topic badge

3.01 Income

Lesson
The most common sources of income people receive are wages and salaries. Let's start by defining these terms.

A wage is an amount paid to an employee at a certain rate. It may be calculated at an hourly rate, or based on the quantity of work done (e.g. piecework), or calculated at a certain percentage of commission based on sales.

Often people who earn wages are paid a set rate per hour to work a certain number of hours per week. If they work more than this set number of hours, they are entitled to receive overtime, which means that their rate of pay increases. This is done as an incentive to get people to work outside "normal" hours, such as on weekends or public holidays, or for working longer hours in one day than normal.

 

Common overtime rates

Time-and-a-half: payment to a worker at $1.5$1.5 times their usual hourly rate.

Double time: payment to a worker at $2$2 times their usual hourly rate.

Double time-and-a-half: payment to a worker $2.5$2.5 times their usual hourly rate.

Triple time: payment to a worker $3$3 times their usual hourly rate.

In contrast, a salary is a fixed income that is based on a fixed number of working hours. People normally sign a contract with an agreed salary amount before they start working with a company. An employee may receive a weekly, fortnightly or monthly salary. The employee will receive the same amount of income each time period, regardless of whether they work more or less hours, as it is thought this will be averaged out.

A salary is normally written as an annual amount. However, it can be helpful to calculate how much money will be receive each week, to help set a budget for spending. Similarly, it may be useful to work out how much money will be earned in a year based on the amount paid fortnightly. So let's look at some examples of how to solve questions involving salaries and wages.

 

Remember
  • There are $52$52 weeks in a year.
  • There are $2$2 weeks in a fortnight.
  • There are $26$26 fortnights in a year.
  • Annually means yearly.

 

Note
  • It is more accurate to say that there are $52.18$52.18 weeks in a year, however in order to make calculations easier we will often just use $52$52
  • Similarly, it is more accurate to say that there are $26.09$26.09 fortnights in a year

 

Practice questions

Question 1

Neil is a web designer who runs his own small business. He charges an hourly rate of $\$145$$145.

  1. On call outs between 5pm and 11pm, Neil charges a time and a half rate. What is his hourly rate for callouts between 5pm and 11pm?

  2. How much would Neil charge for a callout between 6pm and 11pm?

  3. On weekends and public holidays Neil charges a double time rate. What is his hourly rate for callouts on weekends and public holidays?

  4. How much would Neil charge for a callout between 10am and 5pm on a Sunday.

Question 2

Han earned $\$53030$$53030 in one year, and worked an average of $25$25 hours per week.

  1. What hourly wage did he earn, to the nearest cent?

  2. How much would his annual salary have to increase by for his equivalent hourly wage to increase by $\$4$$4?

Question 3

Han's annual salary is $\$32630$$32630. Calculate his fortnightly pay after receiving a $5$5% pay rise. Write your answer correct to two decimal places.

Question 4

Calculate the income earned for the following amounts of piecework:

  1. washing $24$24 cars at $\$10$$10 per car

  2. baking $26$26 croissants at $\$7$$7 per croissant

  3. sewing $84$84 shirts at $\$8$$8 per shirt

Question 5

James is paid a monthly base salary of $\$500$$500 and a commission of $5%$5% of the value of the products he sells. Calculate his gross income if his sales for the month are $\$347000$$347000:

So far, we have looked at the two most common income types: wages and salaries, both of which are based on time, that is, work done per hour or per year. We have also looked at additional payments, often associated with wages and salaries, including annual leave loading, overtime, bonuses and special allowances.

Other payment methods are based on an employee's performance at work, rather than time spent on the job. Three types of payments associated with performance are commission, piecework, and royalties.

 

Commission

A commission is paid to an employee based on how much they sell. It is a typical payment method for salespeople selling items like furniture, cars or real estate. Commission is usually paid as a percentage of an employee's total sales. The percentage may be a fixed rate or based on a sliding scale, depending on the amount sold.

Commission payments can make up an employee's entire income, or they can be paid on top of a regular income, called a retainer.  Retainers are typically paid to less experienced salespeople, or in situations where the volume of sales is low, or inconsistent over each pay cycle.

 

Worked examples

Example 1

Stewart is a car salesman and receives a retainer of $\$450$$450 per week and a commission of $3%$3% of the price of every car he sells.

Calculate Stewart's income in a week he sells one car priced at $\$35000$$35000.

Solution:

Stewart's weekly income is the sum of his retainer and his commission. 

weekly income $=$= $\text{retainer }+\text{commission }$retainer +commission
  $=$= $450+\left(3%\quad\text{of}\quad35\quad000\right)$450+(3% of 35 000)
  $=$= $450+\left(0.03\times35000\right)$450+(0.03×35000)
  $=$= $450+1050$450+1050
  $=$= $\$1500$$1500
 
Example 2

Ellie works as a real-estate agent. Her commission is based on a sliding scale of $2.5%$2.5% on the first $\$500000$$500000 of her sales, $1.5%$1.5% on the next $\$1000000$$1000000, and $1%$1% thereafter.

Calculate Ellie's commission on a sale of:

  1. $\$400000$$400000
  2. $\$1250000$$1250000
  3. $\$2150000$$2150000

Solution:

  1. The sale is less than $\$500000$$500000, so we only need to find $2.5%$2.5% of the sale.
    Commission $=$= $2.5%$2.5% $\text{of }$of  $400000$400000
      $=$= $0.025\times400000$0.025×400000
      $=$= $\$10000$$10000
  2. This sale amount is between $\$500000$$500000 and $\$1000000$$1000000, so we find $2.5%$2.5% of the first $\$500000$$500000 and $1.5%$1.5% of the remaining value.
    Commission $=$= $2.5%$2.5% $\text{of }$of $500000$500000 $+$+ $1.5%$1.5% $\text{of }$of  $\left(1250000-500000\right)$(1250000500000)
      $=$= $\left(0.025\times500000\right)+\left(0.015\times750000\right)$(0.025×500000)+(0.015×750000)
      $=$= $12500+11250$12500+11250
      $=$= $\$23750$$23750
  3. This sales is more than $\$1500000$$1500000, so we need to find $2.5%$2.5% of the first $\$500000$$500000, $1.5%$1.5% of the next $\$1000000$$1000000 and $1%$1% of the remaining value.
    Commission $=$= $2.5%$2.5% $\text{of }$of $500000$500000 $+$+ $1.5%$1.5% $\text{of }$of  $1000000$1000000$+$+$1%$1%$\text{of }$of $\left(2150000-1500000\right)$(21500001500000)
      $=$= $\left(0.025\times500000\right)+\left(0.015\times1000000\right)+\left(0.01\times650000\right)$(0.025×500000)+(0.015×1000000)+(0.01×650000)
      $=$= $12500+15000+6500$12500+15000+6500
      $=$= $\$34000$$34000

 

Practice questions

Question 6

James is paid a monthly retainer of $\$800$$800 and a commission of $2%$2% of the value of the products he sells. Calculate his gross income if his sales for the month are $\$261000$$261000:

Question 7

Sally is paid $7%$7% on the first $\$2600$$2600 of goods sold and $1%$1% on any value thereafter. Goods to the value of $\$15400$$15400 are sold.

  1. Calculate the commission earned on the first $\$2600$$2600 of goods sold.

  2. Calculate the commission earned on the amount of goods sold in excess of $\$2600$$2600.

  3. What is the total commission earned?

Piecework

Piecework, as the name implies, is a fixed rate of pay, per item (or piece) produced. This payment method is used as an alternative to a wage or salary for workers who pick, pack, prune or make things.

Workers who earn their income through piecework might include clothing makers or fruit and vegetable pickers. 

 

Worked example

Example 3

Charlotte delivers leaflets in her neighbourhood. She is paid $\$0.13$$0.13 for every leaflet she delivers. How much does Charlotte earn if she delivers $640$640 leaflets?

Solution:

Multiply the piecework rate by the number of leaflets.

Earnings $=$= $0.13\times640$0.13×640
  $=$= $\$83.20$$83.20

 

Practice question

Question 8

Calculate the income earned for the following amounts of piecework:

  1. washing $24$24 cars at $\$10$$10 per car

  2. baking $26$26 croissants at $\$7$$7 per croissant

  3. sewing $84$84 shirts at $\$8$$8 per shirt

Royalties

A royalty is a payment made to the owner of a piece of intellectual property for the ongoing use of their work or product.

For example, a musician may receives royalties for sales of their music or whenever their music is used in a film or performance.

Royalties are paid either as a percentage of total sales or as a fixed price per item. 

Intellectual property

Intellectual property is a legal term that refers to an individual or company's right to ownership of a work or product that they have created. The three main forms of intellectual property protection are patents, trademarks and copyright.

 

Worked example

Example 4

Ainslie is an author. She receives $7.5%$7.5% of the sales of her books. If her book is developed into a film, she receives a further $2.5%$2.5% of the film's gross takings.

If $50000$50000 copies of her book are sold at a price of $\$24.95$$24.95 each, and her book is created into a film which generates $\$650000$$650000 at the movie box-office, calculate her total earnings from royalties.

Solution:

Add together the royalties she receives from book sales and movie takings.

Total earnings  $=$= $\text{royalties from book sales }+\text{royalties from film takings }$royalties from book sales +royalties from film takings
  $=$= $7.5%$7.5% $\text{of }$of $\left(24.95\times50000\right)$(24.95×50000) $+$+ $2.5%$2.5% $\text{of }$of $650000$650000
  $=$= $\left(0.075\times1247500\right)+\left(0.025\times650000\right)$(0.075×1247500)+(0.025×650000)
  $=$= $93562.50+16250$93562.50+16250
  $=$= $\$109812.50$$109812.50

 

Practice question

Question 9

Tina is a musician. Her royalty payments and music sales for one month are outlined in the following table.

Usage Royalty payment Number of downloads/plays
Online store $\$0.49$$0.49 per song purchased $2518$2518
Streaming service $\$0.0079$$0.0079 per song streamed $6670$6670
Radio station $\$7$$7 per song played $29$29
  1. Calculate her income in royalties for the month.

    Give your answer in dollars, to the nearest cent.

Outcomes

1.1.1.2

calculate weekly or monthly wages from an annual salary, and wages from an hourly rate, including situations involving overtime and other allowances and earnings based on commission or piecework

What is Mathspace

About Mathspace