Formulas and Equations

This Year 11 focus area is about using formulas carefully: put the numbers in the right places, use the calculator properly, and give the answer with units and sensible rounding. It was worth 5 marks in NESA's sample HSC paper for the new syllabus, and the same skills are used in almost every other topic.

• Substitution: put values into linear and non-linear formulas and evaluate.
• Speed, distance and time, including changing units, and stopping distances of vehicles.
• Blood alcohol content (BAC): use the BAC formulas, interpret the result, and explain the limitations of BAC estimates.
• Time for BAC to reach zero.
• Medication dosages for children using Fried's, Young's and Clark's formulas.
• Changing the subject of linear formulas and formulas like $y=a{x}^2+c$.
• Word problems: write an equation, solve it and answer in context.
• Spreadsheets: use formulas in a spreadsheet to do repeated calculations.

A formula is a rule linking quantities, such as $A=lw$. The quantity on its own on the left is the subject. To evaluate a formula:

1. Write the formula.
2. Substitute the values, each in brackets if it is negative or a fraction.
3. Evaluate with your calculator in one go, using brackets around a whole numerator or denominator.
4. Round only at the end and write the units.

For a fraction such as $\frac{10N-7.5H}{6.8M}$, type the numerator in brackets, divide, then type the denominator in brackets. Leaving out the brackets is the most common source of wrong answers in this topic.

The units must match. If speed is in km/h, distance must be in km and time in hours. To change km/h to m/s, divide by 3.6; to change m/s to km/h, multiply by 3.6.

A time like 1.875 hours is not 1 hour 87 minutes. Convert the decimal part: $0.875\times 60=52.5$ minutes. So 1.875 h is 1 h 52.5 min, or 1 h 52 min 30 s. The degrees-minutes-seconds key on your calculator does this for you.

Stopping distance

When a driver sees danger, the car keeps moving during the reaction time before the brakes go on. Then it travels a further braking distance.

The reaction distance is speed × reaction time, so the speed must be in m/s if the time is in seconds.

Worked example: stopping distance

A car is travelling at 72 km/h. The driver's reaction time is 1.2 seconds and the braking distance at this speed is 30 m. Find the stopping distance.

Change the speed to m/s first.

The car travels 54 m before stopping. Notice that almost half of it happens before the brakes are even pressed, which is why tiredness, phones and alcohol (all of which slow reactions) are so dangerous.

Blood alcohol content is the amount of alcohol in the blood, measured in grams per 100 mL. In NSW a full-licence driver must have a BAC under 0.05, and learner and provisional drivers must have zero. Drivers of heavy vehicles, buses, taxis and dangerous-goods vehicles must be under 0.02. The estimates are:

Here $N$ is the number of standard drinks, $H$ the number of hours of drinking and $M$ the body mass in kilograms. If the formula gives a negative value, the BAC is 0. To find the time for the BAC to fall to zero:

Worked example: BAC and time to reach zero

A woman with a mass of 64 kg drinks 4 standard drinks over 3 hours. Estimate her BAC at the end of the 3 hours, and how long after that it will take to reach zero.

Use the unrounded BAC in the second step (your calculator's Ans key). Her estimate is just under 0.05, but that does not mean she is safe to drive.

Limitations of BAC estimates

Questions often ask you to explain why a calculated BAC is only an estimate. Good reasons include:

• Real drinks are often bigger than one standard drink, so $N$ is underestimated.
• Food in the stomach, fitness, health, medication and tiredness all change how alcohol is absorbed.
• The formula assumes alcohol leaves the body at a fixed rate, but this varies between people.
• The formulas use averages for males and females and only body mass, not body composition.

Children need smaller doses than adults. Each formula gives the child's dose as a fraction of the adult dose.

Worked example: three dosage formulas

The adult dose of a medicine is 500 mg. Find the dose for a 6-year-old child using Young's formula, and for a child of 28 kg using Clark's formula.

For Fried's formula, a baby of 15 months with an adult dose of 600 mg would get $\frac{15}{150}\times 600=60$ mg. If the age is given in years for Fried's, change it to months first.

To change the subject, undo the operations in reverse order, doing the same thing to both sides. For a formula like $y=a{x}^2+c$, subtract $c$, divide by $a$, then take the square root. In a practical problem, keep only the positive root if the quantity (such as a length or speed) cannot be negative.

Worked example: making a formula work backwards

The braking distance $d$ metres of a truck is modelled by $d=0.01v^2+5$, where $v$ is its speed in km/h. Find the speed if the braking distance is 41 m.

The negative root, $-60$, is rejected because speed is positive here. Say so in your answer.

Word problems

1. Choose a pronumeral and say what it stands for, with units.
2. Write an equation from the words.
3. Solve it, one step per line.
4. Answer the question in a sentence, in context.

Example. An electrician charges an $85 call-out fee plus $72 per hour. A bill came to $337. Let $h$ be the number of hours worked. Then $85+72h=337$, so $72h=252$ and $h=3.5$. The electrician worked 3.5 hours (3 h 30 min).

In a spreadsheet, the same calculation would be typed once as a formula, such as =85+72*A2, and filled down to cost many jobs at once.

• Not using brackets on the calculator for a whole numerator or denominator.
• Mixing units: multiplying km/h by seconds, or using minutes as hours.
• Writing 2.5 hours as 2 h 50 min. It is 2 h 30 min.
• Using the male BAC formula for a female, or the wrong drinks and hours in the wrong places.
• Rounding the BAC early and then using the rounded value to find the time to zero.
• Using Fried's formula with age in years instead of months.
• Keeping a negative square root for a length, speed or time.

• Calculate questions usually give a mark for the correct substitution. Always write the formula with the numbers in it before the answer.
• Explain or interpret questions about BAC want a sentence linked to the context, such as whether the person is under the legal limit and why the estimate may be unreliable.
• Follow the rounding instruction exactly. If none is given, round sensibly for the context (money to cents, doses to 1 decimal place, times to the nearest minute).
• Formulas like BAC and the dosage rules are usually given in the question. Learn what each letter means so you can use them quickly.

• Linear relationships: a formula like $C=72h+85$ is a straight-line model.
• Algebraic relationships (Year 12): non-linear models, including quadratics and exponentials, use the same substitution skills.
• Ratios and rates (Year 12): speed is a rate, and rate conversions build on the unit changes here.
• Earning money: wage and tax calculations are substitution into simple formulas.