High School Math for Life: Making Sense of Earnings

Salary

Salary refers to the fixed amount of money that an employer pays an employee at regular intervals, typically on a monthly or biweekly basis, for the work they perform. It is a predetermined compensation agreed upon in an employment contract and is not dependent on the number of hours worked or the specific tasks completed, in contrast to hourly wages. Salaries provide employees with financial stability and predictability, making it easier to budget and plan their finances. Employers may base salaries on various factors, including the employee’s qualifications, experience, job responsibilities, and the prevailing market rates for similar positions.

A salary is a fixed annual (yearly) amount that can be paid weekly (\(52\) weeks per year), fortnightly (\(26\) fortnights per year) or monthly (\(12\) months per year). There is no extra pay for hours outside the normal work period but time off in lieu may be arranged.

Question 1

Eva is paid a salary. She receives \( \$530 \) per week. Calculate Eva’s annual salary.

\( \begin{align} \text{1 year} &= \text{52 weeks} \\ \text{Annual salary} &= \$530 \times 52 \\ &= \$27 \ 560 \end{align} \)

Question 2

Tory receives \( \$2140 \) per fortnight. Calculate Tory’s annual salary.

\( \begin{align} \text{1 year} &= \text{26 fortnights} \\ \text{Annual salary} &= \$2140 \times 26 \\ &= \$55 \ 640 \end{align} \)

Question 3

Daniel is paid \( \$5100 \) monthly. Calculate Daniel’s annual salary.

\( \begin{align} \text{1 year} &= \text{12 months} \\ \text{Annual salary} &= \$5100 \times 12 \\ &= \$61 \ 200 \end{align} \)

Question 4

Find the following if the weekly payment is \( \$600 \).

(a)     fortnightly payment

\( \$600 \times \text{2 weeks} = \$1200 \)

(b)     annual payment

\( \$600 \times \text{52 weeks} = \$31 \ 200 \)

(c)     monthly payment

\( \$600 \times \text{52 weeks} \div \text{12 months} = $2600 \)

Question 5

Find the following if the monthly payment is \( \$4200 \).

(a)     annual payment

\( \$4200 \times \text{12 months} = \$50 \ 400 \)

(b)     weekly payment

\( \$50 \ 400 \div \text{52 weeks} = \$969.23 \)

Question 6

Find the following if the fortnightly payment is \( \$1500 \).

(a)     yearly payment

\( \$1500 \times \text{26 fortnights} = \$39 \ 000 \)

(b)     monthly payment

\( \$39 \ 000 \div \text{12 months} = \$3250 \)

Question 7

Find the following if the yearly payment is \( \$62 \ 400 \).

(a)     monthly payment

\( \$62 \ 400 \div \text{12 months} = \$5200 \)

(b)     weekly payment

\( \$62 \ 400 \div \text{52 weeks} = \$1200 \)

Question 8

Tom’s annual salary is \( \$55 \ 620 \). He receives a pay rise of \( 5\% \).

(a)     Find his new annual salary.

\( \$55 \ 620 \times 105\% = \$58 \ 401 \)

(b)     Find his new weekly payment.

\( \$58 \ 401 \div \text{52 weeks} = \$1123.10 \)

(c)     Find how much more he receives per week.

\( \$1123.10-\$55 \ 620 \div \text{52 weeks} = \$53.48 \)

Question 9

John’s monthly salary is \( \$4500 \). He receives a pay reduced by \( 8\% \).

(a)     Find his new annual salary.

\( \$4500 \times \text{12 months} \times 92\% = \$49 \ 680 \)

(b)     Find his new fortnightly payment.

\( \$49 \ 680 \div \text{26 fortnights} = \$1910.77 \)

(c)     Find how much less he receives per fortnight.

\( \$4500 \times \text{12 months} \div \text{26 fortnights}-\$1910.77 = \$166.15 \)

Wage

Wage refers to the payment that an employer provides to an employee in exchange for their labour or services. Unlike a fixed salary, wages are typically calculated based on an hourly rate or a per-unit-of-work basis, such as piecework. Wages are common in industries where employees’ working hours can vary, and they are paid for the actual time spent working. Hourly wages are often subject to overtime pay for hours worked beyond a standard workweek. Wages can vary depending on the job, industry, location, and skill level of the worker, making them a flexible form of compensation.

A wage is a fixed amount paid for each hour’s work for a specified number of hours per week.

Question 10

Jenny is paid \( \$15.20 \) per hour. Find her weekly wage of a \( 30 \) hour week.

\( \$15.20 \times 30 \text{ hours} = \$456 \)

Question 11

Find the number of hours worked if John is paid \( \$412.50 \) and the hourly rate is \( \$16.50 \).

\( \$412.50 \div \$16.50 = 25 \text{ hours} \)

Question 12

Find the hourly rate for a \( 40 \) hour work if the wage is \( \$620 \).

\( \$620 \div 40 \text{ hours} = \$15.50 \)

Question 13

Mike works from \(9.30\)am to \( 2.00 \)pm each day, from Monday to Friday. Find his fortnightly pay if he earns \( \$16.80 \) per hour.

The duration between \( 9.30\)am to \(2.00\)pm is \(4.5\) hours.
\( \$16.80 \times 4.5 \text{ hours} \times 5 \text{ days} \times 2 \text{ weeks} = \$756 \)

Question 14

Kim earns \( \$14.50 \) per hour for a \( 35 \) hour week. Find how much her hourly rate is increased to earn \( \$1120 \) per fortnight.

\( \begin{align} \$1120 \div \text{2 weeks} \div \text{35 hours} &= \$16.00 \\ \$16.00-\$14.50 &= \$1.50 \text{ increase} \end{align} \)

Question 15

Amy is paid \( \$14.20 \) per hour during the week and \( \$18.60 \) per hour on weekends.

\( \begin{array}{|c|c|c|} \hline & \text{Friday} & \text{Saturday} \\ \hline \text{hours worked} & \text{9.30am}-\text{3.45pm} & \text{12.30pm}-\text{4.00pm} \\ \hline \end{array} \)

Find her pay for a week.

The number of hours on Friday is \(6.25\).
The number of hours on Saturday is \( 3.5\).
\( \$14.20 \times \text{6.25 hours}+\$18.60 \times \text{3.5 hours} = \$153.85 \)

Question 16

Tim earns \( \$12.50 \) per hour for a \(20\) hour week. Find how many more hours he needs to work if he wishes to earn \( \$343.75 \) in total per week.

\( \begin{align} \$343.75 \div \$12.50 &= \text{27.5 hours} \\ 27.5-20 &= \text{7.5 hours more} \end{align} \)

Question 17

Ted earns \( \$10.80 \) per hour for a \( 30 \) hour week. He receives a \( 5\% \) increase.

(a)     Find his new hourly rate.

\( \$10.80 \times 1.05 = \$11.34 \)

(b)     Find how much he earns more per week.

\( \$11.34 \times \text{30 hours}-\$10.80 \times \text{30 hours}=\$16.20 \)

Question 18

Kate usually earns \( \$16.00 \) per hour for a \( 19 \) hour week. Find how many more hours she has to work to earn the same weekly pay if she receives a \( 5\% \) pay cut.

\( \begin{align} \text{usual weekly wage: } \$16.00 \times \text{19 hours} &= \$304 \\ \text{new hourly rate: } \$16.00 \times 0.95 &= \$15.20 \\ \$304 \div \$15.20 &= \text{20 hours} \\ \text{20 hours}-\text{19 hours} &= \text{1 hour more} \end{align} \)

Gross Payments, Net Payments and Wage Deductions

Gross payments refer to the total earnings an employee receives before any deductions are made. It includes the employee’s base salary or wages, any overtime pay, bonuses, or other forms of compensation.

Net payments, on the other hand, represent the actual amount an employee takes home after all deductions have been subtracted from their gross earnings. Deductions can include taxes (like income tax and payroll tax), Social Security contributions, retirement fund contributions, health insurance premiums, and any other mandated or voluntary deductions.

Wage deductions, a subset of deductions, specifically refer to the amounts withheld from an employee’s wages. These deductions can vary widely depending on factors like tax laws and benefit programs, and they significantly impact the net payment an employee receives.

Income Tax is the amount the government requires employees to deduct from their gross pay.

Superannuation is money set aside by an employee for their retirement.

$$ \large \text{Gross Payment}-\text{Deduction}=\text{Net Payment}$$

Question 19

John earns a weekly wage of \( \$500 \) and a travel allowance of \( \$40 \) per week. He needs to pay his income tax of \( \$32 \) and superannuation of \( \$45 \).

(a)     Find his gross payment.

\( \$500 + \$40 = \$540 \)

(b)     Find the amount of deductions.

\( \$32+\$45=\$77 \)

(c)     Find his net payment.

\( \$540-\$77=\$463 \)

Question 20

Fill in the blank.

\( \begin{array}{|c|c|c|c|} \hline & \ \ \ \ \text{gross} \ \ \ \ & \text{deduction} & \ \ \ \ \ \ \ \ \text{net} \ \ \ \ \ \ \ \ \\ \hline \text{(a)} & \$560 & \$76 & \\ \hline \text{(b)} & \$871 & & \$764 \\ \hline \text{(c)} & & \$45 & \$513 \\ \hline \end{array} \)

(a)     \( \text{Gross Payment}-\text{Deduction}=\text{Net Payment} \)

\( \$560-\$76=\$484 \)

\( \require{AMSsymbols} \begin{array}{|c|c|c|c|} \hline & \ \ \ \ \text{gross} \ \ \ \ & \text{deduction} & \ \ \ \ \ \ \ \ \text{net} \ \ \ \ \ \ \ \ \\ \hline \text{(a)} & \$560 & \$76 & \bbox[yellow,3px]{\$484} \\ \hline \text{(b)} & \$871 & & \$764 \\ \hline \text{(c)} & & \$45 & \$513 \\ \hline \end{array} \)

(b)     \( \text{Deduction}=\text{Gross Payment}-\text{Net Payment} \)

\( \$871-\$764= \$107 \)

\( \require{AMSsymbols} \begin{array}{|c|c|c|c|} \hline & \ \ \ \ \text{gross} \ \ \ \ & \text{deduction} & \ \ \ \ \ \ \ \ \text{net} \ \ \ \ \ \ \ \ \\ \hline \text{(a)} & \$560 & \$76 & \$484 \\ \hline \text{(b)} & \$871 & \bbox[yellow,3px]{\$107} & \$764 \\ \hline \text{(c)} & & \$45 & \$513 \\ \hline \end{array} \)

(c)     \( \text{Gross Payment}=\text{Net Payment}+\text{Deduction} \)

\( \$45+\$513=\$558 \)

\( \require{AMSsymbols} \begin{array}{|c|c|c|c|} \hline & \ \ \ \ \text{gross} \ \ \ \ & \text{deduction} & \ \ \ \ \ \ \ \ \text{net} \ \ \ \ \ \ \ \ \\ \hline \text{(a)} & \$560 & \$76 & \$484 \\ \hline \text{(b)} & \$871 & \$107 & \$764 \\ \hline \text{(c)} & \bbox[yellow,3px]{\$558} & \$45 & \$513 \\ \hline \end{array} \)

Question 21

Fill in the blank.

\( \begin{array}{|c|r|r|r|} \hline \text{weekly wage} & \$600 & \$750 & \text{(c)} \\ \hline \text{income tax} & \$33 & \$38 & \$35 \\ \hline \text{superannuation} & \$81 & \text{(b)} & \$65 \\ \hline \text{health insurance} & \$25 & \$29 & \$27 \\ \hline \text{union fee} & \$12 & \$15 & \$14 \\ \hline \text{net payment} & \text{(a)} & \$601 & \$579 \\ \hline \end{array} \)

\( \begin{align} \text{(a)} &= \$600-\$33-\$81-\$25-\$12 \\ &= \$449 \end{align} \)

\( \require{AMSsymbols} \begin{array}{|c|r|r|r|} \hline \text{weekly wage} & \$600 & \$750 & \text{(c)} \\ \hline \text{income tax} & \$33 & \$38 & \$35 \\ \hline \text{superannuation} & \$81 & \text{(b)} & \$65 \\ \hline \text{health insurance} & \$25 & \$29 & \$27 \\ \hline \text{union fee} & \$12 & \$15 & \$14 \\ \hline \text{net payment} & \bbox[yellow,3px]{\$449} & \$601 & \$579 \\ \hline \end{array} \)

\( \begin{align} \text{(b)} &=\$750-\$38-\$29-\$15-\$601 \\ &= \$67 \end{align} \)

\( \require{AMSsymbols} \begin{array}{|c|r|r|r|} \hline \text{weekly wage} & \$600 & \$750 & \text{(c)} \\ \hline \text{income tax} & \$33 & \$38 & \$35 \\ \hline \text{superannuation} & \$81 & \bbox[yellow,3px]{\$67} & \$65 \\ \hline \text{health insurance} & \$25 & \$29 & \$27 \\ \hline \text{union fee} & \$12 & \$15 & \$14 \\ \hline \text{net payment} & \$449 & \$601 & \$579 \\ \hline \end{array} \)

\( \begin{align} \text{(c)} &= \$35+\$65+\$27+\$14+\$579 \\ &= \$720 \end{align} \)

\( \require{AMSsymbols} \begin{array}{|c|r|r|r|} \hline \text{weekly wage} & \$600 & \$750 & \bbox[yellow,3px]{\$720} \\ \hline \text{income tax} & \$33 & \$38 & \$35 \\ \hline \text{superannuation} & \$81 & \$67 & \$65 \\ \hline \text{health insurance} & \$25 & \$29 & \$27 \\ \hline \text{union fee} & \$12 & \$15 & \$14 \\ \hline \text{net payment} & \$449 & \$601 & \$579 \\ \hline \end{array} \)

Question 22

The regular hourly rate of a \( 40 \) hour of work is \( \$16 \). He has deducted \( \$44.80 \) in weekly tax.

(a)     Find his gross payment.

\( 40 \times \$16 = \$640 \)

(b)     Find his net payment.

\( \$640-\$44.80=\$595.20 \)

(c)     Find the percentage of his gross wage paid in tax.

\( \displaystyle \frac{\$44.80}{\$640} \times 100\% = 7\% \)

Question 23

The monthly salary is \( \$4800 \).

(a)     Find the amount of the income tax if the tax is \( 6\% \) of the gross payment.

\( \$4800 \times 0.06=\$288 \)

(b)     Find the amount of superannuation which is \( 9\% \) of the gross payment.

\( \$4800 \times 0.09=\$432 \)

(c)     Find the net payment.

\( \$4800-\$288-\$432=\$4080 \)

Overtime Payments

Overtime payments are additional earnings provided to employees for hours worked beyond their regular, predetermined work hours. These extra hours are typically paid at a higher rate, often referred to as an overtime rate, which can be one and a half times (1.5x) or even double (2x) the regular hourly wage. Overtime is usually mandated by labour laws to compensate employees for their extra effort and to encourage fair working conditions. It’s important to note that overtime payments are part of an employee’s gross earnings and are subject to taxation. Properly calculating and managing overtime is crucial for both employees and employers to ensure compliance with labour regulations and fair compensation.

Question 24

Kevin’s normal working hours are from \( 9 \)am to \( 5 \)pm with site allowance of \( \$30 \) per day. Kevin earns \( \$12 \) per hour with his overtime paid at the double time rate.

\( \begin{array}{|c|c|c|c|c|} \hline \text{days} & \text{work hours} & \text{normal} & \text{overtime} & \text{allowance} \\ \hline \text{day 1} & \text{9am to 5pm} & \text{8 hours} & \text{0 hours} & \$30 \\ \hline \text{day 2} & \text{9am to 6pm} & \text{8 hours} & \text{1 hour} & \$30 \\ \hline \text{day 3} & \text{9am to 7pm} & \text{8 hours} & \text{2 hours} & \$30 \\ \hline & \text{total} & \text{24 hours} & \text{3 hours} & \$90 \\ \hline \end{array} \)

(a)     Find his total gross payment.

\( 24 \times \$12 + 3 \times 2 \times $12 + \$90 = \$450 \)

(b)     He pays \( 9\% \) of his gross wage, not including overtime pay, into a superannuation fund each week. Find his amount of superannuation.

\( (24 \times \$12 + \$90) \times 0.09 = \$34.02 \)

Piecework

Piecework is a compensation method where employees are paid based on the number of items or tasks they complete, rather than receiving a fixed salary or hourly wage. In piecework systems, the more work an individual produces, the higher their earnings. This approach is commonly used in manufacturing, agriculture, and other industries where output can be quantified easily. Piecework can provide incentives for productivity but also poses challenges, as it may encourage rushed work or quality compromises. Accurate measurement and fair piece rates are essential to ensure that workers are compensated fairly and motivated to maintain both quantity and quality standards

With piecework, the employee is paid a fixed amount for each item produced. The more items a worker produces, the greater the pay for that pay period.

Question 25

A factory worker is paid \( 50 \) cents per product. Calculate the total payment for \( 240 \) products.

\( \$0.50 \times 240 = \$120 \)

Question 26

Karen is paid \( 30 \) cents per product that she makes. Find the number of products she needs to make to receive \( \$75 \).

\( \$75 \div \$0.30 = 250 \)

Question 27

A printer charges \( 20 \) cents per colour page and \( 12\) cents per black and white page. Find the total price for printing \( 300 \) colour pages and \( 500 \) black and white pages.

\( \$0.20 \times 300 + \$0.12 \times 500 = \$120 \)

Question 28

Joy charges \( \$12 \) per assembly \( A \) that she makes and \( \$20 \) per assembly \( B \). Altogether she was paid \( \$900 \) for \( 30 \) assembly \( A \) and some assembly \( B \). Find how many assemblies \( B \) she made.

\( \begin{align} \$12 \times 30 + \$20 \times B &= \$900 \\ \$360 + \$20 \times B &= \$900 \\ \$20 \times B &= \$900-\$360 \\ &= \$540 \\ \therefore B &= \$540 \div \$20 \\ &= 27 \end{align} \)

Question 29

Deb makes toys. She is paid \( \$15 \) per a toy for the first \( 30 \) toys, then \( \$18 \) for each toy thereafter. How much will she be paid in a month when she makes \( 56 \) toys?

\( \begin{array}{|c|c|c|} \hline \text{Level} &1-30 \text{ toys} & 30+ \text{toys} \\ \hline \text{per Toy} & \$15 & \$18 \\ \hline \text{Toy} & 30 & 26 \\ \hline \end{array} \)

\( \$15 \times 30 + \$18 \times 26 = \$918 \)

Question 30

A plumber charges \( \$60 \) for the first hour and \( \$80 \) per hour thereafter. Find how much he earned for the following works last week.

\( \begin{array}{|c|c|c|} \hline \text{hours} & \$60 \text{ work} & \$80 \text{ work} \\ \hline 2 & 1 & 1 \\ \hline 3 & 1 & 2 \\ \hline 2 & 1 & 1 \\ \hline 4 & 1 & 3 \\ \hline \text{total} & 4 & 7 \\ \hline \end{array} \)

\( 4 \times \$60 + 7 \times \$80 = \$800 \)

Question 31

A real estate agent charges \( \$200 \) per each sale plus \( 5\% \) on the first \( \$200 \ 000 \) and \( 3\% \) on the remaining value of the property sale. She sold \( 3 \) properties last month with the following values. Find her total earnings for the month.

\( \begin{array}{|c|c|c|} \hline \text{sales} & 5\% & 3\% \\ \hline \$180 \ 000 & \$180 \ 000 & \$0 \\ \hline \$350 \ 000 & \$200 \ 000 & \$150 \ 000 \\ \hline \$490 \ 000 & \$200 \ 000 & \$290 \ 000 \\ \hline \text{total} & \$580 \ 000 & \$440 \ 000 \\ \hline \end{array} \)

\( \$200 \times 3 + \$580 \ 000 \times 0.05 + \$440 \ 000 \times 0.03 = \$428 \ 000 \)

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