A rental agency has $80,000 to invest in the purchase of 13

Question

A rental agency has $80,000 to invest in the purchase of 13 cars of two types, compact or fullsize. the cost per compact car is 5000 and the cost per full size car is 8000. the profit from renting the fullsize car is 3500 and for the compact car 2500. how many of each type should be purchased to maximize the profit?

coffeeloverfreak · Answer

First thing to notice is that the profit for a compact car is 2500/5000 or 50%, and the profit from a fullsize car is 3500/8000 or 7/16, which is 1/16 less than 50%. In other words, the compact cars are more profitable than the fullsize cars.

If the company invests its entire $80,000 in compact cars, it should make more profit. It can buy 16 compact cars for $80,000, which would net a profit of 2500x16=$40,000.

(Note: for comparison, if it had bought 10 fullsize cars instead, it would make 3500x10=$35,000 in profit only).

The answer is 16 compact cars, 0 fullsize cars.

(Usually these problems have additional constraints, like, profit for compact cars goes down if the company can't attract people with promos to upgrade to fullsize, etc.)

ranga41 · Answer

if they are to buy 13 cars then 5 fullsize and 8 compact

coffeeloverfreak · Answer

Sorry, didn't read the question carefully. :oops:

Ranga is right. 13 cars means that you want to maximize the number of compact cars and then buy fullsize with the rest. But you need a multiple of 8,000 for the amount to spend on fullsize cars. So 8 compact and 5 fullsize.

anandsebastin · Answer

nice explanation, CLF

duttsit · Answer

Reminds me of optimization problems in operations

popew626 · Answer

Can you explain more on this? I'm still not getting how this optimizes profit. Thanks

FN · Answer

hmm

I set up the following

c for compact, f for fullsize

C+F=13

5C+8F=80

solve for C, since it has higher profit return...

so 5C+8(13-C)=80

5C+104-8C=80

-3C=-24

c=8, then f=5

how is this approach any different from your maximizing profit approach?

duttsit · Answer

the problem cab be stated as:
manimise 5C + 8F - 90 = 0
constrain to: C + F = 13

in this case, both these linear equations can be graphed as stright line. 
Solving these two simulteneous equations, as you did, gives the point where they insersect. This is where BOTH equations are satisfied. 

Dependent on the problem, this may be maxima/minima point.
In this case, this is maxima and your ans is correct.

remember, however, that this method will only be aplicable if equations are linear (not quadratic). In latter case, you might have to use calculas (differentiation) to solve for maxima.

coffeeloverfreak · Answer

Yeah, but calculus and derivates are beyond the scope of GMAT math.

Anyway, I think most of these problems have simple enough numbers that you can use logic to solve them, as opposed to formulas.

jrabenho · Answer

Has anyone noticed that this problem basically is presenting two equations and asking you to solve for both variables... There is really no porift maximization. It's really just finding the one combination that works... I don't like this problem, there should be multiple answers and we should required to find an actual profit maximization point.

John

A rental agency has $80,000 to invest in the purchase of 13

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