If 100 units have a unit cost of
$18and 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.
On
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.
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.