October 14, 2019

Some organizations are looking to make a lot of money off of the flu vaccine that is in such high demand.  A survey done last week reported that 55% of 677 hospital pharmacists who responded reported being offered the vaccine at highly inflated prices.  Of those, 84% said they would not buy the vaccine at those prices.  How many of the 677 hospital pharmacists did not say that they would not buy the vaccine at the high prices?  Express your answer to the nearest whole number.

We know that only 55% of the pharmacists had the offer of vaccines at a high price. This is (0.55)(677) = 372.35. Of these 372.35 pharmacists, 84% said that they would not buy the vaccine, so we are looking at the 16% who did not say that. This would be (0.16)(372.35) = 59.576 or 60 pharmacists, to the nearest whole number. 

Before the shortage of the vaccine was announced, hospitals could purchase a 10-dose vial for about $80.  Then patients were charged about $15 per dose.  As you can see, patients paid more for the 10 doses than it cost the hospital, and we’ll refer to this extra money as the hospital’s profit.  If the same vial is now selling for $400, how much would the hospital have to charge each patient per dose in order to make the same percent of profit on the vial?

We’re told that before the shortage, a vial cost $80 and could be used by 10 patients. The patients would pay a total of 10 ´ 15 = $150 for the vial. Notice that the hospital is making $70 on the vial and it only cost $80. We can see that the percent profit, then, is close to 100%. It is actually 70 ¸ 80 = 87.5% profit. The vial now costs $400. The hospital would need to make 87.5% of $400 in profit, which is 0.875 ´ 400 = $350. This means that the patients would have to pay a total of $400 + $350 = $750. When divided evenly among 10 patients, that's $75 per dose. 

One organization was selling vials of the vaccine for $85 on October 1.  After the shortage was announced, the price was up to $900 by October 8.  If the cost of the vaccine on each day from October 1 through October 8 forms an arithmetic sequence, what was the cost on October 5?  Express your answer as a decimal to the nearest hundredth.

In an arithmetic sequence, the terms of the sequence increase by a constant amount each time. We know that the price on October 1 (or 10/1) was $85. Now we can set up the following: 10/1 = 85; 10/2 = 85 + x; 10/3 = 85 + 2x; 10/4 = 85 + 3x; 10/5 = 85 + 4x; 10/6 = 85 + 5x; 10/7 = 85 + 6x; and 10/8 = 85 + 7x. We also know, though, that the price on 10/8 was $900. Therefore, 85 + 7x = 900. Subtracting 85 from both sides of the equation and then dividing by 7 yields x = 116.42857. We can also see from our listing of possibilities that on 10/5, the expression for the cost of the vaccine is 85 + 4x, which is now 85 + 4(116.42857) = $550.71

Page 1 of the linked PDF contains PROBLEMS & SOLUTIONS.

Page 2 contains ONLY PROBLEMS.