1. Metrocard Bonuses Getting a Fare Deal

2. 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

3. The above 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?

4. How many rides are you able to take with this MetroCard? 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 How much will you have left on your card after taking those rides? 7 rides cost $19.25, so you would have $21.09 - $19.25 = $1.84 remaining How much did you end up paying per ride? You paid $19 and got 7 rides. That works out to $19 / 7 ≈ $2.71 per ride

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

6. What total dollar value will you need on your MetroCard to be able to take that many rides? You will need 9 * $2.75 = $24.75 Fill in the table below

7. Based on your chart, what is the optimal amount to pay for the MetroCard in this situation? The optimal amount is $22.30

8. 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? We need $24.75 on the MetroCard to pay for 9 fares X * 1.11 = 24.75 X = 24.75 / 1.11 ≈ 22.30

9. 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 X be in order to get exactly N fares on your MetroCard? Y * 1.11 = N*2.75 Y = N*2.75 / 1.11 ≈ N*2.477 In your answers above, what is the price per ride? Approximately $2.48 per ride

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

12. What are some possible reasons that the MTA would suggest MetroCard values that lead to a remaining balance?