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# Math Lecture NotesArithmetic Formulas

## Calories

### Basal Metabolic Rate

In 1918, two scientists named J. Arthur Harris and Francis G. Benedict discovered that how many calories a person burns (in one day) while at rest is a feature of metabolism that can be predicted quite accurately by measuring the surface area of the person's skin. They named this resting amount of calorie use the Basal Metabolic Rate, or BMR.

These two scientists then found an estimate for the BMR based on weight, height, age, and gender. This was much more useful because measuring the surface area of a person's skin is difficult. Their formulas were used for more than sixty years.

The Harris-Benedict BMR Equations

Women's BMR = (weight × 4.3) + (height × 4.3) − (age × 4.7) + 655

Men's BMR = (weight × 6.2) + (height × 12.7) − (age × 6.8) + 65

These formulas use weight measured in pounds and height measured in inches. The numbers would be different if we used weight measured in kilograms and height measured in centimeters!

In 1990, these formulas were updated by Mark D. Mifflin and Sachiko T. St. Jeor. For most people these equations are the most accurate. Use these on all class work.

The Mifflin-St.Jeor BMR Equations

Women's BMR = (weight × 4.55) + (height × 15.88) − (age × 5) + 5

Men's BMR = (weight × 4.55) + (height × 15.88) − (age × 5) − 161

These formulas use weight measured in pounds and height measured in inches. The numbers would be different if we used weight measured in kilograms and height measured in centimeters!

Here is a list of these and other BMR formulas translated into SI units, with notes about their original research populations. For example, the The Mifflin-St.Jeor BMR Equations do tend to overestimate the BMR for young Hispanic women and for Asian women.

After strenuous exercise your metabolism stays elevated for a while. This his little long-term effect on younger people. But for older people (over 50) doing resistance exercise as well as aerobic exercise noticeably raises the Basal Metabolic Rate. The BMR for older people can increase by as much as 30% if they are active.

However, be wary of hoping a new exercise program will boost your BMR and DCI (even if you are older). For many people doing additional exercise causes an increase in appetite that nearly cancels out the extra calories burned in the new exercise program.

### Daily Calorie Intake

Fortunately, we are not completely lazy and do move about during the day.

The DCI is how many calories the person would eat each day to maintain their current weight. If the person eats less than this they lose weight; if they eat more than this they gain weight.

A person's DCI is always bigger than their BMR because they move about during the day.

The World Health Organization has discovered that if you ask people to rate their typical activity level that set of categories can be used to pick an appropriate scale factor that turns the BMR into their Daily Calorie Intake, or DCI.

That rating scale even works reasonably well when people are only provided with three options. Here are the scale factors:

couch potatonormal/moderateathletic
Women1.561.641.82
Men1.551.782.10

Thus an athletic woman could eat 1.82 × BMR and maintain her current weight.

Without any fancy equipment you can keep track of what you eat for a few days and determine if you are eating too much or too little.

(For the sake of completeness, know that none of these BMR or DCI formulas work for professional atheletes. Doing that much exercise changes how metabolisms function on a deeper level. The "athletic" category is for people who exercise quite often but are not professional atheletes.)

## Body Fat

### Body Mass Index

Even if you have never heard of BMR and DCI, you have probably seen different acronym about personal health at the doctor's office or on health insurance paperwork.

Your Body Mass Index, or BMI, is your weight (in kilograms) divided by the square of your height (in meters).

We can divide by 2.2 to change a weight in kilograms into a weight in pounds. We can divide by 39.37 to change a height in inches into a height in meters.

Body Mass Index (Ugly Formula)

BMI = (weight ÷ 2.2) ÷ (height ÷ 39.37)2

This formula uses weight measured in pounds and height measured in inches. The weight and height would not be divided by anything if we used weight measured in kilograms and height measured in meters!

Fortunately, this formula has no addition or subtraction. That means we can do some math to isolate the variables of weight and height, and move all of the plain numbers to the end of the formula.

Body Mass Index (Nice Formula)

BMI = weight ÷ height2 × 703

This formula uses weight measured in pounds and height measured in inches. The ×703 would disappear if we used weight measured in kilograms and height measured in meters!

The developers of BMI never intended it to be used to assess individuals. It was designed as a tool to measure statistical differences in groups.

BMI ignores how much body weight is from muscle versus fat, and whether people are "apple shaped" or "pear shaped" (abdominal fat is more strongly corrolated to health issues than thigh fat). For some people BMI is a useful predictor of future health problems, and for other people it simply means they do not skip Leg Days when strength training. But for most people a change in BMI that does not come from increasing exerise should be considered a wake-up call.

Even the Federal government admits that "BMI can be used to screen for weight categories that may lead to health problems but it is not diagnostic of the body fatness or health of an individual."

Nevertheless, some doctors and health insurance providers consider a "healthy" BMI to be between 18.5 and 25.

### Percent Body Fat

Even though it is difficult to measure precisely, percent body fat is a useful enough concept that estimates have been developed. Here is a commonly used formula based on BMI.

Percent Body Fat Estimation

Women = (1.2 × BMI) + (0.23 × age) − 5.4

Men = (1.2 × BMI) + (0.23 × age) + 5.4

This formula uses weight measured in pounds and height measured in inches. The numbers would be different if we used weight measured in kilograms and height measured in centimeters!

Be skeptical about this formula, since it is based upon BMI and BMI was never intended to assess individuals. There are more complicated formulas available online. But none will reliably get closer than ±4%.

## Exercise Pulse Rate

There are standard formulas for estimating a person's heart rate categories. Your maximum safe heart rate decreases with age. When doing aerobic exercise you want your heart rate between 50% and 85% of this maximum.

Heart Rate Formulas

maximum safe heart rate = 220 − age

upper limit for aerobic exercise = (220 − age) × 0.85

lower limit for aerobic exercise = (220 − age) × 0.5

This formula is not accurate for all people. An actual value can be measured at a doctor's office by doing different exercises while monitored by an ECG machine.