a client.  How many teaspoons should be given?

SF = 1 Tbsp

AU = tsp

Equivalent:

3 tsp = 1 Tbsp

Equation:

Set up the problem as 1 tablespoon over 1 times 3 teaspoons over 1 tablespoon. Since the unit tablespoon is in both the numerator and the denominator, tablespoon can be crossed out. That leaves you with the unit teaspoon, which is your answer unit. When you work out the equation 1 over 1 times 3 over 1 you get a final answer of 3 teaspoons.     

 

4. A severe wound measures 0.45 feet.  How many inches long is the wound?

SF = 0.45 ft

AU = in

Equivalent:

12 in = 1 ft

Equation:

Set up the equation as 0.45 feet over 1 times 12 inches over 1 foot. Since foot is in both the numerator and denominator, you can cross out foot which leaves you with inches. Inches is your answer unit, so you know you have the equation set up correctly. When you work out the equation, you get a final answer of 5.4 inches.      

 

5. A container of gelatin on a client’s food tray measures 0.5 pints.  How many ounces are contained in the container?

SF = 0.5 pt

AU = oz

Equivalent:

16 oz = 1 pt

Equation:

Set up the equation as 0.5 pint over 1 times 16 ounces over 1 pint. Since pint is in both the numerator and denominator, you can cancel (cross out) pint. When you work out the equation, you get a final answer of 8 ounces.

 

6. How many ounces are in 3 qts?

SF = 3 qt

AU = oz

Equivalent:

1 qt = 32 oz

Equation:

Set up the equation as 3 quarts over 1 times 32 ounces over 1 quart. Cancel out quarts, which leaves you with ounces. Ounces is your answer unit. Solve the equation to get 96 ounces.

 

7. How many pints are in 5 qts?

SF = 5 qt

AU = pt

Equivalent:

2 pt = 1 qt

Equation:

Set up the equation as 5 quarts over 1 times 2 pints over 1 quart. Cancel quarts. That leaves you with pints, which is your answer unit. Work out the equation to get a final answer of 10 pints.        

 

8. A client is exercising daily by walking two miles.  How many feet does the client walk daily?  (1 mile = 5280 ft)

SF = 2 mi

AU = ft

Equivalent:

1 mi = 5280 ft

Equation:

Set up the equation as 2 miles over 1 times 5280 feet over 1 mile. Cancel (cross out) miles in both the numerator and denominator. Solve the equation to get 10,560 feet.

 

9. A nurse opens a roll of cloth tape containing 12 yards.  How many feet are in the roll of tape? (1 yard = 3 ft)

SF = 12 yd

AU = ft

Equivalent:

1 yd = 3 ft

Equation:

Set up the equation as 12 yards over 1 times 3 feet over 1 yard. Cancel out yard from both the numerator and denominator. That leaves you with the answer unit of feet. Solve the equation to get an answer of 36 feet.

 

10. A client needs to lie flat for 1.5 hours after a medical procedure is performed.  For how many minutes must the client lie flat?  (60 min = 1 hr)

SF = 1.5 hr

AU = min

Equivalent:

1 hr = 60 min

Equation:

Set up the equation as 1.5 hours over 1 times 60 minutes over 1 hour. Cancel hour from the numerator and denominator. That leaves you with minutes. Solve the equation to get a final answer of 90 minutes.

 

11. A child weighs 820 ounces.  How many pounds does the child weigh?

SF = 820 oz

AU = lb

Equivalent:

1 lb = 16 oz

Equation:

Set up the equation as 820 ounces over 1 times 1 pound over 16 ounces. Cancel out ounces. Solve the equation to get 51.25 pounds which rounds to 51.3 pounds.

Use the rounding rule for numbers >1.

 

12. The dosage strength of a medication is listed as 100 units in one ounce.  How many ounces would the nurse give to the client to administer 75 units?

SF = 75 units

AU = oz

Equivalent:

100 units = 1 oz

Equation:

Set up the equation as 75 units over 1 times 1 ounce over 100 units. Cancel out units, which leaves you with ounce. Ounce is your answer unit, so you know you have the equation set up correctly. Solve the equation to get a final answer of 0.75 ounces.

Use the rounding rule for numbers <1.  Make certain that you use the leading zero.

 

13.  A client is being fed liquid nourishment by a tube through the throat to the stomach.  The rate of feeding is 30 ounces per hour.  How many ml of liquid will the client be fed in one hour?

SF = 30 oz

AU = ml

Equivalent:

1 oz = 30 ml

Equation:

Set up the equation as 30 ounces over 1 times 30 milliliters over 1 ounce. Cancel ounces. That leaves you with milliliters. Milliliters is your answer unit. Solve the equation. The client is to be fed 900 ml in one hour.

 

14. A nurse is to give one tablespoon of medication to a client.  How many ounces will the nurse give?

SF = 1 Tbsp

AU = oz

Equivalent:

2 Tbsp = 1 oz

Equation:

Set up the equation as 1 tablespoon over 1 times 1 ounce over 2 tablespoons. Cancel tablespoons, which leaves ounces. Ounces is your answer unit. Solve the equation. The final answer is 0.5 ounces.

 

15.  A nurse has a package of elastic bandage that measures 3 feet.  How many inches of elastic bandage are contained in the package? 

 SF = 3 ft

AU = in

Equivalent:

1 ft = 12 in

Equation:

Set up the equation as 3 feet over 1 times 12 inches over 1 foot. Cancel feet. That leaves inches which is your answer unit. Solve the equation. The final answer is 36 inches.

 

16. A package of gauze for a dressing weighs 6 ounces.  How many packages of gauze will be contained in a box containing 5 pounds of gauze?  (Round to a whole number)

SF = 5 lb

AU = packages

Equivalents:

1 lb = 16 oz

6 oz = 1 package

Equation:

Set up the equation as 5 pounds over 1 times 16 ounces over 1 pound times 1 package over 6 ounces. Pound is in both the numerator and denominator, so it can be cancelled. Ounces is in both the numerator and denominator, so it can be cancelled. That leaves packages, which is your answer unit. Solve the equation to get 80 packages over 6, which works out to 13.33 and rounds to 13 packages.

Following the instructions in the problem, the answer is a whole number.

 

17.  A fluid is to be transferred from a one-quart bottle to smaller containers each containing one cup.  How many of the one cup containers will be used?

SF = 1 qt

AU = container (cont)

Equivalents:

1 qt = 32 oz

8 oz = 1 cup

1 container = 1 cup

Equation:

Set up the equation as 1 quart over 1 times 32 ounces over 1 quart times 1 cup over 8 ounces times 1 container over 1 cup. Cancel quarts, ounces, and cups since all three occur in both the numerator and denominator. That leaves you with packages. Solve for the packages. The equation works out to 32 containers over 8 which equals 4 containers.

 

18.  A dose of medication can be given every 15 min.  How many doses of the medication can be given in four hours?

SF = 4 hrs

AU = dose

Equivalents:

1 hr = 60 minutes (min)

1 dose = 15 minutes

Equation:

The equation is 4 hours over 1 times 60 minutes over 1 hour times 1 dose over 15 minutes. Cancel hours and minutes, leaving doses. When you solve the equation you get 240 doses over 15, which works out to 16 doses.

 

19. In the first four to six months when a baby isn't eating any solids, here's a simple rule: Offer 2.5 ounces of infant formula per pound of body weight each day.  A newborn baby weighs 6.5 lb.  How many ounces of formula should be offered every day?

SF = 6.5 lb

AU = oz 

Equivalent:

2.5 oz  = 1 lb

Equation:

The equation is 6.5 pounds over 1 times 2.5 ounces over 1 pound. Cancel pounds. That leaves ounces. Solve for ounces. The problem works out to 16.25 ounces which rounds to 16.3 ounces.

Use the rounding rule for numbers >1.

 

20. The skin is the largest organ of the body, with a total area of about 20 square feet.  A client has a burn that measures 24 square inches.  What percentage of the client’s body is burned?  (1 square foot = 144 square inches)  Find your answer to four decimal places, convert to a percentage, then use the rounding rule for numbers >1 or numbers <1 as appropriate.

SF = 24 sq in

AU = body area %

Equivalent:

1 body area = 20 sq ft

1 sq ft = 144 sq in

Equation:

The equation is 24 square inches over 1 times 1 square foot over 144 square inches times 1 body area over 20 square feet. Cancel square inches and square feet, which leaves you with body area. Solve the equation. Directions said to take the math out 4 places past the decimal point. The answer is 0.0083 body area, which equals 0.83% body area.

To state a decimal number as a %, move the decimal place two numbers to the right.

In this case, use the rounding rule for numbers <1.  Don’t forget your leading zero.

 

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