Summer Job Pay Comparison
Two summer job offers arrive on the same day. Job A pays $11.50 per hour plus a 3% commission on weekly sales, but commission is paid only on the first $2,000 of sales each week. Job B pays $9.00 per hour plus a 6% commission on all weekly sales and offers a $300 signing bonus after eight weeks. Both jobs expect 30 hours per week. A typical week is expected to produce $2,500 in sales, but a slow week might produce only $1,500. A student writes expressions for weekly pay and sets them equal to find the sales amount at which the jobs pay the same. The student's parent points out that the signing bonus and the commission cap change the comparison and asks for a recommendation based on both a typical week and a slow week over the eight-week period.
1. Write a weekly pay expression for each job as a function of weekly sales s, including the cap for Job A. Then compute each job's weekly pay for s = $2,500 and s = $1,500.
Read the worked answer
Job A: base pay is 30(11.50) = $345. Commission is 3% of only the first $2,000, so the expression is A(s) = 345 + 0.03 * min(s, 2000). Job B: base pay is 30(9.00) = $270, with 6% of all sales, so B(s) = 270 + 0.06s. For s = 2,500, Job A pays 345 + 0.03(2,000) = 345 + 60 = $405. Job B pays 270 + 0.06(2,500) = 270 + 150 = $420. Job B pays $15 more in a typical week. For s = 1,500, Job A pays 345 + 0.03(1,500) = 345 + 45 = $390. Job B pays 270 + 0.06(1,500) = 270 + 90 = $360. Job A pays $30 more in a slow week. The commission cap makes Job A less sensitive to high sales.
2. Find the weekly sales amount at which the two jobs pay the same, considering the cap. Interpret your answer and explain why there may be more than one algebraic case.
Read the worked answer
Because Job A's commission changes at s = 2,000, compare in two cases. If s <= 2,000, set 345 + 0.03s = 270 + 0.06s. Then 75 = 0.03s, so s = 2,500, which is outside the case s <= 2,000, so reject it. If s > 2,000, Job A pays a flat $405, so 405 = 270 + 0.06s. Then 135 = 0.06s, so s = 2,250, which is greater than 2,000 and valid. The jobs pay the same at $2,250 in weekly sales. For sales below $2,250, Job A pays more; for sales above $2,250, Job B pays more. The cap creates the two cases: the first equation assumed the uncapped commission, which only applies below $2,000, so its solution cannot be accepted.
3. Over eight weeks, assuming the same sales every week, compare total pay including the signing bonus for a typical week ($2,500) and a slow week ($1,500). Recommend a job and state one assumption that could change the recommendation.
Read the worked answer
Typical week: Job A pays $405 per week, so over eight weeks it pays 8(405) = $3,240. Job B pays $420 per week plus the $300 bonus, so it pays 8(420) + 300 = 3,360 + 300 = $3,660. Job B is ahead by $420. Slow week: Job A pays $390 per week, so 8(390) = $3,120. Job B pays $360 per week plus the bonus, so 8(360) + 300 = 2,880 + 300 = $3,180. Job B is ahead by only $60. Under both assumptions, Job B pays more over eight weeks, mainly because of the signing bonus. A key assumption is that the bonus is actually paid on time. If the bonus is delayed or not guaranteed, the slow-week comparison changes: without the bonus, Job B pays $2,880 and Job A pays $3,120, so Job A would be better. The recommendation should be Job B if the bonus is reliable, but Job A may be safer if sales are likely to be slow and the bonus is uncertain.