Blood alcohol content

Standard drinks, the BAC formulas for males and females, how long until BAC returns to zero, NSW limits, and why a formula can only ever give a rough estimate.

This is the section where the maths is easy and the stakes are high. Alcohol is a factor in many fatal crashes in NSW every year. The formulas here show how BAC builds up and how slowly it comes down. They also show why no formula can tell you it's safe to drive.

Standard drinks

BAC is the mass of alcohol in your blood: a BAC of $0.05$ means $0.05$ g of alcohol in every $100$ mL of blood. In Australia, one standard drink contains $10$ g of pure alcohol. Drinks come in all sizes and strengths, so the label shows how many standard drinks are in the container. You can also work it out:

Standard drinks

$$S = V \times A \times 0.789$$

$S$ is the number of standard drinks, $V$ is the volume in litres and $A$ is the alcohol content as a percentage (the number in front of the % sign). The $0.789$ comes from the density of alcohol: $1$ mL of it has a mass of $0.789$ g.

Example 1●

How many standard drinks are in a $375$ mL can of beer that is $4.8$% alcohol?

$S = 0.375 \times 4.8 \times 0.789$

V must be in litres: 375 mL is 0.375 L. A is 4.8.

$S \approx 1.4$ standard drinks

Evaluate and round.

Estimating BAC

BAC estimates

$$\text{BAC}_{\text{male}} = \frac{10N - 7.5H}{6.8M} \qquad \text{BAC}_{\text{female}} = \frac{10N - 7.5H}{5.5M}$$

$N$ is the number of standard drinks, $H$ is the number of hours since drinking started, and $M$ is body mass in kg. The top line is grams of alcohol drunk ($10$ per drink) minus the amount the liver has cleared ($7.5$ g an hour). The bottom line reflects how much body water the alcohol spreads through. Women have on average less body water per kilogram, so the same drinks give a higher BAC.

Example 2●●

Jess weighs $60$ kg and has $4$ standard drinks over $2$ hours. Estimate her BAC to three decimal places.

$\text{BAC} = \dfrac{10 \times 4 - 7.5 \times 2}{5.5 \times 60}$

Use the female formula with N = 4, H = 2 and M = 60.

$= \dfrac{25}{330} \approx 0.076$

Top: 40 minus 15 is 25. Bottom: 5.5 times 60 is 330.

That's well over $0.05$, so she must not drive.

Compare with the 0.05 limit for a full licence.

How long until it's gone?

Your liver clears alcohol at a roughly fixed rate. Coffee, cold showers and sleep don't speed it up. A common estimate is that BAC falls by about $0.015$ each hour, so:

Time for BAC to reach zero

$$\text{hours} = \frac{\text{BAC}}{0.015}$$

Example 3●●

Jess's BAC is about $0.076$ when she stops drinking at $10$ pm. Estimate when it reaches zero.

$\text{hours} = \dfrac{0.0757\ldots}{0.015} \approx 5.05$ h

Use her unrounded BAC of 25/330.

About $5$ h $3$ min after $10$ pm, so around $3{:}03$ am.

0.05 of an hour is 3 minutes.

If she starts work at $7$ am she'll be at zero, but driving home at $1$ am would be illegal and dangerous.

She needs to wait at least this long, and longer to be safe, because this is only an estimate.

InteractiveA night out: BAC hour by hour

Try this

Set up a $65$ kg female who has $2$ drinks in each of the first $3$ hours. Read off when her BAC drops below $0.05$ and when it reaches zero. Now move the same $6$ drinks into the first hour. Does the peak change? Does the time to zero change? Then switch to a $90$ kg male with the same drinks.

NSW limits and the limits of the formula

Limit	Who it applies to in NSW
Zero	Learner, P1 and P2 licence holders, and visiting drivers on an overseas learner or provisional licence
Under $0.02$	Drivers of heavy vehicles, public passenger vehicles such as buses and taxis, and vehicles carrying dangerous goods
Under $0.05$	Full (unrestricted) licence holders

The formula is a rough guide only. It assumes all the alcohol is absorbed straight away and that everyone of the same mass and sex is the same. In reality BAC also depends on:

food in the stomach, which slows absorption;
body build (muscle holds more water than fat), age, fitness and liver health;
medications and other drugs;
how fast the drinks were finished, and how big the "drinks" really were.

Two people with the same formula BAC can be very different in real life. That's why police use breath and blood tests, not formulas. The only way to know you're at zero is not to drink before driving.

Your turn

Tom weighs $80$ kg and has $6$ standard drinks. How many hours after he starts drinking will his BAC fall to $0.05$? Give your answer in hours and minutes.

Show the working

Substitute and solve: $0.05 = \dfrac{60 - 7.5H}{6.8 \times 80} = \dfrac{60 - 7.5H}{544}$. Multiply by $544$: $27.2 = 60 - 7.5H$, so $7.5H = 32.8$ and $H \approx 4.373$ h. That is $4$ h and $0.373 \times 60 \approx 22$ min, so about $4$ h $22$ min.

Flashcards

One standard drink in Australia

$10$ g of pure alcohol

Standard drinks formula

$S = V \times A \times 0.789$, with $V$ in litres and $A$ as a percentage

Hours for BAC to reach zero

$\dfrac{\text{BAC}}{0.015}$

NSW BAC limits

Zero for L, P1, P2. Under $0.02$ for heavy and public passenger vehicles. Under $0.05$ for a full licence.

Practice

How many standard drinks are in a $330$ mL bottle of $5$% cider? Give one decimal place.
Estimate the BAC of a $75$ kg male who has had $3$ standard drinks in $1$ hour, to three decimal places.
Estimate the BAC of a $55$ kg female who has had $3$ standard drinks in $1$ hour, to three decimal places.
A person's BAC is $0.09$. About how many hours until it reaches zero?
What is the BAC limit for a P2 driver in NSW?
Which of these does the BAC formula take into account?
Sam stops drinking at midnight with an estimated BAC of $0.12$. Using $\text{hours} = \dfrac{\text{BAC}}{0.015}$, about when is his BAC back to zero?
How many standard drinks are in a $750$ mL bottle of $12$% wine? Give one decimal place.
Exam style: Mei weighs $62$ kg and has $5$ standard drinks over $3$ hours. Estimate her BAC to three decimal places.
Exam style: A $90$ kg male has $7$ standard drinks. How many hours after he starts drinking will his estimated BAC drop to $0.05$? Give two decimal places.