Help With a Real World Math Word Problem (Does not involve two trains)

March 6, 2018 6:16 AM

Jimmy has leased a large number of copiers. The monthly lease cost of the copier contract is X which includes Y color impressions and Z black impressions per month. Color impressions over Y are billed at Yo per impression Black impressions over Z are billed at Zo per impression. What equation does Jimmy use to determine the actual per page cost for both color and black when given the monthly page counts for color and black.
posted by jmsta to Work & Money (8 answers total)

There isn't a "right" way to do it, or rather, there are some counterintuitive properties (if you only copy one sheet all month, did that one sheet cost X)? What if you don't do any color copying that month?

That said, probably the most natural per-sheet cost would be (total page count in a month) / (total amount paid in a month). Total amount paid in a month for colour is the fraction of X that goes to colour (i.e., X * Y/(Y+Z)), plus any overage cost (i.e., max(0, monthly page count in colour - Y) * Yo). Similarly for B&W.
posted by katrielalex at 6:34 AM on March 6, 2018 [1 favorite]

So, something like 5yo + 10zo = 50? So the X, Y and Z are fixed? I'm making up numbers. You can find a solution, but not necessarily a unique solution. If X varies, then you can solve, but otherwise, you have many possible solutions.
posted by Valancy Rachel at 6:38 AM on March 6, 2018

Total cost divided by total number of copies.

Yo and Xo are the marginal cost, i.e. the cost of one additional copy.
posted by SemiSalt at 6:41 AM on March 6, 2018 [1 favorite]

Let the monthly page count for color be C and for black be B.

Then the total cost is X + (C-Y)*Yo + (B-Z)*Zo. (your base cost, plus extra for each color or black copy over the base amount already included).

The actual per page cost is total cost / number of copies = (X + (C-Y)*Yo + (B-Z)*Zo)/(C+B)

As noted above, there are many combinations of C and B which can lead to the same cost per page.
posted by peacheater at 6:59 AM on March 6, 2018 [3 favorites]

If you have already leased a copier for the month, the cost per page is 0 until you hit included impressions, then Yo and Xo. If you are in the business of leasing "a large number of copiers" it may be that only one type of copy determines how many copiers you wish to hire, in which case the cost of it is the marginal cost and the other's is 0. Are all the copiers in the one place?
posted by hawthorne at 8:09 AM on March 6, 2018

The difficulty to me here is that there's a hidden assumption, which is that there is a correct way to describe value X in terms of Y and Z. The simplest would be to simply do a weighted average: (Y/(Y+Z))X is the proportion of X going to color impressions, (Z/(Y+Z))X is the proportion going to B&W impressions. From there the algebra is pretty simple.

In reality though, that's almost certainly not how the vendor determined their price X, they probably have substantially different toner and upkeep costs between the two kinds of impressions, so one way to approach this might be to get (at least two) other quotes that have different X/Y/Z values to triangulate the relationship among them that really leads to the various X values (though this assumes that all vendors consider the relationship among these variables in the same way).

Or perhaps if we had more information about why Jimmy wants to know this 'actual per page cost' we might be able to come at this a bit differently? Is he comparing leases? Does he already know how many impressions of each kind he wants? Or is he trying to make decisions about how many of each kind to prepare under the current lease? Or something else?
posted by solotoro at 8:23 AM on March 6, 2018 [2 favorites]

Best answer: solotoro is thinking the same things I am. Jimmy first needs to decide how $X is apportioned between Y and Z e.g. if there were one color and one black copy all month are you going to cost them out at $X/2? My guess is that a formula that uses Yo and Zo will be more 'representative'.

I'm betting Jimmy manages the lease expenses and is being asked to apportion the copier costs out to departments that are using shared copiers, and when they are making copies they are plugging department codes into the copiers. At the end of the billing period Jimmy gets a report of how many color & black copies are being made by each department and now has to split out all those costs.

All black pages printed in a month = Ztotal = Z + (black impressions over Z)
Black cost factor (how you're apportioning base $X cost) = Zcf (I like Zo/(Zo+Yo)); remember (Zcf + Ycf) must equal 1]

Cost per black page = Ztotal / ((X*Zcf) + ((Ztotal-Z)*Zo)) {black pages divided by base cost for black + incremental costs for black overages}

The tricky part is apportioning X; what do you do in months where you go over Z but not Y or vice versa? What about months when there are zero Z or Y copies?
posted by achrise at 9:06 AM on March 6, 2018

My guess is that a formula that uses Yo and Zo will be more 'representative'. .... Zo/(Zo+Yo)

Yes, of course! We actually do have information about how the vendor sees the relative value between B&W and color copies. I'm frustrated with myself for missing that!

One small suggestion, offered with the caveat that we're at the end of the day so maybe my brain is hazy: I think Ztotal / ((X*Zcf) + ((Ztotal-Z)*Zo)) gives pages per dollar (or whatever unit of currency), and you'll actually want the inverse to get cost per page.
posted by solotoro at 1:37 PM on March 6, 2018