Wages and Salaries
A wage is paid by the hour, while a salary is a fixed amount per year, usually paid weekly, fortnightly or monthly. This NSW Year 9 Advanced maths subtopic calculates and compares both (NESA outcome MA5‑FIN‑C‑01), using \(1\) year \(= 52\) weeks.
You will find weekly pay from hours and rates, including weekend and after-midnight rates and unpaid breaks, and convert a salary to weekly, fortnightly and monthly amounts. You will compare jobs paid in different ways, find an equivalent hourly rate, work out hours from total pay, and show why one monthly pay is more than two fortnightly pays. It is part of Year 9 Advanced Earning Money.
Every question with a fully worked solution.
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Theory
A wage is paid for each hour worked; a salary is a fixed amount for a year. This Year 9 Advanced maths page shows how to work out pay from rosters, convert a salary to weekly, fortnightly and monthly pay using \(1\) year \(=52\) weeks, compare jobs fairly, and set up equations for unknown hours or pay.
A wage is paid at an hourly rate: pay \(=\) rate \(\times\) hours worked. Wages are usually paid weekly or fortnightly.
A salary is a fixed amount per year, written p.a. (per annum). It is the same however many hours are worked, and is usually paid fortnightly or monthly.
For pay calculations, \(1\) year \(=52\) weeks \(=26\) fortnights \(=12\) months. A month is not four weeks, so go through the year to change between weeks and months.
Permanent workers are paid for holidays; casual workers usually earn a higher hourly rate but are paid only for the hours and weeks they work.
To compare jobs, put both on the same basis (per hour, per week or per year) before subtracting.
| From a yearly salary | Calculation |
|---|---|
| weekly pay | salary \(\div 52\) |
| fortnightly pay | salary \(\div 26\) |
| monthly pay | salary \(\div 12\) |
| monthly to weekly | \(\times 12\), then \(\div 52\) |
| equivalent hourly rate | weekly pay \(\div\) hours per week |
How to work out pay
- Find the hours: finish time minus start time, less unpaid breaks, for each day.
- Group hours paid at the same rate and multiply each group by its rate.
- Add the amounts, and round money to the nearest cent at the end.
- Convert to the period asked for using \(52\) weeks, \(26\) fortnights or \(12\) months in a year.
Comparing and working backwards
- Hours from pay: hours \(=\) pay \(\div\) rate.
- Comparing jobs: convert both to the same period first (Example 3). A casual job may run for fewer than \(52\) weeks.
- Two rates and an unknown split: let one group of hours be a pronumeral, write the other as total minus it, form an equation and solve (Example 4).
- Break-even hours: set hourly pay \(\times\) hours equal to the other job's pay. For “earn more than”, solve the inequality and round up to the next whole hour.
- Pay rise: find the increase, divide by the old pay and multiply by \(100\%\); then convert the new salary to the period asked for.
- Monthly versus fortnightly: one month's pay is \(\dfrac{1}{12}\) of a year, which is more than two fortnights, \(\dfrac{2}{26}\) of a year.
Find the paid hours each day first, then the week, then multiply by the rate.
6:45 am to 3:15 pm is \(8\) hours \(30\) minutes \(= 8.5\) h. The break is \(45\) min \(= 0.75\) h.
| \(\text{paid hours a day}\) | \(=\) | \(8.5-0.75 = 7.75\) |
| \(\text{hours a week}\) | \(=\) | \(7.75\times 4 = 31\) |
| \(\text{pay}\) | \(=\) | \(31\times 27.60\) |
| \(\) | \(=\) | \(\mathbf{\$855.60}\) |
Change minutes to a decimal by dividing by \(60\): \(45\) min is \(0.75\) h, not \(0.45\) h.
(i) each week
(ii) each fortnight
(iii) each month.
(iv) He works a \(38\)-hour week. What is his equivalent hourly rate?
| \(\text{(i) weekly}\) | \(=\) | \(83\,720\div 52 = \mathbf{\$1610}\) |
| \(\text{(ii) fortnightly}\) | \(=\) | \(1610\times 2 = \mathbf{\$3220}\) |
| \(\text{(iii) monthly}\) | \(=\) | \(83\,720\div 12 = 6976.666\ldots\) |
| \(\) | \(\approx\) | \(\mathbf{\$6976.67}\) |
| \(\text{(iv) hourly}\) | \(=\) | \(1610\div 38 = 42.368\ldots\) |
| \(\) | \(\approx\) | \(\mathbf{\$42.37}\) |
A fortnight is two weeks, so there are \(26\) fortnights in a year. The monthly pay is more than four weeks' pay because a month is about \(4.33\) weeks.
Compare on the same basis: a year.
| \(\text{Job A}\) | \(=\) | \(6240\times 12 = \$74\,880\) |
| \(\text{Job B}\) | \(=\) | \(44.80\times 35\times 50\) |
| \(\) | \(=\) | \(\$78\,400\) |
| \(\text{difference}\) | \(=\) | \(78\,400-74\,880 = \mathbf{\$3520}\) |
Job B pays \(\$3520\) more. A casual job has no paid holidays, so use the weeks actually worked, \(50\), not \(52\).
Let \(s\) be the Sunday hours. Then he worked \(30-s\) hours on weekdays.
| \(26(30-s)+39s\) | \(=\) | \(871\) |
| \(780-26s+39s\) | \(=\) | \(871\) |
| \(780+13s\) | \(=\) | \(871\) |
| \(13s\) | \(=\) | \(91\) |
| \(s\) | \(=\) | \(\mathbf{7}\) |
He worked \(7\) hours on Sunday. Check: \(23\times 26+7\times 39 = 598+273 = \$871\) ✓
Common pitfalls
Frequently asked questions
What is the difference between a wage and a salary?
A wage is paid for each hour worked, so the pay changes with the hours. A salary is a fixed yearly amount, paid in equal parts whatever the hours.
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 do you convert monthly pay to weekly pay?
Multiply by \(12\) to get the yearly amount, then divide by \(52\). Do not divide by \(4\), because a month is a little more than four weeks.
How do you work out hours worked from start and finish times?
Find the time from start to finish, change any minutes to a decimal of an hour by dividing by \(60\), then subtract unpaid breaks.
Is a casual job better paid than a permanent job?
The hourly rate is usually higher, but casual workers are not paid for holidays. Compare the yearly totals using the weeks each job is actually paid for.