Lesson 1.1 · Unit 1 · N-Q.1-3
What you earn for the hours you work
Most first jobs pay by the hour, and the arithmetic looks too easy to need a lesson. It is not the multiplication that goes wrong. It is overtime, which applies to some hours and not others, and the number of pay periods in a year, which is not the same for every employer. Both of those change the answer, and both are set by rules rather than by arithmetic.
- Gross pay is hours times rate. A worker paid \$15.50 an hour for 32 hours earns \( 15.50 \times 32 = 496.00 \) dollars before anything is taken out.
- Federal law requires overtime of one and a half times the regular rate for hours over 40 in a workweek. California adds its own daily rule. Every problem in this course states which rule applies and uses the weekly rule unless told otherwise.
- Overtime applies only to the extra hours. Pay is \( 40r + 1.5r(h - 40) \) for \( h \gt 40 \) hours at rate \( r \). The first 40 hours are paid at the regular rate.
- The pay period is how often you are paid. Weekly is 52 periods a year, biweekly 26, semimonthly 24 and monthly 12.
- Biweekly and semimonthly are different. Biweekly is every two weeks, which is 26 checks. Semimonthly is twice a month, which is 24 checks. A \$52,000 salary gives \$2,000.00 biweekly and \$2,166.67 semimonthly.
- Annual pay is pay per period times periods per year. From an hourly rate, a full-time year is \( r \times 40 \times 52 = 2080r \).
- The average hourly rate with overtime is total pay divided by total hours. It is always between the regular rate and one and a half times the regular rate.
- Estimate before you compute. Overtime pay for a week must exceed pay for 40 hours and stay below pay for all the hours at time and a half. If it does not, the setup is wrong.
Where students lose marks: the wrong base. Applying the 1.5 to every hour instead of only the hours over 40 makes the answer too large, and no amount of careful arithmetic afterward repairs it. Name which hours are regular and which are overtime before multiplying anything.
The problem. A warehouse worker earns \$18.40 an hour and works 47 hours in a week. (a) Find the gross pay. (b) Find the average hourly rate for the week. (c) Find the biweekly pay if every week is the same. (d) Find the annual pay if every week is the same.
Step one: name the givens. The regular rate is \$18.40. Hours are 47, so 40 are regular and 7 are overtime. The overtime rate is \( 1.5 \times 18.40 \).
Step two: pay the regular hours. \( 40 \times 18.40 = 736.00 \).
Step three: pay the overtime hours. The overtime rate is \( 1.5 \times 18.40 = 27.60 \) dollars an hour, and \( 7 \times 27.60 = 193.20 \).
Step four: add for (a). Gross pay is \( 736.00 + 193.20 = 929.20 \) dollars. Estimate check: 40 hours alone is \$736.00 and all 47 hours at time and a half would be \$1,297.20, and \$929.20 sits between them.
Step five: average rate for (b). \( \dfrac{929.20}{47} = 19.77 \) dollars an hour. This is above the regular \$18.40 and below the overtime \$27.60, as it must be.
Step six: biweekly pay for (c). Two identical weeks give \( 2 \times 929.20 = 1{,}858.40 \).
Step seven: annual pay for (d). Fifty-two identical weeks give \( 52 \times 929.20 = 48{,}318.40 \).
Step eight: state the answers with their meaning. Gross weekly pay is \$929.20, the average rate is \$19.77 an hour, biweekly gross is \$1,858.40 and the annual gross is \$48,318.40. The annual figure assumes 47 hours every week, which a careful answer says, because few jobs offer overtime every week of the year.
Problems 1 to 5 are routine. Problems 6 to 10 are multi-step or ask for reasoning.
- A cashier earns \$15.50 an hour and works 32 hours. Find the gross pay.
Show the full solution
\( 15.50 \times 32 = 496.00 \). There is no overtime because 32 is under 40. \$496.00
- A worker earns \$22.25 an hour for exactly 40 hours. Find the gross pay.
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Forty hours is the limit, so none is overtime: \( 22.25 \times 40 = 890.00 \). \$890.00
- The regular rate is \$16.00 an hour. What is the overtime rate?
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\( 1.5 \times 16.00 = 24.00 \). \$24.00 an hour
- How many pay periods are there in a year when an employer pays biweekly?
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Fifty-two weeks divided into two-week periods gives \( 52 \div 2 = 26 \). 26
- A job pays \$21.50 an hour for 40 hours a week, 52 weeks a year. Find the annual gross
pay.
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A full-time year is \( 40 \times 52 = 2080 \) hours, and \( 21.50 \times 2080 = 44{,}720 \). \$44,720
- A stock clerk earns \$14.80 an hour and works 44 hours in one week. Find the gross pay.
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Regular: \( 40 \times 14.80 = 592.00 \). The overtime rate is \( 1.5 \times 14.80 = 22.20 \), and the 4 extra hours give \( 4 \times 22.20 = 88.80 \). Total \( 592.00 + 88.80 = 680.80 \). \$680.80
- A biweekly paycheck for a 40-hour week with no overtime is \$1,120.00 gross. Find the
hourly rate.
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Two weeks of 40 hours is 80 hours, so the rate is \( \dfrac{1120}{80} = 14.00 \). \$14.00 an hour
- A worker earning \$19.00 an hour works 52 hours in a week. Find the gross pay and the
average hourly rate.
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Regular: \( 40 \times 19.00 = 760.00 \). Overtime rate \( 28.50 \), for 12 hours: \( 12 \times 28.50 = 342.00 \). Gross \( 760.00 + 342.00 = 1{,}102.00 \). The average is \( \dfrac{1102}{52} = 21.19 \), between 19.00 and 28.50 as it must be. \$1,102.00; \$21.19 an hour
- An employee has a \$52,000 annual salary. Find the gross pay per check if paid biweekly
and if paid semimonthly, and say why they differ.
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Biweekly: \( \dfrac{52000}{26} = 2000.00 \). Semimonthly: \( \dfrac{52000}{24} = 2166.67 \). The annual total is the same, but it is divided into 26 checks in one case and 24 in the other, so each semimonthly check is larger. \$2,000.00 biweekly; \$2,166.67 semimonthly
- A student works 50 hours at \$12.00 an hour and computes the gross pay as
\( 50 \times 1.5 \times 12 = 900 \). Find the error and the correct pay.
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The student applied time and a half to all 50 hours. Only the 10 hours over 40 earn the premium. Regular: \( 40 \times 12 = 480 \). Overtime: \( 10 \times 18 = 180 \). Total \( 660 \). The wrong method overstated the pay by \$240. \$660.00