Real Life Math Puzzle: Membership Length

[livingdeb]

I almost never find opportunities to use serious math to help me solve problems in my real life. But today I joined a professional organization and that provided me with such an opportunity. Let's see if y'all get the same answer I do. Here's the situation:

If you join for one year, you pay full price. If you join for two years, you get ten percent off both years. If you join for three years, you get fifteen percent off all three years. Which deal is the best deal?

And let's throw in some simplifying assumptions. First, assume that you want to be in the organization indefinitely. Then assume that you have enough money right now to choose any of these offers, and that any extra money you don't spend now you will invest for later. Assume also that the current full price and options will never change. And add in any other obvious assumptions I'm forgetting about that will make the answer easier to calculate.

(My answer is in the comments.)

Comments

My Answer

[livingdeb.livejournal.com]

Two years. Ah, one of those tricky problems.

If you join for one year, you pay x now. That amount of money is spent regardless of which of the three options you choose.

If you join for two years, you are paying for an additional year now but getting a 10% discount. So your total cost is 0.9x + 0.9x = 1.8x. By giving up an extra 0.8x now, you save 0.2x one year from now. This is a 25% annual return on your extra money.

That's an awfully good guaranteed return, better than anything else I can find.

If you join for three years, your total cost is 3(0.85x) = 2.55x. Compared to the one-year option, you are giving up an extra 1.55x now to save 0.45x over two years. That's a return of 0.45/1.55 = just over 29% in two years. Assuming no compounding, that's an annual return of less than 15%. That's not very precise, but it's precise enough to convince me that two years is the better deal.

Re: My Answer

[fraeuleinchen.livejournal.com]

I love it, livingdeb! You're right, I think, that we don't have many opportunities to use real math in daily life... I very seldom pull out an algebraic equation to help me make a decision. Oh, my rusty algebra!