Mathematics of payroll systems
The mathematics of payroll systems encompasses the calculations and methodologies used to determine employee compensation for work performed, which has historical roots dating back to ancient times. Initially, compensation was provided in goods, but over the centuries, it transitioned to monetary payments. Modern payroll practices involve various payment structures, such as hourly wages, salaries, and commissions. Calculating total earnings often includes considerations for overtime pay, where employees receive additional compensation for hours worked beyond a standard workweek, typically calculated as "time and a half."
Payroll systems also integrate complex elements like tax withholdings, where both employers and employees contribute to social security and Medicare taxes. The timing and frequency of paychecks vary, with common practices including weekly, bi-weekly, and monthly payments. Payroll calculations can be further complicated by voluntary deductions for benefits like health insurance or retirement savings. As the field continues to evolve, especially with advancements in technology, roles related to payroll analysis and management have emerged as viable career paths for mathematics majors, highlighting the importance of mathematical skills in effectively managing payroll operations.
Mathematics of payroll systems
Summary: Various payroll systems employ different mathematical calculations.
A variety of pay practices date back to ancient times, including compensation for services in the form of food, commodities, land, or livestock. Payroll systems are connected with the history of bookkeeping, which can be traced back to 4000 b.c.e. Paymasters were responsible for paying workers. Governments kept financial records called “pipe rolls” at least as early as the eleventh century. In 1494, Franciscan friar and mathematician Luca Pacioli published the book Summa de Arithmetica, Geometria, Proportioni et Proportionalita, which contained double-entry bookkeeping. The term payroll dates back to the seventeenth century, and compensation gradually changed from goods to money. In the mid-twentieth century, mathematician Grace Murray Hopper developed a compiler, later known as the FLOW-MATIC, which could be used for payroll calculations. When the U.S. Navy could not develop a working payroll plan, they called Hopper back to active duty. In the early twenty-first century, a payroll specialist is listed by some schools as a career option for mathematics majors. Accountants and actuaries calculate quantitative measures and predictions based on historic payroll information and salary increases. For example, the pensionable payroll is calculated as an integral that takes salary increases into account. In payroll analysis, the impact of changing salary expenses is compared to other factors, such as sales or profit.
Frequency
Some employees are paid each day they work; however, in many cases, an employer will withhold daily earnings and pay the cumulative amount earned at a later time as a lump sum. Common payroll frequencies include weekly, bi-weekly (every other week), semi-monthly (twice a month), and monthly. Each of these frequencies would correspond to receiving 52, 26, 24, and 12 paychecks each year, respectively, assuming a full year of work. Some seasonal jobs pay only for part of the year, but still use the standard payroll frequencies. For example, teachers often receive pay for only nine months. Some schools offer for that pay to be spread over a full year to guarantee consistent income during the summer months when teachers are not actually working.
On payday, the employee will receive earned wages for the previous pay period. Rather than receiving cash, sometimes an employee will receive a check that can be exchanged for an equivalent amount of cash. Other times, an employee will receive income as a “direct deposit” where the income is automatically deposited into the employee’s checking or savings account.
Earning Money
Some employees work for an hourly wage—for every hour of work they perform, they get paid a specified amount of money. Suppose that a worker had an hourly wage of $10 and worked for 20 hours. To find the total amount of the paycheck, the worker would multiply the hourly wage by the number of hours worked. For example, $10×20=$200.
Sometimes, contracts or laws dictate the number of hours a person can work per week and—should they work more than that amount—his or her income increases. For example, in the United States, 40 hours is a common workweek. A person working over 40 hours often gets paid “time and a half” or “wage and a half” for the number of hours over 40 that he or she works (called “overtime”). Again, assuming an hourly wage of $10, an employee who worked 48 hours in one week would earn $10×40=$400 for the first 40 hours they worked. The eight hours he or she worked beyond 40 hours would earn him or her extra money. If the employee earns “time and a half,” the time would be multiplied by 1.5 before being multiplied by his or her hourly wage. If he or she earns “wage and a half,” the wage would be multiplied by 1.5 before being multiplied by the number of hours worked beyond 40. In reality, the method of calculating overtime earnings is irrelevant since multiplication is associative. Time and a half would be calculated as $10 × (1.5 × 8) = $10 × 12 = $120, and wage and a half would be calculated as ($10 × 1.5) × 8 = $15 × 8 = $120. The total earnings for that week would be found by taking the sum of these wages: $400+$120=$520.
Another method for earning money is a salary. Unlike the hourly wage, a salary is a predetermined amount of money that the worker earns regardless of how long (or how short) it takes the worker to accomplish those tasks. Often, salary is determined based on how much a person will make over a year’s time. However, rarely does a person only receive one paycheck a year. The amount of money earned on each paycheck is calculated by taking the salary and dividing it by the number of pay periods in a year. That number will vary depending on how often a person gets paid. Suppose an employee agreed to work for a salary of $31,200 each year. Looking at the common pay periods, weekly, bi-weekly, semi-monthly, and monthly, this employee would earn $600, $1,200, $1,300, or $2,600, respectively, for each paycheck during the year.
A worker earning commission does not actually get paid based on how long it takes to do the job, but by how productive the worker is (oftentimes based on the amount of items the worker sells). Sometimes, commission is a flat fee per item sold, other times, it is a percentage of sales. For example, if an employee earned 7% commission on sales and sold $1,250 worth of merchandise on a given day, then pay would be calculated $1,250×7%=$1,250×0.07=$87.50. Some jobs combine an hourly rate and commission—the employee earns a certain amount of money for every hour they are at the job, but then also earns commission on top of that wage to determine the total money earned.
Payroll Withholdings
Upon receipt of a paycheck or notice of direct deposit, usually the amount paid to the employee (the net pay) is less than what is calculated as his or her earnings for the pay period (the gross pay). Before being issued money, an employee may have his or her income reduced by certain amounts—some voluntary, others involuntary. In order to pay for various levels of government (and the benefits they offer), income and payroll taxes are frequently withheld from earnings. Some employees pay premiums for different insurances (such as medical, life, or disability) from their pay. Sometimes, money is withheld as a long-term savings for eventual retirement of the employee. Job-related expenses can also be withheld, such as for dues or charges for employee uniforms.
Bibliography
Booth, Phillip, Robert Chadburn, Steven Haberman, Dewi James, Zaki Khorasanee, Robert Plumb, and Ben Rickayzen. Modern Actuarial Theory and Practice. 2nd ed. Boca Raton, FL: CRC Press, 2004.
Bragg, Steven M. Essentials of Payroll: Management and Accounting. Hoboken, NJ: Wiley, 2003.
Haug, Leonard. The History of Payroll in the U.S. San Antonio, TX: American Payroll Association, 2000.