Unfortunately, I have some painfully bad news. You will be **required** to do math in respiratory therapy school.

I know, right? The agony.

This includes learning the **formulas and calculations** that are provided here in this guide.

However, I do have some good news as well. These calculations are very easy to perform as long as you know the formula.

If you have the correct formula, you can easily calculate the correct answer simply by plugging the numbers in.

No joke, it’s really that simple.

So, if you’re ready, let’s get started. Keep reading to learn about the **formulas, calculations, and equations** that are required for respiratory therapy students.

Grab your FREE digital copy of this study guide now — no strings attached.

## Respiratory Therapy Formulas and Equations:

**Note:** Respiratory care formulas and equations may vary from one publication to another. However, we have attempted to provide the ones that will be most helpful for the national board exams.

**Minute Ventilation (VE)**

VE = Respiratory Rate x Tidal Volume

**Alveolar Minute Ventilation (VA)**

VA = Respiratory Rate x (Tidal Volume – Deadspace)

**Airway Resistance (Raw)**

Raw = (PIP – Plateau pressure) / Flow

**Mean Airway Pressure (Paw)**

Paw = ((Inspiratory Time x Frequency) / 60) x (PIP – PEEP) + PEEP

**Work of Breathing (WOB)**

WOB = Change in Pressure x Change in Volume

**Alveolar-Arterial Oxygen Tension Gradient (P(A-a)O2)**

P(A-a)O2 = PAO2 – PaO2

**Alveolar Oxygen Tension (PAO2)**

PAO2 = (PB – PH2O) x FiO2 – (PaCO2 / 0.8)

**Arterial/Alveolar Oxygen Tension (a/A) Ratio**

(a/A) Ratio = PaO2/PAO2

**Arterial Oxygen Content (CaO2)**

CaO2 = (Hb x 1.34 x SaO2) + (PaO2 x 0.003)

**End-Capillary Oxygen Content (CcO2)**

CcO2 = (Hb x 1.34 x SaO2) + (PAO2 x 0.003)

**Mixed Venous Oxygen Content (CvO2)**

CvO2 = (Hb x 1.34 x SvO2) + (PvO2 x 0.003)

**Shunt Equation (QS/QT)**

QS/QT = (CcO2 – CaO2) / (CcO2 – CvO2)

**Modified Shunt Equation (QS/QT)**

QS/QT = ((PAO2 – PaO2) x 0.003) / ((CaO2 – CvO2) + (PAO2 – PaO2) x 0.003)

**Arterial-Mixed Venous Oxygen Content Difference (C(a-v)O2)**

C(a-v)O2 = CaO2 – CvO2

**Oxygen-to-Air Entrainment Ratio (O2:Air)**

O2:Air = 1 : (100 – FiO2) / (FiO2 – 2)

**Arterial Oxygen Saturation Estimation (SaO2)**

SaO2 = PaO2 + 30

**PaO2/FiO2 Ratio (P/F Ratio)**

P/F Ratio = PaO2 / FiO2

**Oxygenation Index (OI)**

OI = ((Paw x FiO2) / PaO2) x 100

**Oxygen Consumption (VO2)**

VO2 = Cardiac Output x C(a-v)O2

**Oxygen Extraction Ratio (O2ER)**

O2ER = (CaO2 – CvO2) / CaO2

**FiO2 Estimation for Nasal Cannula**

FiO2 = 20 + (4 x Liter Flow)

**Oxygen Cylinder Duration**

Duration = (Gauge Pressure x Tank Factor) / Liter Flow

**Liquid Oxygen System Duration**

Duration = (344 x Liquid Weight) / Flow

**Cardiac Index (CI)**

CI = Cardiac Output / Body Surface Area

**Cardiac Output (QT)**

QT = Heart Rate x Stroke Volume

**Cardiac Output (CO) Fick’s Method**

CO = (O2 Consumption / CaO2 – CvO2)

**Cerebral Perfusion Pressure (CPP)**

CPP = Mean Arterial Pressure – Intracranial Pressure

**Mean Arterial Pressure (MAP)**

MAP = (Systolic BP + (2 x Diastolic BP)) / 3

**Stroke Volume (SV)**

SV = Cardiac Output / Heart Rate

**Maximum Heart Rate (HRmax)**

HRmax = 220 – Age

**Heart Rate on an EKG Strip (HR) **

HR = 300 / # of large boxes between R waves

**Respiratory Quotient (RQ)**

RQ = VCO2 / VO2

**Systemic Vascular Resistance (SVR)**

SVR = (MAP – CVP) x (80 / Cardiac Output)

**Pulmonary Vascular Resistance (PVR)**

PVR = (MPAP – PCWP) x (80 / Cardiac Output)

**Static Compliance (Cst)**

Cst = Tidal Volume / (Plateau Pressure – PEEP)

**Dynamic Compliance (Cdyn)**

Cdyn = Tidal Volume / (Peak Pressure – PEEP)

**Deadspace to Tidal Volume Ratio (VD/VT)**

(VD/VT) = (PaCO2 – PECO2) / PaCO2

**Children Dosage Estimation**

Child Dose = (Age / Age + 12) x Adult Dose

**Infant Dosage Estimation**

Infant Dose = (Body Weight in lbs / 150) x Adult Dose

**Infant and Children Dosage Estimation (Fried’s Rule)**

Infant or Child Dose = (Age in Months / 150) x Adult Dose

**Anion Gap**

Anion Gap = Na+ – (Cl- + HCO3-)

**Body Surface Area (BSA)**

BSA = ((4 x Body Weight) + 7) / (Body Weight + 90)

**Elastance**

Elastance = Change in Pressure / Change in Volume

**Smoking Use Calculation (Pack Years)**

Pack Years = (Packs Smoked per Day) x (Number of Years Smoked)

**Suction Catheter Size Estimation**

Catheter Size = (Internal Diameter / 2) x 3

**Endotracheal Tube Size Estimation in Children**

Tube Size = (Age + 16) / 4

**Boyle’s Law**

P1 x V1 = P2 x V2

**Charles’s Law**

V1 / T1 = V2 / T2

**Gay-Lussac’s Law**

P1 / T1 = P2 / T2

**LaPlace’s Law**

P = (2 x Surface Tension) / Radius

**Celsius to Fahrenheit Temperature Conversion**

˚F = (˚C x 1.8) + 32

**Fahrenheit to Celsius Temperature Conversion**

˚C = (˚F – 32) / 1.8

**Celsius to Kelvins Temperature Conversion**

K = ˚C + 273

**Helium/Oxygen Conversion (He/O2)**

Actual Flow = Given Flow Rate x Factor

**Total Lung Capacity (TLC)**

TLC = IRV + VT + ERV + RV

TLC = VC + RV

TLC = IC + FRC

**Vital Capacity (VC)**

VC = IRV + VT + ERV

VC = IC + ERV

VC = TLC – RV

**Inspiratory Capacity (IC)**

IC = IRV + VT

IC = TLC – FRC

IC = VC – ERV

**Functional Residual Capacity (FRC)**

FRC = ERV + RV

FRC = TLC – IC

**Time Constant (t)**

t = Compliance x Resistance

**Ideal Body Weight (IBW)**

IBW = 50 kg + (2 x Number of Inches over 5 feet)

**Tidal Volume (VT)**

VT = Flow rate x Inspiratory Time

**Exhaled Tidal Volume (VT)**

VT = Minute Ventilation / Frequency

**Corrected Tidal Volume (VT)**

VT = Expired Tidal Volume – Tube Volume

**Pressure Support Ventilator Setting (PSV)**

PSV = ((Peak Pressure – Plateau Pressure) / Set Flow) x Peak Flow

**Rapid Shallow Breathing Index (RSBI)**

RSBI = Rate / Tidal Volume

**Endotracheal Tube Size Estimation in Children**

Tube Size = (Age + 16) / 4

**Minimum Flow Rate in Mechanical Ventilation**

Flow Rate = Minute Ventilation x I:E Ratio Sum of Parts

**Winters’ Formula**

Expected PaCO2 = (1.5 x HCO3-) + 8 ± 2

## Respiratory Therapy Calculations Practice Questions:

**Note:** This section contains additional practice questions for you to practice and test your knowledge. Going through practice problems is an effective technique for learning and memorizing all the equations.

**1. What is the most important calculation for the TMC Exam?**

Ideal Body Weight (IBW) because it is needed to determine the patient’s initial tidal volume setting for mechanical ventilation.

You will need to calculate a patient’s IBW multiple times on the TMC Exam. We break this down even further inside our **Hacking the TMC Exam** video course.

This course shares our best tips, tricks, and insights to boost your chances of passing the exam.

**2. A patient receiving mechanical ventilation has a PIP of 60 cmH2O and a plateau pressure of 45 cmH2O. The ventilator flow rate is set at 60 L/min. What is the patient’s airway resistance?**

15 cm H2O/L/sec

**3. What is the A-a gradient, and why is it significant?**

The A-a gradient is the alveolar-arterial oxygen gradient and represents the driving force of oxygen from the alveolar sac into the artery.

**4. What is the normal value of the A-a gradient?**

The A-a gradient is usually around 10 mmHg.

**5. After obtaining a patient’s PFT results, they have a VC of 3.4, FRC of 5.8, and an ERV of 1.2. What is the patient’s TLC? **

8.0

**6. The physician has requested the dynamic compliance measurement for an adult patient who is receiving mechanical ventilation. This value can be obtained by dividing the patient’s tidal volume by what?**

(PIP – PEEP)

**7. The doctor has requested the static compliance measurement for an adult patient who is receiving mechanical ventilation. This value can be obtained by dividing the patient’s tidal volume by what?**

(Pplat – PEEP)

**8. What is normal carbon dioxide production?**

200 mL/min

**9. What is the formula for alveolar partial pressure of carbon dioxide (PACO2)?**

PACO2 = VCO2 X 0.863 / VA

**10. What is the formula for deadspace/tidal volume ratio?**

VD/VT = (PaCO2 – PeCO2) / PaCO2

**11. What is the patient’s VD/VT if their PaCO2 is 40 mmHg with a mixed expired CO2 of 28 mm Hg?**

0.3

**12. What is the patient’s VD/VT if their PaCO2 is 58 mmHg with a mixed expired CO2 of 32 mmHg?**

0.45

**13. What is the formula for Boyle’s Law?**

P1 x V1 = P2 x V2

**14. What is the formula for Charles’s law?**

V1 / T1 = V2 / T2

**15. What is the formula for combined gas law?**

(P1 x V1 ) / T1 = (P2 x V2 ) / T2

**16. An H cylinder is half full (full = 2,200), and the patient is receiving oxygen via nasal cannula at 3 L/min. How long will the cylinder last in minutes and in hours? **

1,151 minutes and 19.18 hours

**17. An E cylinder is at 1,400 psi, and the flow rate is 2.5 L/min. How many minutes will the tank last?**

156.8 minutes

**18. If a patient has smoked 2 packs of cigarettes daily for the past 35 years, what would their pack year history be?**

70 pack years

**19. What is the formula for minute ventilation?**

MV = Respiratory Rate x Tidal Volume

**20. A 36-year-old female patient has a respiratory rate of 12 and a tidal volume of 500 mL. What is the minute ventilation?**

6 L/min

**21. A 78-year-old male patient with a history of COPD has a respiratory rate of 20 and tidal volume of 650 mL. What would his minute volume be in Liters?**

13 L/min

**22. What is the formula for partial pressure?**

Partial Pressure = Barometric Pressure x Fractional concentration of Gas

**23. What is the PO2 in dry air at a barometric pressure of 760 mmHg?**

760 x 0.21 = 159.6

**24. How do you calculate the PO2 of humidified air?**

Partial Pressure = (Barometric Pressure – Water Vapor Pressure) x Fractional concentration of gas

**25. What is the PO2 of humidified tracheal air?**

(760 – 47) x 0.21 = 149.7

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**26. If the alveolar gas has a PO2 of 100 mmHg, what is the PO2 of the pulmonary capillary blood?**

100 mmHg

**27. In air, what is the mol percentage of nitrogen?**

78%

**28. In air, what is the mol percentage of oxygen?**

21%

**29. In air, what is the mol percentage of argon?**

1%

**30. At a normal body temperature, what is the partial pressure of water vapor?**

47 mmHg

**31. PA and Pa in the alveolar gas equation represent the gas pressures in what locations?**

PA represents the gas pressure in the alveoli. Pa represents the gas pressure in the artery.

**32. In the alveolar gas equation, what does R represent?**

R represents the V:Q ratio of carbon dioxide and is dependent upon the type of metabolism that a person is undergoing.

**33. What two factors determine cardiac output?**

Heart Rate and Stroke volume

**34. What is the formula for cardiac index?**

CI= Cardiac Output / BSA

**35. A cardiac index below what value can be life-threatening?**

< 2.2

**36. What is the formula for stroke index?**

SI = Stroke Volume / BSA

**37. What is the fick equation?**

CO = VO2 / CaO2 – CvO2

**38. What does the deadspace-to-tidal volume ratio measure?**

It measures the percentage of the tidal volume that is dead space which does not participate in gas exchange.

**39. What is the normal value for (Vd/Vt)?**

20 to 40% (or up to 60% for patients on the ventilator)

**40. What is the formula to Vd/Vt?**

Vd/Vt = (PaCO2 – PeCO2 / PaCO2) x 100

**41. What is the average PCO2 of the exhaled air that can be measured by a capnograph?**

PeCO2

**42. If you know the patient’s tidal volume, but the deadspace must be calculated, what formula should be used?**

Vd/Vt x Vt

**43. What is the tubing compliance when the measured volume is 100 mL, and the static pressure is 65 cm H2O?**

1.5 mL/cm H2O

**44. While setting up a new patient on the ventilator, the plateau pressure is 47 cmH2O, and the tidal volume is set at 100 mL. The average PIP reached during the delivery of a breath is 28 cmH2O. What amount of volume was lost in the ventilator tubing?**

60 mL

**45. What is the average tidal volume for a patient who has a minute ventilation of 10 L/min and a rate of 12/min?**

833 mL

**46. What is the inspiratory time when the tidal volume is set at 800 mL and a flow rate of 40 L/min?**

1.2 seconds

**47. What is the I:E ratio for a ventilator that is set to deliver a tidal volume of 850 mL at a frequency of 15/min with a flow rate of 45 L/min?**

1:2.5

**48. What is the expiratory time when the rate is set to 25/min and the inspiratory time is 0.75 seconds?**

1.65 seconds

**49. What flow rate would be necessary to deliver a tidal volume of 600 mL with a constant waveform at a respiratory rate of 15/min with an I:E ratio of 1:4? **

45 L/min

**50. What tidal volume setting for mechanical ventilation would be appropriate for a 5’2″ female patient with normal lungs?**

400 mL

**51. How should the initial minute ventilation setting be adjusted for an adult patient with a body temperature of 40° C?**

Increase it by 30% because the minute ventilation would have to be increased by 10% for each degree above 37° C.

**52. A 52-year-old female patient has been smoking 1.5 packs of cigarettes per day for 30 years. Her smoking history would be recorded as:**

45 pack years

**53. A patient is receiving 3 L/min of oxygen from an E-cylinder at 1,200 psi. What is the approximate duration of flow?**

112 minutes

**54. A 5-foot, 6-inch-tall 130-lb. female patient with normal lungs has a tidal volume of 480 mL and is breathing at a rate of 12 breaths/min. What is her alveolar ventilation?**

4.20 L/min

**55. A 43-year-old male patient is receiving volume-controlled ventilation at a rate of 12/min. The expiratory time is 3.30 seconds. What is the inspiratory time?**

1.70 seconds

#### Example TMC Practice Question:

**A 61-year-old male patient who weighs 165 lbs is receiving volume control A/C ventilation with a tidal volume of 500 mL. He has the following data:**

PEEP 5

PIP 35

Pplat 30

**What is the patient’s static compliance?**

A. 16.7 mL/cm H2O**B. 20.0 mL/cm H2O**

C. 25.7 mL/cm H2O

D. 30.0 mL/cm H2O

For the TMC Exam, you must know how to calculate both static and dynamic compliance. That means you should know the formulas for both.

**Static Compliance**= Exhaled VT / (Pplat – PEEP)**Dynamic Compliance**= Exhaled VT / (PIP – PEEP)

So for this one, the question asks for the patient’s static compliance. All you have to do is plug the numbers in the formula to get the correct answer:

Static Compliance = 500 / (30 – 5)

Static Compliance = 20

The correct answer is:B. 20.0 mL/cmH2O

This practice question was taken from our **TMC Test Bank**. It’s a massive bank of premium practice questions that cover every topic students must know in order to pass the TMC Exam.

As you can see, each question comes with a detailed rationale that explains exactly why the answer is correct. This is critical when it comes to learning the information.

If you’re interested in getting access to more practice questions (like this one), definitely consider checking it out.

## Final Thoughts

Math doesn’t have to be all that bad! Sure, there are quite a few calculations and equations that you must be familiar with.

However, as previously mentioned, all you have to do is learn and memorize the formulas. Then you can easily plug the numbers in to quickly calculate the correct answer.

This is a requirement for your exams in respiratory therapy school, and it’s also something that you’ll see on the **TMC Exam**. Keep practicing, I know you can do it! Thanks for reading and, as always, breathe easy, my friend.

**Medical Disclaimer:** This content is for educational and informational purposes only. It is not intended to be a substitute for professional medical advice, diagnosis, or treatment. Please consult with a physician with any questions that you may have regarding a medical condition. Never disregard professional medical advice or delay seeking it because of something you read in this article. We strive for 100% accuracy, but errors may occur, and medications, protocols, and treatment methods may change over time.

### References

The following are the sources that were used while doing research for this article:

- Maury, Bertrand. The Respiratory System in Equations (MS&A). 2013th ed., Springer, 2013.
- Chang, David. Respiratory Care Calculations. 4th ed., rtexam, 2018.
- Chang, David. Clinical Application of Mechanical Ventilation. 4th ed., Cengage Learning, 2013.
- Faarc, Kacmarek Robert PhD Rrt, et al. Egan’s Fundamentals of Respiratory Care. 11th ed., Mosby, 2016.

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**Medical Disclaimer:** The information provided by Respiratory Therapy Zone is for educational and informational purposes only. It should not be used as a substitute for professional medical advice, diagnosis, or treatment. Please consult with a physician with any questions that you may have regarding a medical condition.