# Yes, Algebra is Necessary, and There are Lots of Ways to Do It

This awesome post that a friend of mine linked to on Facebook this morning has a great real world application of simple algebra.

The set up is that The Interview made \$15M over Christmas weekend from both rentals and ticket sales, and about 2M people either rented it for \$6 or bought a ticket for \$15. Exactly how many rentals there were vs. ticket sales is left as a mystery.

The point of the post I linked to, as well as the picture above, is that you can solve the mystery using algebra. The standard way is to say r = number of rentals, s = number of tickets sold, and to solve the system

r + s = 2,000,000                  (total number of rentals and sales was 2M)

6r + 15s = 15,000,000          (total revenue from rentals and sales was \$15M)

But even if you have a trouble setting up and solving a system of linear equations, you can still use math to understand what’s going on! I’d be perfectly happy with a kid, or a grown-up, who went about it like this:

• If every one of the 2M gate receipts (= tickets sold + rentals) were a \$6 rental, the total revenue would have been \$12M.
• If every one of the 2M gate receipts were a \$15 sale, the total revenue would have been \$30M.
• The actual revenue was \$15M. Which is a lot closer to \$12M than to \$30M. So there must have been a lot more rentals than sales. I’d actually be pretty happy even if you stopped there, because you’ve already grasped the main quantitative point: most people rented the movie rather than buying a ticket.
• The actual revenue of \$15M should be a weighted average of the two extreme scenarios of \$12M and \$30M. Since \$15M is 1/6 of the way from \$12M to \$30M (a step of \$3M toward a total distance of \$18M), we can guess that 1/6 of the people did one thing and the rest (5/6 of the people) did the other. Since we know most people rented, we’re guessing that 5/6 of 2M people (or 1 2/3 M) rented and 1/6 of 2M people (or 1/3 M) bought a ticket. (Note: rather than guessing, we can also get to the answer by carefully thinking through how averages work. But the guess here seems pretty intuitive to me, and it sure is quick — though anytime we make a guess, we need to check it.)
• Let’s check: 1 2/3 M people renting for \$6 is \$10M of revenue, and 1/3 M people buying a \$15 ticket is \$5M of revenue. Add them up: \$15M on the dot!

Now look at the post I linked above again. Guess what: the math you go through to solve the system of equations actually mirrors the heuristic argument pretty closely. For example, my first bullet point is equivalent to the equation 6r + 6s + 12,000,000. The heuristic argument didn’t have any equations or unknowns, but at the heart of it we were still doing algebra!

The arguments aren’t exactly the same, and that’s OK: they both have their benefits, and one complements the other. The system of equations gives you a more systematic way of getting to the answer: you don’t have to guess (or make your way through some slightly twisted logic about averages). The heuristic argument gets you to the main point — more rentals than sales — more quickly and transparently. But, at their heart, they are both about using arithmetic operations (adding, subtracting, multiplying, dividing), and known relationships between quantities, to tease out even more information about those quantities. That’s algebra. And it’s valuable to be able to do it, one way or another.