Earning an Income
Earning an income covers the different ways people are paid: an hourly wage with overtime at time-and-a-half or double time, a salary, commission on sales, a retainer, piece rates and annual leave loading. In NSW Year 10 Advanced maths these are combined in multi-step problems and compared to choose the better deal (NESA outcome MA5‑FIN‑C‑01).
You will calculate weekly pay with overtime and Saturday rates, work backwards from total pay to an hourly rate or to hours of overtime, and find sales from a commission. You will work out 17.5% leave loading, convert between weekly and monthly pay, compare a permanent job with a casual one, and decide whether a pay rise keeps up with rising prices. Real estate agents, couriers, tilers and car salespeople all feature. It is part of Year 10 Advanced Earning Money.
Theory
Earning an income covers the main ways Australians are paid: hourly wages with overtime, annual salaries, commission and piecework. This Year 10 Advanced maths page covers penalty rates, leave loading, tiered commission, successive pay rises and working backwards from a total pay.
A wage is paid per hour worked. Hours above the normal week (often \(38\)) or on weekends attract penalty rates: time-and-a-half pays \(1.5\) times the normal rate and double time pays \(2\) times. A casual loading is an extra percentage paid to casual workers instead of paid leave.
A salary is a fixed amount per year, paid in equal parts weekly (\(\div 52\)), fortnightly (\(\div 26\)) or monthly (\(\div 12\)). Many employees also get four weeks of paid annual leave and leave loading, usually \(17.5\%\) of their normal pay for those four weeks. Superannuation is paid by the employer on top of the salary, as a percentage of it, into a retirement fund.
Commission is a percentage of the value of sales, sometimes with different rates on different tiers of sales. A retainer is a fixed amount paid as well as commission. Piecework pays a set amount for each item completed.
Junior rates are a percentage of the adult rate, depending on age.
| Type | Pay |
|---|---|
| Wage | hours \(\times\) hourly rate |
| Overtime | hours \(\times\) rate \(\times 1.5\) or \(\times 2\) |
| Salary | weekly \(= \dfrac{\text{annual}}{52}\), fortnightly \(= \dfrac{\text{annual}}{26}\), monthly \(= \dfrac{\text{annual}}{12}\) |
| Leave loading | \(17.5\%\times 4\times\) weekly pay |
| Commission | rate \(\times\) sales in each tier, plus any retainer |
| Piecework | number of items \(\times\) rate per item |
| Pay rise of \(r\%\) | new pay \(=\) old pay \(\times\left(1+\dfrac{r}{100}\right)\) |
How to calculate a week's pay
- Find the hours: from start and finish times, subtract unpaid breaks.
- Split the hours into normal time and each penalty rate, as the award describes.
- Multiply each group by its rate and add, or use equivalent hours.
- Add any loading, bonus, retainer or commission, and round to the nearest cent at the end.
Working backwards and comparing
- From total pay to hours or rate: subtract the known parts, then divide by the rate that applies to the rest (Example 3).
- Tiers and thresholds: first work out the pay at the threshold, to see which tier the answer is in.
- Least number of items: if the division is not exact, round up, because rounding down would not reach the target.
- Break-even: to find when two pay options are equal, write each as an expression in the same pronumeral and solve the equation.
- Comparing jobs: compare the whole year: salary, leave loading, casual loading for the weeks worked, and superannuation.
- Successive rises: multiply by each multiplier in turn; the overall rise is not the sum of the percentages (Example 4).
Weekday hours first. 7:45 am to 4:15 pm is \(8.5\) h; take off the unpaid break.
| \(8.5-0.75\) | \(=\) | \(7.75\text{ h a day}\) |
| \(5\times 7.75\) | \(=\) | \(38.75\text{ h}\) |
So \(38\) normal hours and \(0.75\) h at time-and-a-half.
| \(38\times 30.80\) | \(=\) | \(\$1170.40\) |
| \(0.75\times 1.5\times 30.80\) | \(=\) | \(\$34.65\) |
| \(3\times 2\times 30.80\) | \(=\) | \(\$184.80\) |
| \(\text{Pay}\) | \(=\) | \(\mathbf{\$1389.85}\) |
Check with equivalent hours: \(38+0.75\times 1.5+3\times 2 = 45.125\) normal-rate hours, and \(45.125\times 30.80 = 1389.85\).
(i) his weekly and fortnightly pay
(ii) his monthly pay
(iii) his leave loading of \(17.5\%\) on \(4\) weeks' pay.
A year has \(52\) weeks, \(26\) fortnights and \(12\) months.
| \(\text{(i) weekly}\) | \(=\) | \(83\,720\div 52 = \mathbf{\$1610}\) |
| \(\text{fortnightly}\) | \(=\) | \(2\times 1610 = \mathbf{\$3220}\) |
| \(\text{(ii) monthly}\) | \(=\) | \(83\,720\div 12 = 6976.666\ldots\) |
| \(\) | \(\approx\) | \(\mathbf{\$6976.67}\) |
(iii) Leave loading is an extra percentage of the normal pay for the leave weeks.
| \(\text{loading}\) | \(=\) | \(0.175\times 4\times 1610\) |
| \(\) | \(=\) | \(\mathbf{\$1127}\) |
Do not divide by \(4\) to get a monthly amount: a month is about \(4.33\) weeks, not \(4\).
(i) Find his pay in a week with sales of \(\$52\,000\).
(ii) What sales give a pay of \(\$2050\)?
(i) Split the sales at the threshold.
| \(380+0.02\times 45\,000+0.035\times 7000\) | \(=\) | \(380+900+245\) |
| \(\) | \(=\) | \(\mathbf{\$1525}\) |
(ii) Find the pay at exactly \(\$45\,000\) first, to see which tier the sales reach.
| \(380+0.02\times 45\,000\) | \(=\) | \(\$1280\) |
| \(2050-1280\) | \(=\) | \(\$770\) |
| \(770\div 0.035\) | \(=\) | \(22\,000\) |
| \(\text{sales}\) | \(=\) | \(45\,000+22\,000 = \mathbf{\$67\,000}\) |
\(\$770\) was earned at \(3.5\%\), so it came from \(\$22\,000\) of sales above the threshold.
(i) Find her new hourly rate, to the nearest cent.
(ii) Find the overall percentage increase.
(i) A rise of \(r\%\) multiplies the rate by \(1+\dfrac{r}{100}\). The second rise applies to the already-increased rate.
| \(31.20\times 1.028\times 1.035\) | \(=\) | \(33.1961\ldots\) |
| \(\) | \(\approx\) | \(\mathbf{\$33.20}\) |
(ii) The overall multiplier is
| \(1.028\times 1.035\) | \(=\) | \(1.06398\) |
so the overall increase is \(\mathbf{6.398\%}\), slightly more than \(2.8+3.5 = 6.3\%\).
Common pitfalls
Frequently asked questions
How do you calculate time and a half?
Multiply the normal hourly rate by \(1.5\), then by the number of overtime hours. At \(\$30\) an hour, time-and-a-half is \(\$45\) an hour.
How is annual leave loading calculated?
Leave loading is usually \(17.5\%\) of the normal pay for the \(4\) weeks of annual leave. Multiply the weekly pay by \(4\), then by \(0.175\).
How do you convert an annual salary to weekly pay?
Divide the annual salary by \(52\). For fortnightly pay divide by \(26\), and for monthly pay divide by \(12\).
How does tiered commission work?
Different rates apply to different parts of the sales. Split the sales at each threshold, multiply each part by its own rate, then add the parts and any retainer.
What is the difference between a wage and a salary?
A wage is paid for each hour worked, so pay changes with the hours, and overtime is paid at penalty rates. A salary is a fixed yearly amount, paid in equal parts, whatever the hours.