The problem says that both “start at the same time” and “after two hours.” Therefore the two times are the same and both equal 2 hours.
Perhaps I was wrong to say that the two equations are obvious. I have been thinking like this for a lifetime almost, and it is new to you.
I start by thinking about relevant quantities when faced with a quantitative problem and I assign letters to them in writing so I can think and talk about them without getting mixed up about what exactly I am talking about. Some call that naming the variables; others talk about labelling. I don’t care what you call it; it is a way to guide you through the initial mental fog that a hard problem represents. That to me is the first step. It is not hard. You just ask yourself what quantities may be relevant. Then you see which ones are known. This tells you how many equations you need.
The second step is where you do most of the hard thinking. I call it translation, but I have had students who prefer to think of it as a mystery story. You need to find a certain number of clues in English that you then must translate into mathematic notation. Your job is to search out the clues and then do the translation. In this case the clue to the fourth equation is that they start from the same place but go in perpendicular directions. If you draw a diagram, you get a right triangle and then you can begin to search for helpful mathematical ideas relevant to right triangles.
Once you have your system of equations, your third step is to just follow the mechanical steps to get a solution. No deep thought required; just careful attention to details.
Your fourth step is to check your work.
That method will let you solve not only word problems but real life quantitative problems.
Perhaps I was wrong to say that the two equations are obvious. I have been thinking like this for a lifetime almost, and it is new to you.
I start by thinking about relevant quantities when faced with a quantitative problem and I assign letters to them in writing so I can think and talk about them without getting mixed up about what exactly I am talking about. Some call that naming the variables; others talk about labelling. I don’t care what you call it; it is a way to guide you through the initial mental fog that a hard problem represents. That to me is the first step. It is not hard. You just ask yourself what quantities may be relevant. Then you see which ones are known. This tells you how many equations you need.
The second step is where you do most of the hard thinking. I call it translation, but I have had students who prefer to think of it as a mystery story. You need to find a certain number of clues in English that you then must translate into mathematic notation. Your job is to search out the clues and then do the translation. In this case the clue to the fourth equation is that they start from the same place but go in perpendicular directions. If you draw a diagram, you get a right triangle and then you can begin to search for helpful mathematical ideas relevant to right triangles.
Once you have your system of equations, your third step is to just follow the mechanical steps to get a solution. No deep thought required; just careful attention to details.
Your fourth step is to check your work.
That method will let you solve not only word problems but real life quantitative problems.
