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(11) Simplify the problem if possible. Take advantage of symmetries which often exist.
Keep in mind that your first try may not work. But don't get discouraged. If one approach doesn't work, try another. You have to keep trying different approaches, different ideas. As you gain experience, your problem solving skills improve and you tend to find the right approach sooner.
Let us now look at some examples to illustrate the topics discussed above.
Example 1
This is an example in which you can find a solution once you analyze and understand the unknowns and data.
Problem: A survey of TV viewers shows the following results:
To the question "Do you watch comedies?", 352 answered "Yes".,
To the question "Do you watch sports ?", 277 answered "Yes", and
To the question "Do you watch both comedies and sports ?", 129 answered "Yes".
Given these data, find, among people who watch at least one of comedies and sports, percentages of people who watch at least one of comedies and sports watch only comedies, only sports, and both comedies and sports.
Let us try to solve this problem following the framework presented above.
Understanding the Problem: This is a "find" type problem. So we try to identify unknowns, data and conditions.
The unknowns are the percentage of people who watch only comedies, the percentage of people who watch only sports, and the percentage of people who watch both comedies and sports.
The data are the three numbers: 352, 277 and 129, representing the number of people who watch comedies, sports, and both comedies and sports, respectively. Note that 352 includes people who watch both comedies and sports as well as people who watch only comedies. Similarly for 277.
The conditions are not explicitly given in the problem statement. But one can see that the percentages must add up to 100, and they must be nonnegative.
Devising a Solution Plan: Here we first examine the principal parts in detail.
First let us consider the unknowns in more detail. To calculate the percentage of the people who watch only comedies, for example, we need the number of people who watch at least one of comedies and sports, and the number of people who watch only comedies. Thus actually two unknowns are involved in each of the required percentages, and the real unknowns are the number of people in each of the categories, and the number of people who watch at least one of comedies and sports.
Next let us look at the data. First the number 352 is the number of people who watch comedies. But that is not necessarily that of the people who watch only comedies. It includes that and the number of people who watch both comedies and sports. Similarly for the second number 277.
Let us use symbols to represent each of the unknowns: Let C represent the number of people who watch only comedies, S that of the people who watch only sports, and T that of the people who watch at least one of those programs.
Then we have the following relationships among the unknowns:
C + 129 = 352
S + 129 = 277
C + S + 129 = T
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