Calculate earnings from wages, salaries and overtime, including conversion between pay periods, hourly rates, penalty rates and allowances

What this dot point is asking

NESA wants you to work out how much a person earns from their job. That involves
a few skills. You need to convert a salary between different pay periods (yearly,
monthly, fortnightly, weekly). You need to find an hourly rate from a weekly wage,
or work the other way. You also need to calculate gross pay when some hours are
paid at higher overtime rates such as time-and-a-half or double time. Finally, you
need to handle the extras that sit on top of ordinary pay: shift loadings (penalty
rates) and allowances for difficult or expensive working conditions.

The answer

There are two ways most Australians are paid, and the maths differs slightly for
each.

• A salary is a fixed amount for a year's work, split into equal payments. A
  salaried worker is paid the same each period regardless of the exact hours
  worked, and usually receives no overtime. Teachers, nurses and accountants are
  typically salaried.
• A wage is paid for each hour worked, at an hourly rate. Work more hours and
  you earn more; work fewer and you earn less. Wage earners are the ones who
  attract overtime. Retail staff, apprentices and casual hospitality workers are
  typically waged.

The single most useful fact is that a year contains a fixed number of each pay
period, and converting between periods is just multiplying or dividing by those
numbers.

Notice that a month is not the same as a fortnight, and four weeks is not a
month. A common slip is to treat a monthly salary as four weekly payments. This is
wrong, because $52 \div 12 \approx 4.33$ weeks make up an average month, not $4$.
Always go through the annual amount. The fan below shows the same $84500
salary converted into each of the three common pay periods.

Here $84500 a year is $7041.67 a month, $3250.00 a fortnight, or
$1625.00 a week. The fortnightly figure is exactly twice the weekly figure,
but the monthly figure is more than four times the weekly figure, because a month
is longer than four weeks.

Converting between pay periods

The conversions all come from the table of periods in a year.

To go the other way (a weekly wage up to a yearly figure), multiply by $52$. The
direction of the arithmetic follows from the period getting longer (multiply) or
shorter (divide).

Hourly rate

The hourly rate is the pay for one hour of ordinary work. From a weekly wage,
divide by the number of hours worked that week:

A standard full-time week in Australia is $38$ hours, so a wage of $1024.80
for a $38$ hour week is an hourly rate of $1024.80 \div 38 = 26.97$, i.e.
$26.97 per hour. Going the other way, an hourly worker on $32.50 per hour
for $38$ hours earns $32.50 \times 38 = 1235.00$, i.e. $1235.00 a week, which
is $1235.00 \times 52 = 64220.00$, i.e. $64220.00 a year.

Overtime: penalty rates

When a wage earner works beyond their ordinary hours, those extra hours are
usually paid at a higher penalty rate. The two standard rates are:

• Time-and-a-half: the ordinary rate multiplied by $1.5$.
• Double time: the ordinary rate multiplied by $2$.

Gross pay for a week with overtime is the sum of the ordinary pay and each block
of overtime, each computed at its own rate:

where $R$ is the ordinary hourly rate. The crucial point is that the penalty
multiplier always applies to the base rate, never to an already-loaded rate.
Time-and-a-half means $1.5 \times R$, full stop; it is not $1.5$ times some
higher figure.

Building gross pay stage by stage

A tradesperson is paid a base rate of $34.60 per hour. In one week they work
the ordinary $38$ hours, plus $4$ hours at time-and-a-half and $3$ hours at
double time. The three panels below build the gross pay one block at a time, so
you can see each penalty rate add to the running total.

Stage 1, the ordinary pay. The $38$ ordinary hours are paid at the base rate:
$38 \times 34.60 = 1314.80$, i.e. $1314.80. This is the foundation that the
overtime sits on top of.

Stage 2, add the time-and-a-half. The $4$ overtime hours at time-and-a-half
earn $4 \times 34.60 \times 1.5 = 207.60$, i.e. $207.60, lifting the running
total to $1314.80 + 207.60 = 1522.40$, i.e. $1522.40. The time-and-a-half
rate per hour is $34.60 \times 1.5 = 51.90$, i.e. $51.90.

Stage 3, add the double time. The $3$ hours at double time earn
$3 \times 34.60 \times 2 = 207.60$, i.e. $207.60, for a final gross of
$1522.40 + 207.60 = 1730.00$, i.e. $1730.00. The double time rate per hour is
$34.60 \times 2 = 69.20$, i.e. $69.20. Notice that $3$ hours of double time
earns the same as $4$ hours of time-and-a-half here, because $3 \times 2 = 4
\times 1.5 = 6$ in base-rate terms.

There is a neat shortcut hiding in this. Counting in base-rate units, the week is
worth $38 + 1.5 \times 4 + 2 \times 3 = 50$ ordinary hours, and
$50 \times 34.60 = 1730.00$ confirms the gross in one line. Converting overtime
into "equivalent ordinary hours" is a fast self-check.

Shift loadings and allowances

On top of ordinary and overtime pay, two more extras appear in NESA questions.

• A shift loading (or penalty loading) is a percentage added for working at
  unsociable times, such as a $15\%$ loading on a night shift. A $15\%$ loading
  turns a base rate of $28.40 into a loaded rate of $28.40 \times 1.15 =
  32.66$, i.e. $32.66 per hour.
• An allowance is an extra payment for difficult, dangerous or costly
  conditions: a height allowance, a remote-area allowance, a meal or tool
  allowance. It can be a flat amount per day or per shift, or a rate per hour.

The method is always the same: work out each piece over a consistent time period
and add them up. The trap is that a percentage loading is taken on the ordinary
pay, while a flat allowance is simply added on.

How exam questions ask about wages

The wording shifts but each version maps to one operation. Learn to translate:

• "Calculate the weekly / fortnightly / monthly pay from a salary." Divide
  the annual salary by $52$, $26$ or $12$.
• "Find the annual salary" from a weekly wage. Multiply the weekly figure by
  $52$.
• "Calculate the hourly rate." Divide the weekly wage by the hours worked.
• "Find the gross pay / weekly earnings" with overtime. Add ordinary pay plus
  each overtime block at its penalty rate; the multiplier always hits the base
  rate.
• "... plus an allowance of $X per day / per hour." Compute the allowance
  over the same period as the rest of the pay and add it on.
• "... with a $15\%$ shift loading." Multiply the ordinary pay for the loaded
  hours by $1.15$ (or add $0.15$ times the ordinary pay).
• "How many hours of overtime did they work?" Work backwards: subtract the
  ordinary pay from the gross, then divide the remainder by the overtime rate per
  hour (not the base rate).
• "Which job pays more?" Reduce both to the same period (usually per week or
  per year) and compare, stating any assumption about regular overtime.

Edge case: a month is not four weeks

It is tempting to find a monthly salary by multiplying the weekly figure by $4$,
but that undercounts. Take the $84500 salary. The true monthly amount is
$84500 \div 12 = 7041.67$, i.e. $7041.67. By contrast, $4$ weeks would give
only $4 \times 1625.00 = 6500.00$, i.e. $6500.00. That is a gap of $541.67
every month. The fix is to always route a conversion through the annual amount and
divide by $12$ for months, never to approximate a month as four weeks.