# Drinking Math

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The difficulty lies not so much in developing new ideas as in escaping from old ones.

— John Maynard Keynes

## Introduction

Figure 1: As a boy, I used to love to see people make layered drinks. The first layered drink I saw as a boy was a zombie. (Source)

As a non-drinker, I have never had much interest in alcohol. That said, alcohol has been a big part of my life. My mother earned her living working in bars, starting as a waitress and eventually managing them. Even in retirement, she works part-time selling pull tabs in bars for local sports groups. I have also been the designated driver for literally hundreds of social occasions. At work, I have had numerous co-workers who have derived much pleasure brewing beer and wine. As a boy, I did find the layered drinks interesting (e.g. Figure 1), but only because they looked cool.

But alcohol must be used in moderation. To help guide people in their consumption, there are numerous charts and tables that tell people how much they can drink and remain under the legal limits. While prowling around the Wolfram Alpha web site, I noticed that they have a Blood Alcohol Content (BAC) calculator. I started looking at the output and I saw that there was some interesting math going on there.

For those who are interested, I have implemented the formulas below in an Excel spreadsheet here. Let's dig …

## Analysis

### BAC Definition

The Wikipedia provides a very complete definition of BAC and I will start there with a slight paraphrasing of their definition.

BAC is the concentration of alcohol in a person's blood. It is most commonly used as a metric of intoxication for legal or medical purposes. It is usually expressed as a fractional percentage in terms of volume of alcohol per liter of blood in the body.

So BAC is unitless quantity because it is the ratio of two volumes. This definition of BAC is not universal. For example, California actually uses grams (gm) of alcohol per deciliter (100 mL = dL) of blood. For this post, we will use the more common ratio of volumes, which is expressed mathematically in Equation 1.

 Eq. 1 $BAC\triangleq \frac{{{V}_{Ethanol}}}{{{V}_{Blood}}}$

### Key Analysis Assumptions

The analysis assumptions in the creation of the BAC tables vary because a person's response to alcohol varies by person. Here are the assumptions that I could identify.

• The alcohol evenly spreads through all the water of the body, not just the blood.
• The amount of water in the body is proportional to body weight.
This assumption is highly variable between individuals. Quoting from the Wikipedia, "In a newborn infant, this may be as high as 75 percent of the body weight, but it progressively decreases from birth to old age, most of the decrease occurring during the first 10 years of life. Gender also affects the percentage of water for an individual. This is because women, on average, have a higher body fat percentage. Higher body fat percentage correlates with lower body water percentages. As such, obesity decreases the percentage of water in the body, sometimes to as low as 45 percent."
• Every drink has the same amount of alcohol.
In fact, the amount of alcohol varies by drink. To simplify the discussion, most tables assume a drink contains about 0.5 oz of alcohol by weight. Wolfram Alpha assumes 0.533 gm per drink. Since ethyl alcohol (ethanol) has a density of 0.789 g/cm3, 0.533 oz (mass) of alcohol is equivalent to 0.648 fluid ounces. This amount of alcohol corresponds roughly to the following common drinks:

• 12 fluid ounces of beer
• 1.25 fluid ounces of 100 proof liquor
• 4 fluid ounces of 25 proof table wine
• The body eliminates alcohol at a fixed rate.
Quoting the Wikipedia, "The rate of elimination in the average person is commonly estimated at .015 to .020 gm/dL per hour, although again this can vary from person to person and in a given person from one moment to another." To convert the units from gm/(dL hr) to the BAC's unitless over volume/volume, we need to apply the density of ethanol as follows: $\left( 0.015-0.020 \right)\cdot \frac{\text{gm}}{\text{dL}\cdot \text{hr}}\cdot \frac{\text{c}{{\text{m}}^{3}}}{\text{0}\text{.789}\cdot \text{gm}} = \left( 0.19\%-0.25\% \right)\frac{1}{\text{hr}}$.

### Modeling BAC

Everything I have been able to find on the web uses the model shown in Equation 2 with different parameters.

 Eq. 2 $BAC=\frac{n\cdot \gamma }{\frac{w\cdot \alpha }{{{\rho }_{{{H}_{2}}O}}}}-\beta \cdot t$

where

• w is the weight in pounds
• n is the number of drinks
• t is the time since consumption (hours)
• α is the percentage of water in a body (weight of water/body weight = ~75%)
• γ is volume of ethanol in a drink (0.648 fluid ounces)
• β is the rate of elimination (~0.021% per hour)
• ρH20 is the density of water (1 gm/cm3 = 0.065198 lb/fluid ounce)

If we substitute the values shown into Equation 2 we get Equation 3.

 Eq. 3 $BAC = 5.633\text{\%} \cdot \frac{n}{w}-0.021\text{\%} \cdot t$

Figure 2 shows the Wolfram Alpha result and Figure 3 shows Equation 3 graphed in Mathcad. The results appear identical.

 Figure 2: BAC Versus Time for 240 lb Man After 6 Drinks. Figure 3: BAC Model in Mathcad.

## Conclusion

I have looked at a number of web sites on BAC levels and they all make different assumptions about their definition of a drink, body water percentages, and elimination rate. The variation in the assumptions all reflect that fact that these characteristics vary by individual. Other important factors, like how long you were drinking and whether you were consuming food while drinking, are completely ignored. Looks to me like it is difficult to know how many drinks you can have and still drive. Sounds like not drinking and driving is still the best way to go.

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### 9 Responses to Drinking Math

1. ellie says:

Hello Mathscinotes,

I am a high school student assigned the task of researching BAC levels for my end of the year pre-calculus project. The formulas you listed are very helpful for me but I was wondering whether you remembered which websites other than Wikipedia that you referenced for those formulas.

Thank you!

• mathscinotes says:

Sure.

All of these web sites discuss what really are slight modifications of Widmark's formula.

Anyway, don't hesitate to ask any questions.

Mathscinotes

2. Jim Larson says:

Way cool.
1. The above is for males. Do you have a formula for females? There is a considerable difference from the many BAC charts I've seen. As for the wolframAlpha.com blood alcohol percentage calculator, for 5 drinks in 3 hours for 180 pound people, it gets 0.09% for males and 0.11% for females. (I wish wolframAlpha gave us at least another digit, e.g. 0.087%, for people who like to play with formulas and compare results, with a disclaimer that these things are off by 0.03 percentage points or whatever, even before considering factors like food intake, age, height, individual variations in metabolic rate, yada).

2. I'm concerned that the Minnesota Drivers Manual says that "If you consume more than one standard drink per hour, your alcohol concentration will increase. The effects of alcohol vary greatly among individuals." (They seem to be using a 0.6 fluid oz of alcohol as the definition of standard drink, which I'll use in the following. I'm using BAC charts in the "Responsible Drinking" book, which, well, for a 180 pound man is BAC = (24 * NumDrinks - 16 * Hours) * 0.001 and for a 130 pound female is BAC = (39.9 * NumDrinks - 16 * Hours) * 0.001 .

Anyone relying on a 1 drink per hour rule is really going to get into trouble -- our 180 pound male drinking one drink/hour will have his BAC increase by 0.01 per hour, so that after 8 hours he will be at the legal limit of 0.08. (whereas he'd think he had a trivial amount of alcohol because he's burning drinks off at the same rate he's drinking them)

Our 130 pound female drinking 1 drink/hour will have her BAC increase by 0.024 per hour; after 8 hours she will be at 0.192 BAC, or 2.4 times the legal limit!!!

I've seen the "1 drink / hour" in other places on the net, and even the intox.com website (maker of breathalysers!), makes the same awful statement: "As a rule of thumb, a person will eliminate one average drink or .5 oz (15 ml) of alcohol per hour" (fortunately their website's calculator is more accurate than that. (The http://intox.com/t-Physiology.aspx page gives the alcohol content of various drinks which one needs to know to do comparison studies, because the BAC calculator asks for the number and type of drinks, e.g. "American beer" or "whiskey sour". I usually use whiskey sour because it is 0.6 fl oz of alcohol -- the definition of "standard drink" I most often use. ).

By the way, I have also made adjustments to the Responsible Drinking formula for a 0.5 oz standard drink, and I get, for 8 drinks in 8 hours,
For the 180# male: BAC = (24*8*(.5/.6) - 16*8)*0.001= 0.032
For the 130# female: BAC = (39.9*8*(.5/.6) - 16*8)*0.001= 0.138 (1.72 X legal limit)

In case you or anyone reading this is as obsessed with this stuff as I am.

• mathscinotes says:

Thanks for the comments. I have not looked at a formula for females. As a non-drinker raised by a woman who managed a bar, I have personally observed the wide variation in alcohol sensitivity between people. Because of this variation, I have not seen a "rule of thumb" that makes me comfortable for everyone. In my own life, I have been the "designated driver" literally hundreds of times. That is the only safe option from my standpoint.

mathscinotes

3. CC says:

When I was in high school I tried to calculate this. I failed to get any kind of answer that made sense; I think now because I didn't know what assumptions the BAC calculations were using. Especially the one about how the alcohol is assumed to spread through the entire water content of the body. That would have explained my order-of-magnitude error!

The main thing I remember from that exercise though, was the relative LD50 toxicity of the various alcohols, and that ethanol was significantly less poisonous than methanol, propanol, butanol, and on up. 1, 3, 4, 5, and higher carbon count alcohols were all poisonous, and 2-carbon alcohol was... well, still poisonous, but much less so.

4. andreas says:

hi im making an app and i need to get the weight part right

• mathscinotes says:

Do you have a specific question?

Mathscinotes

5. Hi there,

I found Jim Larson's second comment especially interesting as I've been thinking about the 1 drink/hour rule as well. My husband likes to quote it, but it just doesn't make sense to me- mathematically or in my own life experiences.

I've made up a spreadsheet to look at the numbers I believe he was discussing, though I used your formula, rather than his. I put in 150 for weight.