if i wanted to practice more problems like this, what particular word problem would this be called? it's not the usual algebra problems i'm used to (like systems of equations) so it threw me for a loop.

Great question.

Your original question was how to think about solving problems. And learning how to solve problems is the value of learning mathematics for everyone who is not a mathematician (me for example).

It is a fact that there are frequently many ways to solve a problem. Some people find one way more natural; others find a different way more natural.

But before plunging into equations, etc, it is always valuable to think. That helps you determine what tools may be valuable. Lev thought about relative speeds, and that led to algebraic tools. I thought about relative times and that led to counting. Both were successful. Neither is better in the abstract.

The quantitative problems you will face in practice do not come with labels like “simultaneous equations” or “counting.” You will have no clue what techniques may be helpful (or even whether the problem is soluble by any technique you already know). What you will always know is what do you need to find out and what information do you already have. It is always helpful to spend some time thinking about how you might be able to get from the known information to the desired information.

Obviously, virtually any problem assigned in an algebra class can be solved by some

**UNKNOWN** combination of algebraic techniques. But the way to start thinking about any mathematical problem is to grasp what is known and what needs to be known and a plan to go from one to the other. I admit that a great way to do well on tests is to memorize a technique that always works on a problem of a recognizable type, but it does not work whenever you run across a type you do not recognize.

True story. About four or five months ago, my son introduced me to a game. After working at it for a few hours, I said that there must be some kind of algebra to solve the game easily. He did not know one, but agreed that one must exist. A day or so ago, I found out that Claude Shannon had figured out in the 1930’s that Boolean algebra, developed in the early 1800’s, was the correct algebra for such problems. Without being skilled in the algebra and without knowing that it applied, I still succeeded at the game. It just took me a long time, over days in fact, to do so. Recognition of patterns is great, but not needing to rely on patterns to solve problems is even greater.

I fear that too much of math education suggests that doing math is to: (1) recognize the type of problem , (2) memorize one technique for each type, and (3) plug and chug. That is not math. That is not problem solving.