**Mathematics Problem**

Students can receive discounts when they ride the bus in three ways: (1) Show your student ID and one trip is 60 cents; (2) Use your student number to buy a 10-ride pass for $6.00; or (3) Buy a semester-long pass that lasts 5 months for $50. Which choice would you prefer?

*Quick answer: Choices 1 and 2 are the same value. Choice 3 will be cheaper if you ride the bus at least 84 times over the five months, so if you plan to ride the bus about 17 times per month, or about 4 times per week, then it will be cheaper to buy a semester pass, so you should prefer to buy the semester pass.*

**Real-world problem**

“Prefer” often indicates in mathematics problems that you should optimize the situation. For this problem, optimization probably means finding the cheapest way. But in the real-world, we have to think about other constraints that might not be named in the problem. Here is where you can ask your students (if they have thought about this in their own life) to share their thinking and interpretation of the problem.

I have thought about this situation each semester in the past four years. I choose not to buy the semester pass, because I will feel pressured to always ride the bus. But I *want* to choose to walk or ride my bike most of the time, except when weather prevents, so that I am incorporating exercise into my daily routine. So my choice really becomes between options 1 and 2.

They seem like the cost should be the same, but in practice option 1 is much more expensive for the following reasons: **First**, I don’t use cash a lot so there are times I don’t have cash at all. When I do, I generally have quarters or dollar bills. If I use quarters or dollar bills, I receive a change card from the driver. Somehow because of the coins I usually have, I end up with lots of nickel change cards. I can only use one change card at a time, so (for me) the nickel change cards only result in more nickel change cards. So I throw those away. **Second**, I sometimes misplace my student ID and so at those times I have to pay $1.25 as the full fare.

For both of those reasons, option 1 is more expensive. And yet – I end up using it much more often than option 2. Why? Because option 2 requires me to buy the 10-ride card online, one at a time, and wait for it to arrive in the mail. So it is not convenient, even though it is cheaper.

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