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If 100 units have a unit cost of $18 and 500 units have a unit cost of $15.5, what is the cost of 2000 units?

I was asked this in an interview and the only thing I could think of was for a 5 multiple of units cost goes down by $2.5 so for 20 multiple of units the price goes down by $50 which is obviously wrong. Please help me with the approach.

Clarinetist
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srip
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  • It was for a software programmer position. I believe he was trying to test my problem solving abilities. – srip Sep 04 '16 at 23:37
  • Ah, perhaps there was a fixed cost. – ZirconCode Sep 04 '16 at 23:38
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    Why would $500$ units cost less than $100$ units? – lulu Sep 04 '16 at 23:39
  • @lulu: Ordering in bulk is often cheaper due, for example, to reduced transport costs per unit. – Will R Sep 04 '16 at 23:40
  • @Will R. Well, yes. Though the OP says "the units cost" as opposed to "the per unit price for N units is...". But there's no way for us to guess the volume discount the seller is willing to offer. I mean...I suppose the unit price decays exponentially to the marginal cost, or to some fixed spread over that. But we know neither the marginal cost nor the fixed spread. – lulu Sep 04 '16 at 23:42
  • @lulu: I see your point, but I think we both know the intended meaning. Regardless, questioning the question is not a job interview technique that I would recommend. :) – Will R Sep 04 '16 at 23:44
  • Ok, technicalities aside: Assume the total cost of $n$ units is linear. So $C(n)=mn+b$. We are given $C(100)=18\times 100=1800$ and $C(500)=15.5\times 500=7750$. These two data points determine $m,b$ and then you can solve for $C(2000)$ and deduce the per unit price from that. – lulu Sep 04 '16 at 23:48
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    @lulu: Are you interpreting the question as "100 units cost $18 each"? That's not how I would interpret the question. – Will R Sep 04 '16 at 23:58
  • @Will R Ah, so we weren't on the same page. How else can you interpret the question? Obviously the absolute cost can not be lower! Nobody would ever buy the $100$ lot! – lulu Sep 04 '16 at 23:59
  • @WillR: Sorry if the question is not clear. But it means if you buy 100 units each unit is going to cost you 18$ – srip Sep 05 '16 at 00:00
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    @rowang Thank you for clarifying. Yes, that's the only rational way I can see to interpret the question (note: that does not preclude those rational ways which I have not spotted). But your phrasing is poor. I recommend editing the question to say "the per unit price on $100$ units is $$18$, while the per unit price on $500$ units drops to $$15.5$ " or something like that. Also, note that the linearity of the total cost function is an extra assumption. – lulu Sep 05 '16 at 00:03
  • Note: as you see from the posted solution by @Clarinetist , even the assumption is ambiguous. I assumed the total cost function was linear. That user supposed the unit price function was linear. I think my assumption is more realistic (the other yields negative prices pretty fast), but both are logically possible within some range. – lulu Sep 05 '16 at 00:08
  • @lulu: I do realize, of course, that, under my proposed interpretation, the cost per unit on $100$ units would be 18 cents, compared to a mere 3 cents per unit when buying $500$ units, so that nobody would ever buy the $100$ unit bundle; but, to be honest, I wouldn't have been surprised if the question really was intended to be that way. Ultimately the question was not intended to test business smarts, but rather to test mathematical skill, which are (perhaps unfortunately) often treated as separate. – Will R Sep 05 '16 at 00:12

2 Answers2

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As clarified in the comments, the costs above are unit costs, rather than total costs.

The quickest way to do this that I would do in an interview is to assume that the unit price were linear. Compute the slope: $$m = \dfrac{15.5-18}{500-100}\text{.}$$ This is the unit price change per unit increase. So, we could just take $$18+ \dfrac{15.5-18}{500-100}(1900)$$ where the $1900$ comes from the difference of $2000$ units and $100$ units. If you have a calculator, this comes out to $\$6.125 \approx \$6.13$. If you're looking for the total cost, multiply this by $2000$.

If we assumed instead that the total cost were linear, then we would have: $$m = \dfrac{15.5(500)-18(100)}{500-100}$$ and similarly, $$18(100)+\dfrac{15.5(500)-18(100)}{500-100}(1900) = \$30,062.50$$ which is the total cost, and for $2000$ units, this is a unit price of $\$15.03$.

I'm assuming the interviewer wasn't interested in the answer itself, but rather, your method.

Clarinetist
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As stated in the comments, there is is considerable ambiguity here. We are given two values of the per unit cost function, let's call it $U(n)$. Logically, we could assume that $U(n)$ were linear but, at best, that could only be true in a range (as it is clearly a decreasing function of $n$, linearity in a large range would allow $U(n)<0$ which is not realistic). But it seems to me that that the only realistic assumption (that we can have enough information to solve for!) is that the total cost function is linear. That is to say, if we define $C(n)$ to be the total amount to be paid (thus $C(n)=U(n)\times n$) then $$C(n)=m\times n + b$$ for suitable constants $m,b$.

Let's see how this plays out. We are given that $$C(100)=18\times 100=1800\;\;\;\;\;\&\;\;\;\;\;C(500)=15.5\times 500=7750$$

A quick calculation to solve for $m,b$ then shows that $$C(n)=14.875n+312.5$$

As noted in the solution by @Clarinetist this lets you solve for $C(2000)=30062.5$ and the associated per unit cost of $15.03125$ .

It is interesting to note that, in the limit as $n\to \infty$ we get the marginal cost of a unit to be $$\lim_{n \to \infty}\;\frac {C(n)}n=14.875$$ Which feels fairly plausible in light of the given data.

lulu
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