i need help. i dont even kno how to START figuring this out.?

Question

You are selling cases of mixed nuts and roasted peanuts. You can order no more than 400 pieces (cans of mixed nuts nad packages of roasted peanuts) and spend no more than $500. Mixed nuts cost $24/case, contain 12 cans and give you a profit of $20/case. Roasted peanuts cost $15/case, contain 20 packages and give a profit of $15/case. How many cases of each should you buy to maximize profit? What is your maximum profit?

Answer 1

Although the mixed nuts give a higher profit PER CASE they do not give a higher profit per dollar spent...

Mixed nuts: Spend $24 dollars - make $20 dollars profit (83% profit)
Peanuts: Spend $15 dollars - make $15 dollars profit (100% profit)

But we can't max out on peanuts as then we would go over the maximum total pieces (400) limit.

If M=number of cases of mixed nuts, P=number of cases of peanuts

We have
1) 12M + 20P ≤ 400 (pieces constraint)
2) 24M + 15P ≤ 500 (price constraint)

From 1) P ≤ 20 - (3M)/5

Now we know we want the outer limits of this equation... so let P=20-3M/5 and put it in to 2)

=> 24M +15(20-3M/5) ≤ 500
=> 15M ≤ 200
=> M ≤ 13.33
So it looks like M=13 is our max for M

Feed this back into
1) gives 12M + 20P ≤ 400
=> 156 + 20P ≤ 400
=> 20P ≤ 244
=> P≤12.2
So 12 cases of peanuts

Check equation 2)
2) 24M + 15P ≤ 500 (price constraint)
=> 24.13 + 15.12 = 492 ≤ 500 - OK

Profit made = 13 x $20 (Mixed nuts) + 12 x $15 (Peanuts)
= $440

You can check that this is the best by trying to take away a case of mixed nuts... (the less profitable)
12 cases of nuts - cost is now 12 x $24 = 288 - so can spend $212 and buy $14 cases of peanuts
BUT
12 cases of mixed nuts is 144 pieces and 14 cases of peanuts is 280 pieces = 424 pieces in total - no good... in fact only allowed to buy 12 cases of peanuts to stick within 400 total pieces budget - so lose profit...



EDITED to say - answer above did not check working:
14 mixed nuts cost 14 x $24 = $336
11 peanuts cost 11 x $15 = $165
Total = $501 - OVER BUDGET

Correct answer is still 13 mixed, 12 peanuts with $440 profit.

Answer 2

Looks like all mixed nuts give the most profit, why work for peanuts when you can have Pecans.
Buy 20 cases I guess.
20 cases @ $24 per case is $480 for a profit of $400.

Answer 3

<= $500
<= 400 pieces

Mixed nuts ` ` ` ` $24/case: 12 cans at profit $20/case
Roasted peanut $15/case 20 packages at profit $15/case

Let you buy n cases of mixed nuts, totaling 12 12n pieces
then # of cases of roasted nuts = (400 - 12n)/20

profit , P = $20n + $15 (400 - 12n)/20
Cost , C = $24n + $15 (400 - 12n)/20 = $500
24n + (3/4)(400 - 12n) = 500
24n + 300 - 9n = 500
15n = 200
n = 40/3 = 13.33

since profit is larger for mixed nuts,

try 14 cases of mixed nuts,
then (400 - 12n)/20 ~ 11 cases roast nuts
Profit = 14($20) + 11($15) = $445

or 13 cases of mixed nuts
and (400 -12(13))/20 ~ 12 cases roat nuts
profit = 13($20) + 12($15) = $440

Ans 14 cases of mixed nuts, 11 cases roast nuts

Answer 4

vlee1225 has a very nice method, except that he forgot to check the cost. 14 cases of mixed plus the 11 cases of peanuts will cost $501.

I t looks like 13 cases of mixed plus 12 cases of peanuts will maximize your profit within your purchasing limits.

Ooops, within the time it took me to write up my answer, someone else did it much better!

Answer 5

x= cases of mixed nuts
y = cases of roasted peanuts
x=>0
y=>0
24x + 15y =<500
12x + 20y =<400
Profit = 20x+15y
Plot the lines 15y = -24x+500 and 20y =-12x +400
Find the feasible area.
Check the cornerpoints to find max