Lesson 1.2 · Unit 1 · N-Q.1-3
Pay that depends on what you produce
Not every job pays for time. A salary pays for a role, a commission pays for sales, piecework pays for units made and tips pay for service. To compare them, or to decide which a worker should prefer, every one has to be put onto the same footing: the same pay period and the same hours.
- A salary is a fixed annual amount. Pay per period is the annual salary divided by the periods per year. The hourly equivalent, assuming 40 hours a week, is the salary divided by 2080.
- A straight commission is a rate times sales. A rate of 8 percent on \$12,500 of sales is \( 0.08 \times 12500 = 1000 \) dollars. Convert the percent to a decimal first.
- Base plus commission is a fixed amount plus a rate times sales.
- A graduated commission pays one rate on the first tier of sales and a higher rate only on the sales above it, like layers. The higher rate does not apply retroactively to the lower tier.
- Piecework pays a fixed amount for each unit completed. Pay is the rate per piece times the pieces.
- A tip is a percent of the bill. The tip is the rate times the pre-tip bill, and a server's hourly pay is wages plus tips, divided by hours.
- To compare two plans, put them on the same footing. Convert both to pay per month or per year, or to an hourly equivalent for the hours actually worked.
- The break-even point is the sales level at which two plans pay the same. Set the two pay formulas equal and solve for sales. Below it one plan wins; above it, the other.
Where students lose marks: the percent as a decimal. A commission of 6 percent is \( 0.06 \), not \( 0.6 \) and not \( 6 \). Entering 6 or 0.6 makes the commission ten or one hundred times too large, and the answer looks like a plausible amount of money until you compare it with the sales.
The problem. A furniture salesperson is offered two plans for a month. Plan A is a salary of \$2,600. Plan B is a base of \$1,500 plus 6 percent of all sales. (a) Find Plan B's pay if the month's sales are \$18,500. (b) Find the sales at which the two plans pay the same. (c) Find the pay under a graduated plan that pays 3 percent on the first \$10,000 and 5 percent on sales above that, for the same \$18,500. (d) Find the hourly equivalent of Plan A's yearly pay.
Step one: name the givens. Plan A is \$2,600 a month. Plan B is \( 1500 + 0.06S \) for sales \( S \). Sales this month are \( S = 18500 \).
Step two: evaluate Plan B for (a). \( 1500 + 0.06 \times 18500 = 1500 + 1110 = 2610 \). Plan B pays \$2,610, which is \$10 more than Plan A this month.
Step three: set up the break-even for (b). The plans are equal when \( 1500 + 0.06S = 2600 \).
Step four: solve. \( 0.06S = 1100 \), so \( S = \dfrac{1100}{0.06} = 18{,}333.33 \). Check: \( 1500 + 0.06 \times 18333.33 = 2600 \).
Step five: interpret. Below about \$18,333 in monthly sales Plan A pays more, and above it Plan B pays more. This month's \$18,500 is just over, which matches step two. A salesperson who expects a slow month should prefer the salary.
Step six: graduated commission for (c). The first tier pays \( 0.03 \times 10000 = 300 \). The remaining \( 18500 - 10000 = 8500 \) is paid at 5 percent: \( 0.05 \times 8500 = 425 \). Total \( 300 + 425 = 725 \) dollars.
Step seven: hourly equivalent for (d). A year of Plan A is \( 2600 \times 12 = 31{,}200 \), and \( \dfrac{31200}{2080} = 15.00 \) dollars an hour at 40 hours a week.
Step eight: state the answers. Plan B pays \$2,610 this month. The plans break even at \$18,333.33 in sales. The graduated commission is \$725, and Plan A is worth \$15.00 an hour. Only the commission was computed in step six, so it cannot be compared with a salary until a base is added.
Problems 1 to 5 are routine. Problems 6 to 10 are multi-step or ask for reasoning.
- An annual salary of \$43,200 is paid biweekly. Find the gross pay per check.
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\( \dfrac{43200}{26} = 1661.54 \), rounded to the cent at the end. \$1,661.54
- A salesperson earns 8 percent commission on \$12,500 of sales. Find the commission.
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\( 0.08 \times 12500 = 1000 \). \$1,000.00
- A sewing worker is paid \$0.85 for each unit and finishes 640 units. Find the pay.
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\( 0.85 \times 640 = 544.00 \). \$544.00
- A customer leaves an 18 percent tip on a pre-tip bill of \$64.50. Find the tip.
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\( 0.18 \times 64.50 = 11.61 \). \$11.61
- Find the hourly equivalent of a \$62,400 salary for a 40-hour week.
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\( \dfrac{62400}{2080} = 30.00 \). \$30.00 an hour
- A salesperson earns a base of \$1,800 plus 4.5 percent of sales and sells \$27,000 in a
month. Find the gross pay.
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Commission: \( 0.045 \times 27000 = 1215 \). Total \( 1800 + 1215 = 3015 \). \$3,015.00
- A graduated plan pays 2 percent on the first \$15,000 of sales and 5 percent on sales
above that. Find the commission on \$38,000.
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First tier: \( 0.02 \times 15000 = 300 \). The rest is \( 38000 - 15000 = 23000 \), and \( 0.05 \times 23000 = 1150 \). Total \( 300 + 1150 = 1450 \). \$1,450.00
- A server earns \$16.00 an hour plus tips. In one week she works 30 hours and receives
\$412 in tips. Find her average earnings per hour.
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Wages \( 16.00 \times 30 = 480 \). Total \( 480 + 412 = 892 \). Per hour \( \dfrac{892}{30} = 29.73 \). \$29.73 an hour
- Plan X pays \$2,000 plus 3 percent of sales. Plan Y pays \$1,200 plus 7 percent of sales.
Find the sales at which they pay the same, and say which plan is better above that level.
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Set \( 2000 + 0.03S = 1200 + 0.07S \). Then \( 800 = 0.04S \), so \( S = 20000 \). Above \$20,000 the plan with the larger commission rate, Plan Y, earns more with each extra dollar and ends ahead. Below it, Plan X's higher base wins. \$20,000 in sales; Plan Y is better above it
- A student computes a 6 percent commission on \$3,000 of sales as \( 0.6 \times 3000 =
1800 \). Find the error.
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Six percent is \( 0.06 \), not \( 0.6 \). The correct commission is \( 0.06 \times 3000 = 180 \). The wrong answer is ten times too large, and a commission of \$1,800 on \$3,000 of sales would be 60 percent, which no employer pays. \$180.00