**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**

- 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).
- How many rides are you able to take with this MetroCard?
- How much will you have left on your card after taking those rides?
- How much did you end up paying per ride?

- Suppose you are a tourist visiting New York City and you know that you are going to take exactly 9 rides during your stay.
- What total dollar value will you need on your MetroCard to be able to take that many rides?
- 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 |

- Based on your chart, what is the optimal amount to pay for the MetroCard in this situation?

- We can also figure out the solution algebraically.
- 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?
- 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?
- In your answers to (a) and (b), what is the price per ride?

- Of the four suggested MetroCard values in the picture above, which is the only one that does not lead to a remaining balance?

- 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