The sum of Mike's age and Ed's age is 44 years. Mike is twice as old as Ed was when Mike was half as old as Ed will be when Ed is three times as old as Mike was when Mike was three times as old as Ed.
How old is Mike?
Can you tell by quick inspection who is older?
Just as there is more than one way to skin a fish (I am a cat lover so I changed the expression) there is more than one way to set up a problem. However, simplicity is the hallmark of eloquence so I will strive for it in my setup.
First I will derive equations from the statement of the problem. All numbers (except the ordinal numbers I will use to label each equation) will be quantities of time in units of years.
Let "M" represent Mike's current age and "E" represent Ed's current age. In this problem these are both constants as the statement of the problem is a freeze-frame of the present point in time. Exactly one year (in the future direction) from now Ed will be (E + 1) years old.
I will use quotes around a symbol when I first introduce it and when needed to make the meaning clear.
So most people will probibly start off as I have:
1. M + E = 44
That is equation 1. comprising the complete information of the first sentence of the problem-statement.
Continuing with the 1st phrase of the 2nd sentence of the problem-statement, "Mike is twice as old as Ed was...".
This is the first hurdle for many people because we don't know how many years in the past this reference to Ed's age then, is.
So we might proceed by introducing a new unknown with a new symbol such as "x" or " M' " spoken as "M prime". However, one can tell by the rest of the sentence that we would need to introduce many more symbols in repeating this method. There is no way to not introduce more unknowns like this but we can limit their number if we think of a more eloquent way to represent either Mike's or Ed's age at some other point in time than the present, which we already have the symbols for.
So I propose that we consider an interval of time, in units of years of course, an absolute value of the difference in years between the present and some past or future time.
So let "a" represent that difference, in years (a positive value greater than 0) between the present and that point in time in which, Ed was, the age referenced by the first phrase of the 2nd sentence, "Mike is twice as old as Ed was...".
In short, let (E - a) be Ed's age "a" years ago where "a" is a number greater than 0. It is now a simple matter to derive equation 2 which I shall add below equation 1.
1. M + E = 44
2. M = 2 (E - a) This is a shorter way of writing M = 2 x (E - a). Also, note the parentheses are required as M = 2 x E - a is not an equivalent equation.
Continuing with sentence 2, "...when Mike was half as old as Ed will be...". The adverb "when" connects two events to the same point in time, in this case, "a" years ago. Well, if Ed was (E - a) years old then, how would you represent Mike's age then, "a" years ago? You can do this without introducing a new symbol for an additional unknown.
[I was going over something else when it suddenly dawned on me I had made this false statement in red above. It is a mistake I made and you should totally ignore the sentence in red. The only reason I don't just simply remove it is that many have already seen this page. I can just imagine tomorrow's headlines: AJI MAKES A MISTAKE]
Eventually, after the setup, we will need to make all these equation be true at the same time which is why they are called simultaneous equations. However, don't worry about that now. Strive to get the setup completed without error since it is the foundation upon all your work. If the foundation contains one or more errors it is highly unlikely any successful solution of the simultaneous equations will yield the correct answer. And even if by chance you did get the correct answer(s), you really have not proved that they are the correct answer(s).
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Here is a story about tool and die makers. I apprenticed at my Father's tool and die plant during the summer of 1971. There is always a shortage for highly skilled labor. So during the trip around the world in 1967 in which I accompanyed my Father, he was always looking for skilled tool makers. With the promise of a job in Canada at Paragon Tools just outside Windsor Ontario, it was easy to sponser them to the Canadian government for immigration. My Father was actively involved in this "brain drain" from many different countries of the world. That was only a secondary purpose of his trip though. The primary purpose was to sell licensing of his company's methods of vacuum metalizing to auto suppliers accross Europe and Japan. His purpose in India was to probe the possibility of building a polypropolene refinery there with the sweetener of also producing prophylactics. Now I can't help but mention that as a young boy of 14, I thought how great it would be to have a lifetime source of free condoms.
I will now revert back to tool makers and how this relates to our problem. Paragon Tools primarily made molds for my Father's other companies which would use these in plastics injection molding. They also did lots of other processes after a plastic part was molded. These include hotstamping and vacuum metalizing. But the most crucial part is that the molds be made correctly.
Molds start off as a block of steel, often several thousand pounds in weight. As this block is tooled, much money is required to pay the labor used in making it into a mold. Each later stage in this process becomes more risky because of the amount of labor that went into making it in the previous stages. So even a tiny mistake near the final stages would render a mold useless with all the labor that went before ending up totally wasted. So only the most skilled (and careful) tool makers would work on the final stages of producing a mold. It is a high-stress job as so much is at stake.
* * *
So what is the moral of this story? It is: Be sure the foundation is sound before building the rest of the house. For our purposes with this problem, don't even worry about the final stages. Concentrate on getting the setup correct.
And don't even worry about getting the complete setup at this point. Just see if you can derive equation 3 from what you know thus far.
I had asked a simpler question that was "Can you tell by quick inspection who is older?". I can and so can you. Look at just the last phrase of the 2nd sentence. Now tell me who is older.
As always, do not present your derivation of equation 3 in the comments. If you want me to confirm that it is indeed correct, personal message me by clicking Aji or e-mail me at AjiSabaki@hotmail.com.
Since I took up so much time with the story about tool makers, I will defer until a later part-posting the story of how I came to find this problem and maybe throw in a joke or two, still free of charge.
You could write a book. And I could read your book. I would not consider this interactive. You would have been active in the writing of your book. And I would have been passive in the reading of it. Somewhat interactive would be for you to write a workbook and I would not only read it, but also attempt to solve the many problems you present within it. With the Internet the possibilities for interactivity in social intercourse soar to new heights. This is a big reason why I am so fond of quizzes and problems which involve the listener or reader to guess the answer or solve the problem.
I think that most probibly skip my interactive posts. This is all the more reason that those few of you who enjoy them leave me a short comment. I do this primarily for my own edification. However, it is even more edifying to know that at least one other person has enjoyed one of my postings as well.