Calculating Maximum Heart Rates by Age

Date: 08/08/2005 at 17:46:05
From: Alex
Subject: Heart Rate

Find the heart rate of a 15 year old at 85% intensity.  I'm not sure
if I am supposed to add or multiply.

Date: 08/08/2005 at 22:51:44
From: Doctor Wilko
Subject: Re: Heart Rate

Hi Alex,

Thanks for writing to Dr. Math!

While the true way of measuring maximum heart rate (MHR) probably 
involves stress tests done by cardiologists/exercise physiologists, 
there are formulas we can use as rules of thumb to estimate our 
maximum heart rates, which in turn will allow us to calculate our 
target/training heart rates (in this case, 85%).

A well-known formula that I've seen used for determining maximum heart 
rate is

  A)  220 - (age) = MHR (in beats per minute)

The true maximum heart rates of individuals probably aren't linear as 
this formula suggests.  This formula may overestimate the MHR of 
younger people and underestimate the MHR of older people.

The author of this website,

  Maximum Heart Rate
    http://www.brianmac.demon.co.uk/maxhr.htm 

also references these formulas:

  B)  206.3 - (0.711 * Age) = MHR

  C)  217  - (0.85 * Age)  = MHR

While formula A above will probably give you the answer you're looking 
for, let's calculate the 85% intensity (of MHR) from all three 
formulas to see how they compare.

To do this, we'll calculate

  1.  The MHR using each of the three formulas above, and then

  2.  Take 85% of each MHR

Formula A:

  220 - 15 = 205 (MHR)

             205 * 0.85 = 174.25 (85% of MHR)
                          ======

Formula B:

  206.3 - (0.711 * 15) = 195.64 (MHR)

                         195.64 * 0.85 = 166.29 (85% of MHR)
                                         ======

Formula C:

  217  - (0.85 * 15)  = 204.25 (MHR)

                        204.25 * 0.85 = 173.61 (85% of MHR)
                                        ======


Using these three formulas as an estimate of an average 15-year-old's 
maximum heart rate, you can say that his/her 85% intensity level is 
somewhere between 166 and 175 beats per minute.

Notice that in each case we found 85% by multiplying the MHR by 0.85,
which is the easiest way to find a percent of a number.


Another useful and probably more accurate formula for determining 
one's training heart rate is the Karvonen Formula, which is based on 
an individual's resting heart rate.  For more on this, check out 
this link,

  Understanding Your Training Heart Rate
    http://www.primusweb.com/fitnesspartner/library/activity/thr.htm 

And here are a couple more links you might find interesting:

  The Walking Site
    http://www.thewalkingsite.com/thr.html 

  Measuring Exercise Intensity
    http://www.ginmiller.com/gmf04/artinfo/targetheart1.htm 

Does this help?  Please write back if you have further questions.

- Doctor Wilko, The Math Forum
  http://mathforum.org/dr.math/