Those last few nickels and dimes leftover on your metro card, and the time it takes to add on to it, can really add up. So what's the frugal (and time-crunched) train traveler to do? Mathematics has the answer. [UPDATE]

Top image: Train car in motion / Cec

Over at I Quant NY, Ben Wellington, a visiting professor of city planning at the Pratt Institute, has run the numbers on how New Yorkers are buying their metro tickets and has come up with the most economical and time-efficient method. [Update: Wellington will also be answering questions in the comments, so if you have any, ask away!]

In New York, a local metro fair has a base price of $2.50. By charging up their metro cards, New Yorkers can also get a nice perk: a 5% cash bonus added on to their total charge. But, as Wellington points out, that bonus is not really a bonus at all if you're never able to actually use it.

How would it do that? Well, assume you purchase a $9.00 MTA card which, with the bonus, is now worth $9.45. After 3 trips, you're left with a ticket worth $1.95. At this point, you either need to charge your ticket to take another trip or lose that remainder.

Explains Wellington:

Imagine how many tourists come to NYC and leave with balances that never get used. Imagine how many people lose metro cards with those balances that never get used. And even if it gets used on a later refill, the MTA gets to collect the cash earlier this way! Win win for them, right?

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And the sum of all that unredeemed cash, he says, comes to about $50 million each year.

So how do you avoid your own subway change falling into that statistic? Simple. You pick a ticket that, with the remainder, will leave no remainder, as Wellington has sketched out on the chart below. A ticket worth $11.90 or $19.05 can save you not only money, but time.

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Chart: Ben Wellington / I Quant NY