Believe it or not, **math** plays an essential role in respiratory therapy, serving as a foundation for various diagnostic and therapeutic procedures.

In fact, respiratory therapy students are required to learn **various formulas and calculations** throughout their educational journey, including respiratory therapy school, and for the TMC Exam.

This guide provides a list of practice questions on the various formulas, calculations, and equations required for aspiring respiratory therapists.

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## Is Math Required for Respiratory Therapists?

**Yes, math is required for respiratory therapists. It’s essential for calculating medication dosages, oxygen flow rates, and ventilator settings, among other tasks. These calculations ensure accurate and effective patient care, making math skills crucial for anyone entering the field of respiratory care.**

## Respiratory Therapy Math Calculation Practice Questions

**1. Are math and calculations required in respiratory therapy school?**

Yes, respiratory therapy students are required to learn specific calculations that are necessary for becoming a respiratory therapist. In addition, some calculations are also required for the TMC Exam.

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

Ideal body weight (IBW) is the most important formula for the TMC Exam because it is used to calculate a patient’s initial tidal volume setting for mechanical ventilation. This calculation is performed several times on the TMC Exam.

**3. 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

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

The normal value for A-a gradient is usually around 10 mmHg.

**5. After obtaining a patient’s PFT results, you find that 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 the formula for alveolar partial pressure of carbon dioxide (PACO2)?**

PACO2 = VCO2 x 0.863 / VA

**9. What is the formula for calculating minute ventilation (VE)?**

VE = Respiratory Rate x Tidal Volume

**10. What is the formula for calculating alveolar minute ventilation (VA)?**

VA = Respiratory Rate x (Tidal Volume – Deadspace)

**11. What is the formula for airway resistance (Raw)?**

Raw = (PIP – Plateau pressure) / Flow

**12. What is the formula for calculating mean airway pressure (Paw)**

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

**13. What is the formula for work of breathing (WOB)?**

WOB = Change in Pressure x Change in Volume

**14. What is the formula for alveolar-arterial oxygen tension gradient (P(A-a)O2)?**

P(A-a)O2 = PAO2 – PaO2

**15. What is the formula for alveolar oxygen tension (PAO2)?**

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

**16. What is the formula for the arterial/alveolar oxygen tension (a/A) ratio?**

(a/A) Ratio = PaO2/PAO2

**17. What is the formula for arterial oxygen content (CaO2)?**

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

**18. What is the formula for end-capillary oxygen content (CcO2)?**

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

**19. What is the formula for mixed venous oxygen content (CvO2)?**

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

**20. What is the formula for the shunt equation (QS/QT)?**

QS/QT = (CcO2 – CaO2) / (CcO2 – CvO2)

**21. What is the formula for the modified shunt equation (QS/QT)?**

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

**22. What is the formula for arterial-mixed venous oxygen content difference (C(a-v)O2)?**

C(a-v)O2 = CaO2 – CvO2

**23. What is the formula for the oxygen-to-air entrainment ratio (O2:Air)?**

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

**24. What is the formula for arterial oxygen saturation estimation (SaO2)?**

SaO2 = PaO2 + 30

**25. What is the formula for the PaO2/FiO2 ratio?**

P/F Ratio = PaO2 / FiO2

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

VD/VT = (PaCO2 – PeCO2) / PaCO2

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

0.3

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

0.45

**29. What is the formula for the oxygenation index (OI)**

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

**30. What is the formula for oxygen consumption (VO2)?**

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

**31. What is the formula for the oxygen extraction ratio (O2ER)?**

O2ER = (CaO2 – CvO2) / CaO2

**32. What is the formula for the FiO2 estimation when a patient is wearing a nasal cannula?**

FiO2 = 20 + (4 x Liter Flow)

**33. What is the formula for calculating oxygen cylinder duration?**

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

**34. What is the formula for calculating the duration of a liquid oxygen system?**

Duration = (344 x Liquid Weight) / Flow

**35. What is the formula for cardiac output (QT)?**

QT = Heart Rate x Stroke Volume

**36. What is the formula for calculating cardiac output using Fick’s method?**

CO = (O2 Consumption / CaO2 – CvO2)

**37. What is the formula for cerebral perfusion pressure (CPP)?**

CPP = Mean Arterial Pressure – Intracranial Pressure

**38. What is the formula for mean arterial pressure (MAP)?**

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

**39. What is the formula for stroke volume (SV)?**

SV = Cardiac Output / Heart Rate

**40. What is the formula for calculating a patient’s maximum heart rate (HRmax)?**

HRmax = 220 – Age

**41. What is the formula for determining heart rate by looking at an EKG strip?**

HR = 300 / # of large boxes between R waves

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

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

**43. 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

**44. 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

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

70 pack years

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

MV = Respiratory Rate x Tidal Volume

**47. 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

**48. A 78-year-old male patient with a history of COPD has a respiratory rate of 20 and a tidal volume of 650 mL. What is the minute volume?**

13 L/min

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

Partial Pressure = Barometric Pressure x Fractional Concentration of Gas

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

760 x 0.21 = 159.6

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

Partial Pressure = (Barometric Pressure – Water Vapor Pressure) x Fractional Concentration of Gas

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

(760 – 47) x 0.21 = 149.7

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

100 mmHg

**54. What is the formula for respiratory quotient (RQ)?**

RQ = VCO2 / VO2

**55. What is the formula for systemic vascular resistance (SVR)?**

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

**56. What is the formula for pulmonary vascular resistance (PVR)?**

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

**57. What is the formula for static compliance (Cst)?**

Cst = Tidal Volume / (Plateau Pressure – PEEP)

**58. What is the formula for dynamic compliance (Cdyn)?**

Cdyn = Tidal Volume / (Peak Pressure – PEEP)

**59. What is the formula for the deadspace to tidal volume ratio (VD/VT)?**

(VD/VT) = (PaCO2 – PECO2) / PaCO2

**60. What is the formula for estimating the appropriate medication dosage for children?**

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

**61. What is the formula for estimating the appropriate medication dosage for infants?**

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

**62. What is the formula for estimating the appropriate medication dosage for infants and children using Fried’s rule?**

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

**63. What is the formula for calculating the anion gap?**

Anion Gap = Na+ – (Cl- + HCO3-)

**64. What is the formula for calculating body surface area (BSA)?**

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

**65. What is the formula for elastance?**

Elastance = Change in Pressure / Change in Volume

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

47 mmHg

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

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

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

Heart rate (HR) and stroke volume (SV)

**69. What is the formula for cardiac index (CI)?**

CI= Cardiac Output / BSA

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

Less than 2.2

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

SI = Stroke Volume / BSA

**72. What is the Fick equation?**

CO = VO2 / CaO2 – CvO2

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

It measures the percentage of deadspace in the tidal volume that does not participate in gas exchange.

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

20-40%; or up to 60% for patients on the ventilator

**75. What is the formula for estimating the proper suction catheter size?**

Catheter Size = (Internal Diameter / 2) x 3

**76. What is the formula for estimating the proper endotracheal tube size for children?**

Tube Size = (Age + 16) / 4

**77. What is the formula for Boyle’s law?**

P1 x V1 = P2 x V2

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

V1 / T1 = V2 / T2

**79. What is the formula for Gay-Lussac’s law?**

P1 / T1 = P2 / T2

**80. What is the formula for LaPlace’s law?**

P = (2 x Surface Tension) / Radius

**81. What is the formula for converting Celsius to Fahrenheit?**

˚F = (˚C x 1.8) + 32

**82. What is the formula for converting Fahrenheit to Celsius?**

˚C = (˚F – 32) / 1.8

**83. What is the formula for converting Celsius to Kelvin?**

K = ˚C + 273

**84. What is the formula for calculating the actual flow for heliox (He/O2)?**

Actual Flow = Given Flow Rate x Factor

**85. What are the formulas for total lung capacity (TLC)?**

TLC = IRV + VT + ERV + RV

TLC = VC + RV

TLC = IC + FRC

**86. What are the formulas for vital capacity (VC)?**

VC = IRV + VT + ERV

VC = IC + ERV

VC = TLC – RV

**87. What are the formulas for inspiratory capacity (IC)?**

IC = IRV + VT

IC = TLC – FRC

IC = VC – ERV

**88. What are the formulas for functional residual capacity (FRC)?**

FRC = ERV + RV

FRC = TLC – IC

**89. What is the formula for the time constant (t)?**

t = Compliance x Resistance

**90. What is the formula for ideal body weight (IBW)?**

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

**91. What is the formula for calculating tidal volume (VT)?**

VT = Flow rate x Inspiratory Time

**92. What is the formula for calculating exhaled tidal volume (VT)?**

VT = Minute Ventilation / Frequency

**93. What is the formula for the corrected tidal volume (VT)?**

VT = Expired Tidal Volume – Tube Volume

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

PeCO2

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

Vd/Vt x Vt

**96. What is the tubing compliance when the measured volume is 100 mL and the static pressure is 65 cmH2O?**

1.5 mL/cm H2O

**97. 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 a breath delivery is 28 cmH2O. What amount of volume was lost in the ventilator tubing?**

60 mL

**98. 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

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

1.2 seconds

**100. 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

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

1.65 seconds

**102. 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 and an I:E ratio of 1:4?**

45 L/min

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

400 mL

**104. What is the formula for determining the level of pressure support ventilation (PSV)?**

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

**105. What is the formula for the rapid shallow breathing index (RSBI)?**

RSBI = Rate / Tidal Volume

**106. What is the formula for calculating the minimum flow rate during mechanical ventilation?**

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

**107. What is Winters’ formula for calculating the expected PaCO2 range?**

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

**108. 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.

**109. A 52-year-old female patient has been smoking 1.5 packs of cigarettes per day for 30 years. What is her smoking history in pack year?**

45 pack years

**110. 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

**111. A female patient who is 5’6″ and weighs 130 lbs has normal lungs, 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

**112. 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

**113. A patient has a respiratory rate of 12, tidal volume of 450 mL, deadspace of 147 mL, and a vital capacity of 1.2 L. What is the alveolar minute ventilation?**

3.6 L/min

**114. A patient is receiving 2 L/min of oxygen from an H-cylinder at 1,500 psi. What is the approximate duration of flow?**

39 hours, 15 minutes

**115. What size suction catheter is most appropriate for an infant who is intubated with a size 3.0 mm endotracheal tube?**

5 Fr

## Final Thoughts

Mastering the **mathematical aspects of respiratory therapy** is a pivotal step in preparing for a successful career in the field of respiratory care.

The practice questions provided in this guide are designed to reinforce your understanding of essential formulas, calculations, and equations, ensuring you’re well-equipped for both the TMC Exam and the practical challenges of respiratory therapy school.

Embracing these mathematical principles will not only aid in your academic achievements but also enhance your ability to provide top-notch patient care in your professional journey.

Written by:

**John Landry, BS, RRT**

John Landry is a registered respiratory therapist from Memphis, TN, and has a bachelor's degree in kinesiology. He enjoys using evidence-based research to help others breathe easier and live a healthier life.

### References

- 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.
- Gertler R. Respiratory Mechanics. Anesthesiol Clin. 2021.