Equations

Lesson

A formula is a convenient way of writing down the method for doing a calculation. It shows how some relevant quantities are related to one another without specifying the actual quantities.

If all but one of the quantities mentioned in a formula are known, then the remaining quantity can be calculated by the rules contained in the formula.

For example, we might have some quantities represented by the letters $A,m,x$`A`,`m`,`x` and $p$`p`, and a formula to show how they are related. The formula could be expressed in words: To obtain $A$`A`, multiply $x$`x` by $p$`p` and add the result to half of $m$`m`. This could be written as

$A=\frac{m}{2}+px$`A`=`m`2+`p``x`.

Note that the letters $p$`p` and $x$`x` are written next to each other. The convention is that letters next to each other are to be multiplied.

Now, suppose we are given the information that in a particular situation the quantities are $m=8$`m`=8, $p=1.5$`p`=1.5 and $x=3$`x`=3. The formula tells us how to calculate the value of $A$`A`. We substitute the given numbers for the letters in the formula. So, we have

$A=\frac{8}{2}+1.5\times3$`A`=82+1.5×3

We multiply $1.5$1.5 by $3$3 and add the result to half of $8$8 and obtain the answer: $A=8.5$`A`=8.5.

There are formulas (or formulae) for many different purposes. We consider two of these that are to do with alcohol consumption.

By describing an alcoholic drink in terms of the number of standard drinks it contains, we are able to compare its alcoholic content with that of other drinks. We can also use the number of standard drinks in further calculations to do with blood alcohol content.

The number of Equivalent Standard Drinks in a given drink depends on the volume of the drink and on the concentration of alcohol in the drink.

We use the letter $N$`N` for the number of Equivalent Standard Drinks; we use $V$`V` for the volume of the drink measured in litres, and the letter $A$`A` represents the concentration given as the percentage of alcohol in a litre of the drink. These are combined by the formula

$N=0.789\times V\times A$`N`=0.789×`V`×`A`

What is the equivalent in standard drinks of a $375$375 mL bottle of beer with an alcohol concentration of $4.5%$4.5% ?

Before substituting the numbers into the formula, we have to express the volume in litres, not millilitres. So, we put $V=0.375$`V`=0.375 because there are $1000$1000 mL in $1$1 L. Now, the calculation is

$N=0.789\times0.375\times4.5$`N`=0.789×0.375×4.5

This works out to be $N=1.33$`N`=1.33 standard drinks, to two decimal places.

The Blood Alcohol Content (BAC) formula predicts what a person's blood alcohol content will be if they consume a given number of standard drinks over a certain number of hours. The BAC also depends on the person's weight and on whether the person is male or female.

There could be other factors that influence a particular person's blood alcohol content. The BAC formula gives an estimate based on the main factors. We give two versions of the formula, one for males and one for females.

$BAC_{\text{male}}=\frac{10N-7.5H}{6.8M}$`B``A``C`male=10`N`−7.5`H`6.8`M`

$BAC_{\text{female}}=\frac{10N-7.5H}{5.5M}$`B``A``C`female=10`N`−7.5`H`5.5`M`

In both formulae, the letter $N$`N` stands for the number of standard drinks consumed, the letter $H$`H` stands for the number of hours over which the drinks were consumed, and $M$`M` stands for the person's mass in kilograms.

Note that if the person's weight is given in pounds, this number must be converted to kilograms before it can be used in the formula. The mass in pounds should be multiplied by $0.45$0.45 to convert it to kilograms, since $1$1lb $=$= $0.45$0.45kg.

A female weighing $158$158 pounds consumes $3$3 standard drinks in $2.5$2.5 hours. What is her predicted blood alcohol content?

Her weight in kilograms is $158\times0.45=71.1$158×0.45=71.1. So, according to the formula,

$BAC_{\text{female}}$BACfemale |
$=$= | $\frac{10\times3-7.5\times2.5}{5.5\times71.1}$10×3−7.5×2.55.5×71.1 |

$=$= | $\frac{30-18.75}{391.05}$30−18.75391.05 | |

$=$= | $0.03$0.03 |

To calculate the number of standard drinks we use the formula $N=0.789\times V\times A$`N`=0.789×`V`×`A`; where

$V$`V` is the volume of the drink in litres (L); and

$A$`A` is the percentage of alcohol (% alc/vol) in the drink.

Calculate the number of standard drinks in a $375$375 mL can of pre-mix drink that has an alcohol content of $7%$7% alc/vol to the nearest decimal place.

To calculate the blood alcohol content (BAC) of a person we use the formula $BAC_{male}=\frac{10N-7.5H}{6.8M}$`B``A``C``m``a``l``e`=10`N`−7.5`H`6.8`M` for males and $BAC_{female}=\frac{10N-7.5H}{5.5M}$`B``A``C``f``e``m``a``l``e`=10`N`−7.5`H`5.5`M` for females; where

$N$`N` is the number of standard drinks consumed;

$H$`H` is the number of hours of drinking; and

$M$`M` is the person's mass in kilograms.

Bill is an adult male that weighs $87$87kg and has consumed $4$4 standard drinks in $4$4 hours. Calculate his blood alcohol content correct to three decimal places.

We want to calculate the BAC of a female with a mass of $53$53kg, who drinks three $425$425ml bottles of full strength beer with an alcohol content of $4.8%$4.8% alc/vol, over $3$3 hours.

First calculate the number of standard drinks correct to one decimal place. Remember we use the formula $N=0.789\times V\times A$

`N`=0.789×`V`×`A`; where$V$

`V`is the volume of the drink(s) in litres (L); and$A$

`A`is the percentage of alcohol (% alc/vol) in the drink(s).Using your answer to part (a), calculate her blood alcohol content (BAC) correct to three decimal places. Remember, we use the formula $BAC_{female}=\frac{10N-7.5H}{5.5M}$

`B``A``C``f``e``m``a``l``e`=10`N`−7.5`H`5.5`M` where;$N$

`N`is the number of standard drinks consumed;$H$

`H`is the number of hours of drinking; and$M$

`M`is the person's mass in kilograms.

Use formulae and equations to solve problems in the workplace