Activity: Metrocard Bonuses – Getting a Fare Deal

Learning Objective

Students will be able to perform the appropriate calculations to analyze the MTA’s MetroCard bonus system and determine a solution to a practical everyday problem.

Quantitative Literacy VALUE Rubric – Representation, Calculation, Application/Analysis

Background

A fare on the New York City subway system costs $2.75 as of March 2015. If you purchase a pay-per-ride MetroCard, you are given an 11% bonus on purchases of $5.50 or more. For example, if you pay $10.00, you will receive an $11.10 MetroCard. You can buy a MetroCard for any amount that is divisible by a nickel.

Activity

mta1

  1. The above picture shows suggested card values as presented by a MetroCard vending machine. Suppose you choose the $19.00 option (with a $2.09 bonus).
    1. How many rides are you able to take with this MetroCard?
    2. How much will you have left on your card after taking those rides?
    3. How much did you end up paying per ride?
  1. Suppose you are a tourist visiting New York City and you know that you are going to take exactly 9 rides during your stay.
    1. What total dollar value will you need on your MetroCard to be able to take that many rides?
    2. Fill in the table below.
You Pay Bonus On Card # of Rides Cost of Rides Remainder
22.15 2.44 24.59 8 22.00 2.59
22.20
22.25
22.30
22.35
22.40
22.45
    1. Based on your chart, what is the optimal amount to pay for the MetroCard in this situation?
  1. We can also figure out the solution algebraically.
    1. Let X be the amount that you pay for a MetroCard. Assuming that you get an 11% bonus on what you pay, what must X be in order to get exactly 9 fares on your MetroCard?
    2. Let Y be the amount that you pay for a MetroCard. For any number of rides N, assuming that you get an 11% bonus on what you pay, what must Y be in order to get exactly N fares on your MetroCard?
    3. In your answers to (a) and (b), what is the price per ride?
  1. Of the four suggested MetroCard values in the picture above, which is the only one that does not lead to a remaining balance?
  1. What are some possible reasons that the MTA would suggest MetroCard values that lead to a remaining balance?

Suggested Solutions/Discussion

1.

a. The value of the MetroCard is $19.00 + $2.09 = $21.09. $21.09 / $2.75 ≈ 7.7, so you can take 7 rides with this MetroCard.

b. 7 rides cost $19.25, so you would have $21.09 – $19.25 = $1.84 remaining.

c. You paid $19 and got 7 rides. That works out to $19 / 7 ≈ $2.71 per ride.

2.

a. You will need 9 * $2.75 = $24.75

b.

You Pay Bonus On Card # of Rides Cost of Rides Remainder
22.15 2.44 24.59 8 22.00 2.59
22.20 2.44 24.64 8 22.00 2.64
22.25 2.45 24.70 8 22.00 2.70
22.30 2.45 24.75 9 24.75 0.00
22.35 2.46 24.81 9 24.75 0.06
22.40 2.46 24.86 9 24.75 0.11
22.45 2.47 24.92 9 24.75 0.17

c. The optimal amount is $22.30.

3.

a.

X * 1.11 = 24.75

X = 24.75 / 1.11 ≈ 24.75 * 0.901 ≈ 22.30

b.

Y * 1.11 = N*2.75

Y = N*2.75 / 1.11 ≈ N*2.477

c.

Approximately $2.48 per ride.

4. The only one of the four suggested values that does not lead to a remaining balance is the $27.25 purchase.

You Pay Bonus On Card # of Rides Cost of Rides Remainder
9.00 0.99 9.99 3 8.25 1.74
19.00 2.09 21.09 7 19.25 1.84
27.25 3.00 30.25 11 30.25 0.00
39.00 4.29 43.29 15 41.25 2.04

5. Here are some suggested answers:

-Vending machines accept cash, and even numbers do not require the machine to dispense change (although the suggested value of $27.25 does require the machine to have quarters)

-Remaining balances may encourage customers to refill their cards when they otherwise would have taken other forms of transportation

-The unused balances that expire are a source of tens of millions of dollars a year for the MTA (http://www.nytimes.com/2014/01/17/nyregion/unspent-metrocard-money-means-millions-for-mta.html?_r=2)

Download Content: 

PowerPoint: Metrocard Bonuses

Word Document: Metrocard Bonuses

Sources/Further Reading

http://web.mta.info/metrocard/compare.htm

http://iquantny.tumblr.com/post/96700509489/how-memorizing-1905-can-help-you-outsmart-the