# More Metrocard Calculations

Inspired by the recent increases in fares for public transit in NYC, I used Geogebra to put together a little graph to compare the various consumer options.

The solid red line represents the unlimited monthly card, and this costs \$104 regardless of how many times it is used.  The solid blue line represents a pay-per-ride strategy, plotting the total cost against the number of rides purchased.

These lines intersect at the point (49.46 , 104), meaning that the two plans are equivalent if one rides 49.46 times per month.  Graphically, you can see that pay-per-ride is a better value for less than 49.46 rides (it’s lower than the red line), and is a worse value above that number (higher than the red line).

The dotted lines factor in the discount many New Yorkers enjoy by using pre-tax dollars to purchase transit cards.  The use of pre-tax dollars saves you whatever you would have paid in income taxes on that amount:  for New York City residents, the combination of federal, state, and city taxes is around 35% for typical earners.   The discount affects both plans equally, so the point of intersection of the two dashed lines occurs at the same number of rides as the POI of the solid lines.

An astute observer might wonder why the equation of the solid blue line is not y = 2.25 x.  While the fare is indeed \$2.25 per ride, by pre-purchasing rides in bulk you receive a 7% discount.  This changes the effective fare per ride, which is taken into consideration in the above graph.  A trip over to the metrocard bonus calculator might shed some light on the subject.

Categories: Application

#### patrick honner

Math teacher in Brooklyn, New York

### 4 Comments

#### Leandra Stewart · January 13, 2011 at 9:22 pm

What’s funny is I actually did a similar thing to figure out what the benefit would be of pay-per-ride verses an unlimited. I didn’t plot my results but its nice to see it laid out in this form. I guess math minds think alike.

#### MrHonner · January 13, 2011 at 9:36 pm

I’m trying to think of ways to expand this analysis to include other interesting factors. If you have any ideas, let me know.

#### Danny Kwok · January 15, 2011 at 12:52 am

I’ve heard that the increase of the unlimited card from \$89 to \$104 was due to people helping others swipe in. The intersection of the unlimited and pay-by-ride strategies is basically a way of portraying how each of the two strategies’ costs exceed each other in an interval. If we could find out the probability of people who help others swipe in and on average how many times a card is swiped per day, the worth of the unlimited card will not be \$104 anymore, but something less. In other words, we can find the expected worth per unlimited metro card and find the “true” intersection of that and the pay-by-ride strategy.
We might go about doing this by conducting surveys (since a census is impractical). Have the people in the sample randomly selected and representative of the entire population (MTA users). This also implies that the people in the sample should be of different age groups,ethnicities,etc. We can then ask them how many people they share the card with and we should be able to get an estimate on how many rides per month (or day) for each unlimited metro card and calculate an expected value. From the expected number of rides per month, we can calculate it’s worth.
If one were the share the unlimited metrocard among 3 people, then the \$104 is not considered to be expensive. What we should calculate is worth and by considering all of the factors(or at least most of them), determine the number of pay-per-rides needed to overtake the worth of the unlimited Metro Card.

#### Short Cinema · February 12, 2011 at 12:17 pm

Have you ever used had a monthly card?

It is very challenging to share it on a regular basis. One way to share maybe if you used it during weekdays and let someone else use it on the weekends.

Another reason sharing an unlimited metrocard is difficult because you have to wait 15 minutes before you are allowed to use it again at the same location.

I doubt that sharing is what lead to the increase in fares.

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