Your electronics store sells two types of portable CD players. The first type, A, costs 70 dollars and you make a 25 dollar profit on each one. The second type, B, costs 60 dollars and you make a 20 dollar profit on each one. You expect to sell at least 50 players this month and you need to make at least 1100 dollar profit on them. You must order at least one of each type of player. How many of each type of player should you order to minimize your cost?
A-20 A players, 30 B players
B-1 A player, 1 B player
C-1 A player, 49 B players
D-49 A players, 1 B player
I encountered this question while doing the Systems of Linear Equations and Inequalities test at http://www.classzone.com/books/algebra_2/chapterquiz_national.cfm. To start, I ruled out answer B, since it obviously does not meet the requirements. Next I checked the amount of profit and the cost of the other answers.
Answer A
Profit
20*25+30*20, or 1100 dollars
Cost
20*70+30*60, or 3200 dollars
Answer C
Profit
1*25+49*20, or 1005 dollars
Since this doesn't meet the profit requirements, I ruled it out.
Answer D
Profit
49*25+1*20, or 1245 dollars
Cost
49*70+1*60, or 3490 dollars
Now I got a bit confused here because answer A has the lowest cost, but answer D has the highest profit to cost ratio. Although the question asks how many you should order to minimize the cost, I ended up choosing answer D because it would be a better business decision in real life. However, when I finished the test, it said that answer C, 1 A player and 49 B players, was correct. I'm not sure whether or made a mistake or whether the website is mistaken. I'm leaning towards the latter though, as there was one other problem on the test whose answer I disagreed with. I posted that question here at Find the minimum value of C subject to the given constraints., and apparently the website was mistaken about that question. Hopefully you can help me settle this question as well.
Player B is cheaper than Player A, so it makes sense to buy as many B's as possible, over buying A's, to reach your profit goal. Every A you buy instead of a B increases your cost.
So you buy the least number of A's that will get you to your profit goal, without buying so many that you might not sell all of them.
Actually, C isn't correct, because the profit is only $\$1,005$.
Either A or D will let you reach the needed profit level, but A is the correct answer, because it has the lower cost of the two.