6. Piecework, Tips, and Multi-Part Pay

Not every job pays a flat hourly rate. Some workers are paid piecework — a fixed amount for each unit they produce, which rewards speed and output directly — while many service workers earn a small guaranteed hourly wage plus tips, which can make up the majority of their income.

• Piecework pay: \[ P = r \times u, \] where \(r\) is the rate per unit and \(u\) is the number of units produced.
• Wage-plus-tips pay: \[ P = wh + T, \] where \(w\) is the hourly wage, \(h\) is hours worked, and \(T\) is total tips collected.

Both formulas follow the same basic idea as Session 1's wage formula — a rate multiplied by an amount — but piecework measures output instead of time, and tipped pay adds a second, less predictable income stream on top of a guaranteed base.

Worked example: A seamstress is paid $4.50 per shirt hemmed. She hems 36 shirts in a shift. Pay is \(4.50 \times 36 = 162\) dollars.

Worked example (tips): A server earns $8 per hour and works a 6-hour shift, collecting $97 in tips. Pay is \(8 \times 6 + 97 = 48 + 97 = 145\) dollars.

Case study: Lena assembles jewelry at home for a small company that pays $1.25 per bracelet and $2.00 per necklace, since necklaces take longer to make. In one week she assembles 80 bracelets and 35 necklaces. Her total pay combines two piecework rates: \(1.25 \times 80 = 100\) dollars from bracelets, plus \(2.00 \times 35 = 70\) dollars from necklaces, for a total of \(100 + 70 = 170\) dollars. This shows that piecework problems can involve more than one rate at once — each type of item needs to be multiplied by its own rate before the results are added together.

Worked example (mixed pay with a shift minimum): Some employers guarantee that tipped workers earn at least the state minimum wage even if tips fall short, a rule sometimes called a "tip credit make-up." Suppose the minimum wage is $12 per hour, and a bartender earning a $6 hourly wage works a 5-hour shift but only collects $25 in tips. Wage-plus-tips would give \(6 \times 5 + 25 = 30 + 25 = 55\) dollars, but the guaranteed minimum for 5 hours is \(12 \times 5 = 60\) dollars. Since $55 falls short of $60, the employer must pay the $5 difference, bringing the bartender's total to $60 for the shift.

Common mistakes to avoid:

• Reading a rate as "per hour" when it is actually "per unit," or the reverse — always check the units stated in the problem before multiplying.
• In multi-rate piecework, applying only one rate to the combined total number of items instead of applying each rate to its own item type separately.