Use the blood alcohol content formulae for males and females to estimate BAC by substitution, read BAC from a body-weight table, and use the time formula BAC divided by 0.015 to estimate the hours until BAC reaches zero or a legal limit

What this dot point is asking

Blood alcohol content (BAC) is the topic NESA (the board that runs the HSC) uses to test two skills: whether you can substitute into a given formula and read a value from a table. It does this inside a real and serious context. BAC measures how much alcohol is in the blood. A reading of $0.05$ means $0.05$ grams of alcohol in every $100$ millilitres of blood. You are not asked to work out the formula yourself. You are given the male and female formulae and a separate "hours to wait" formula on the HSC reference sheet. The marks come from putting the right number in the right place, working it out carefully, rounding sensibly and comparing the answer with a legal limit. It also helps to know what the formula is really saying. The top, $10N - 7.5H$, is how much alcohol has gone in (from the $N$ standard drinks) minus how much the body has already removed over the $H$ hours of drinking. The bottom spreads that alcohol through the body, whose size is set by the mass $M$ and a constant that differs by sex. This is why a smaller, lighter person reaches a higher BAC on the same drinks, and why waiting longer lowers it. Marks are lost in a few common ways: reading $H$ as the time since the last drink rather than the hours of drinking, mixing up the male and female constants, forgetting that the "hours to wait" formula divides by $0.015$, and rounding the wait time the wrong way.

The answer

The two BAC formulae, and what each symbol means

NESA gives you two estimation formulae, one for males and one for females, identical except for the constant on the bottom:

Every symbol must be read correctly, because swapping one is the fastest way to lose the marks:

• $N$ is the number of standard drinks consumed. A standard drink contains $10$ grams of pure alcohol; it is not the same as one glass or one can, because a large or strong drink can be more than one standard drink.
• $H$ is the number of hours spent drinking, that is, the length of the drinking session, not the time since the last drink. More hours means the liver has had longer to remove alcohol, which is why $7.5H$ is subtracted on top.
• $M$ is the person's mass in kilograms. A larger mass spreads the alcohol through more body, lowering the BAC, which is why $M$ is on the bottom.
• $\text{BAC}$ is the estimate itself, the grams of alcohol per $100$ mL of blood, normally written to three decimal places.

The result is only an estimate. Real BAC also depends on fitness, food, health and liver function, so the formula gives a guide, not a guarantee, and the safe reading of any borderline answer is to wait longer.

Substituting into a BAC formula

The method is just substitution into a formula, the same skill you use for any other formula on the course. First choose the male or female version to match the person. Then replace $N$, $H$ and $M$ with their numbers. Work out the top and the bottom separately before you divide, because the top is a subtraction and the bottom is a multiplication. Keep all the calculator digits until the last line, then round to three decimal places. Write the formula first, then the line with the numbers in, then the value. That way you earn the method marks even if the final digit slips.

Reading a BAC table

A BAC table saves the substitution by listing the estimate for common combinations of body weight and number of drinks, usually for one hour of drinking. The two inputs are the column (body weight in kilograms) and the row (number of standard drinks); the value where they meet is the BAC. To read it, find the weight column closest to the person, run down to the drinks row, and take the cell value. The table below is built from the male formula with $H = 1$, and the highlighted cell shows that an $80$ kg male who has $4$ standard drinks in an hour is estimated at a BAC of $0.060$.

Two things to notice in any BAC table. Reading across a row, the BAC falls as the body weight rises, because mass is on the bottom of the formula. Reading down a column, the BAC rises sharply with each extra drink, because $N$ is on the top. If a person's exact weight is not a column, use the nearest listed weight, or be guided by the question.

The "hours to wait" formula

The body removes alcohol at a roughly steady rate of about $0.015$ of BAC per hour, no matter how high the BAC started. Because the fall is steady, the time for the BAC to drop back to zero is the starting BAC divided by that rate:

This is why the line in the first diagram is straight, not a curve: a fixed amount, $0.015$, comes off every hour, so the BAC decreases linearly. The same idea finds the wait to reach a legal limit rather than zero: divide the drop you need by $0.015$. For the diagram, a BAC of $0.065$ must lose $0.015$ to reach the $0.05$ limit, which takes $0.015 \div 0.015 = 1$ hour, and must lose the whole $0.065$ to reach zero, which takes $0.065 \div 0.015 = 4.33\ldots$ hours, about $4$ hours and $20$ minutes. Always round a "how long until I can drive" time up, because at the rounded-down time the BAC is still too high.

The NSW legal limits

The wait you calculate only means something against a limit, and in NSW there are three:

• Zero ($0.00$) for all learner and provisional (P1 and P2) licence holders. This means there is no amount they can drink and then legally drive, so for these drivers the relevant wait is always the time for BAC to reach zero.
• Under $0.02$ for drivers of heavy vehicles, public passenger vehicles such as buses and taxis, and vehicles carrying dangerous goods.
• Under $0.05$ for full (unrestricted) licence holders.

A question that says "learner", "P-plater" or "provisional" is signalling the zero limit; "full licence" signals $0.05$. Match the limit to the driver before deciding whether they are under it or how long they must wait.

How exam questions ask about BAC

The wording tells you which formula to use and whether to substitute, read a table, or find a time:

• "Calculate (or estimate) the BAC for ..." with a sex, mass, drinks and hours given means substitute into the matching male or female formula and round, usually to two or three decimal places (do exactly what the question's rounding instruction says).
• "Use the table to find the BAC of ..." means read off the cell where the weight column meets the drinks row; no formula needed.
• "How long should the person wait before driving?" or "estimate the number of hours until the BAC reaches zero" means use $\text{hours} = \dfrac{\text{BAC}}{0.015}$, and round the time up (or to the nearest hour or minute if told).
• "How long until the person can legally drive?" depends on the licence: for a learner or P-plater the limit is zero, so divide the whole BAC by $0.015$; for a full licence the limit is $0.05$, so divide the drop down to $0.05$ by $0.015$.
• "Is the person under the limit?" means compute or read the BAC, then compare it with the right limit ($0.00$, $0.02$ or $0.05$) and state the conclusion.
• "Explain why one person's BAC is higher ..." wants you to point at the formula: a smaller mass $M$ or the female constant $5.5$ (versus $6.8$) gives a smaller denominator and so a larger BAC.

Working out the wait stage by stage

The "hours to wait" calculation is short, but the marks reward laying it out cleanly and choosing the right target. The four stages below take Liam's BAC of $0.065$ from the worked example through to the wait for a full-licence driver, who needs the BAC down to $0.05$, not zero. The key decision is which drop to divide by $0.015$: the whole BAC for a zero limit, or just the part above the limit for a full licence.

Stage 1, find the BAC

From the substitution, Liam's BAC is $0.065$. This is the starting value on the decay line, the height the BAC falls from.

Stage 2, choose the target limit

Liam's friend on a full (unrestricted) licence may drive once the BAC is below $0.05$. So the target is $0.05$, and the drop needed is $0.065 - 0.05 = 0.015$, not the whole $0.065$.

Stage 3, divide the drop by 0.015

The BAC falls $0.015$ each hour, so the time to lose $0.015$ is

Stage 4, interpret and round. A full-licence driver waits about $1$ hour for the BAC to fall from $0.065$ to the $0.05$ limit. A learner or P-plater would instead need the BAC at zero, dividing the whole $0.065$ by $0.015$ to get $4.33\ldots$ hours, about $4$ hours $20$ minutes. Same person, very different wait, because the limit is different. On the decay diagram, the full-licence wait is the short step to where the line meets the dashed $0.05$ guide, while the zero wait runs all the way to the axis.

Why a lighter person reaches a higher BAC

It is worth seeing why the formula behaves the way it does, because "explain" questions ask exactly this. The number of standard drinks $N$ sits on the top, so more drinks raise the BAC directly. The mass $M$ sits on the bottom (the part you divide by), so a heavier person divides the same alcohol by a bigger number and ends up with a lower BAC. The female constant $5.5$ is smaller than the male $6.8$. That makes the bottom of the female formula smaller, so the BAC comes out higher for the same drinks, mass and time. The hours of drinking $H$ appear as $-7.5H$ on the top, so a longer session lowers the estimate, because the body has had more time to remove alcohol. Put all four together. The highest BAC comes from many drinks, a short session, a light body and the female formula. The lowest comes from few drinks, a long session, a heavy body and the male formula.