Overtime
Overtime is paid at a higher rate than normal pay, usually time-and-a-half (\(1.5\) times the normal rate) or double time. This NSW Year 9 Advanced maths subtopic calculates pay that includes overtime and penalty rates (NESA outcome MA5‑FIN‑C‑01).
You will total a week's pay from normal and overtime hours, read timesheets with start and finish times and unpaid lunch breaks, and add allowances. Working backwards, you will find a normal hourly rate or the overtime hours from total pay, find the overtime needed to reach a target, and decide which of two pay offers is better. It is part of Year 9 Advanced Earning Money.
Every question with a fully worked solution.
- Overtime - Video - Hourly Rate and Overtime Rate Watch
Theory
Overtime is work beyond normal hours, paid at a higher penalty rate: time-and-a-half (\(1.5\times\) the normal rate) or double time (\(2\times\)). This Year 9 Advanced maths page covers pay in bands, daily overtime from timesheets, working backwards to the normal rate, and forming equations and inequalities for overtime hours.
The normal rate is the usual hourly rate of pay. Hours beyond the normal hours, or at weekends, at night or on public holidays, are often paid at a penalty rate.
Time-and-a-half pays \(1.5\) times the normal rate. Double time pays \(2\) times the normal rate. Some awards use double time and a half, \(2.5\) times.
Overtime can be counted per week (hours beyond, say, \(38\) a week) or per day (hours beyond, say, \(7.5\) a day). Read the rule carefully.
An allowance is an extra fixed payment, for example for working on site or using your own tools. It is added on, not multiplied by a rate.
Every penalty hour equals some number of equivalent normal hours: one hour at time-and-a-half is worth \(1.5\) normal hours.
| Rate | Multiply the normal rate by |
|---|---|
| normal | \(1\) |
| time-and-a-half | \(1.5\) |
| double time | \(2\) |
| double time and a half | \(2.5\) |
With normal rate \(r\), \(n\) normal hours, \(a\) hours at time-and-a-half and \(d\) hours at double time:
How to work out pay with overtime
- Split the hours into bands, using the rule (per day or per week).
- Find each penalty rate on its own line: \(1.5\times\) and \(2\times\) the normal rate.
- Multiply each band's hours by its rate, add any allowances, and total.
- Check the hours in the bands add up to the hours worked.
Working backwards
- Normal rate from total pay: divide by the equivalent normal hours (Example 3), or let the rate be \(r\) and solve.
- Overtime hours from total pay: subtract the normal pay, then divide by the overtime rate.
- Same pay at a different rate: work out the pay, then divide by the new rate to find the hours.
- Split between two penalty rates: let \(d\) be the double-time hours; the rest are total \(-\,d\). Form an equation and solve.
- Reaching a target: write an inequality and round up to a whole hour (Example 4).
- Comparing offers: work out the weekly pay under each offer; for the break-even, set the two expressions equal.
Split the hours: \(38\) normal, \(4\) at time-and-a-half, and \(46-42=4\) at double time. Work out each penalty rate on its own line.
| \(1.5\times 26.80\) | \(=\) | \(\$40.20\text{ an hour}\) |
| \(2\times 26.80\) | \(=\) | \(\$53.60\text{ an hour}\) |
| \(\text{normal}\) | \(=\) | \(38\times 26.80 = \$1018.40\) |
| \(\text{time-and-a-half}\) | \(=\) | \(4\times 40.20 = \$160.80\) |
| \(\text{double time}\) | \(=\) | \(4\times 53.60 = \$214.40\) |
| \(\text{total}\) | \(=\) | \(\mathbf{\$1393.60}\) |
| Day | Start | Finish | Unpaid lunch |
|---|---|---|---|
| Tue | 7:30 am | 5:00 pm | 30 min |
| Wed | 8:00 am | 4:00 pm | 30 min |
| Thu | 6:30 am | 4:30 pm | 1 h |
Paid hours: Tue \(9.5-0.5=9\), Wed \(8-0.5=7.5\), Thu \(10-1=9\). Overtime is counted day by day: Tue \(1.5\) h, Wed \(0\), Thu \(1.5\) h.
| \(\text{normal}\) | \(=\) | \(3\times 7.5 = 22.5\text{ h}\) |
| \(\) | \(\) | \(22.5\times 28.40 = \$639.00\) |
| \(\text{overtime rate}\) | \(=\) | \(1.5\times 28.40 = \$42.60\) |
| \(\text{overtime}\) | \(=\) | \(3\times 42.60 = \$127.80\) |
| \(\text{total}\) | \(=\) | \(\mathbf{\$766.80}\) |
Wednesday has no overtime: exactly \(7.5\) hours is all at the normal rate.
Turn every hour into equivalent normal hours, then divide.
| \(\text{equivalent hours}\) | \(=\) | \(36+4\times 1.5+2\times 2\) |
| \(\) | \(=\) | \(36+6+4 = 46\) |
| \(\text{normal rate}\) | \(=\) | \(1127\div 46 = \mathbf{\$24.50}\) |
Check: \(36\times 24.50+4\times 36.75+2\times 49\) \(= 882+147+98\) \(= \$1127\) ✓. Or let the rate be \(r\) and solve \(46r = 1127\).
Normal pay is \(38\times 32 = \$1216\) and the double-time rate is \(\$64\). Let \(h\) be the Sunday hours.
| \(1216+64h\) | \(\ge\) | \(1650\) |
| \(64h\) | \(\ge\) | \(434\) |
| \(h\) | \(\ge\) | \(6.78\ldots\) |
\(6\) hours is not enough, so round up: she needs \(\mathbf{7}\) hours. Check: \(6\) h gives \(\$1600\); \(7\) h gives \(\$1664\).
Common pitfalls
Frequently asked questions
What does time-and-a-half mean?
Each hour is paid at \(1.5\) times the normal hourly rate. On a normal rate of \(\$30\) an hour, time-and-a-half is \(\$45\) an hour.
How do you calculate overtime pay?
Multiply the normal rate by the penalty multiplier, \(1.5\) or \(2\), then multiply by the overtime hours. Add this to the pay for the normal hours.
How do you find the normal rate from total pay with overtime?
Convert all the hours into equivalent normal hours, counting each time-and-a-half hour as \(1.5\) and each double-time hour as \(2\), then divide the total pay by that number.
What is a penalty rate?
A higher rate of pay for working outside normal hours, such as overtime, weekends, nights or public holidays.
Is double time twice the hours or twice the pay?
Twice the pay rate. Each hour of double time earns two times the normal hourly rate.