Do not try to solve problem algebraically. Instead, make a guess that satisfies one or more conditions of the problem and then evaluate your guess, as we did in Example.
Solving a Word Problem by Guessing
Suppose that you own a house in a school district that has a yearly budget of $100 million. In order to reduce the portion of the budget borne by property owners, your local taxpayers’ organization has negotiated the following political agreement:
1. The amount funded by the state income tax will be three times the amount funded by property taxes.
2. The amount funded by the state sales tax will be $15 million more than the amount funded by property taxes.
How much of the budget will be funded by property taxes?
SOLUTION:
In a later chapter, we will discuss how to solve problems algebraically, but for now, all we want to do is make several educated guesses and then keep adjusting them until we get an acceptable answer. Let us call the amount of the budget due to property taxes p, the amount due to income taxes i, and the amount due to sales taxes s.
Guesses for p, i, s (in millions of $) | Evaluation of Guess | |
Good Points | Weak Points | |
20, 20, 60 | Total is 100. | Amounts are not different. |
20, 30, 60 | Amounts are different. | Total is not 100. The amount i is not three times p. The amount s is not 15 more than p. |
20, 60, 35 | The amount i is three times p. The amount s is 15 more than p | The total is greater than 100, so we have to reduce the amount p. |
18, 54, 33 | The amount i is three times p. The amount s is 15 more than p | The total is still more than 100. |
17, 51, 32 | We have a solution. The amount from property taxes is $17 million. |
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We will begin by assuming that p = 20, i = 20, and s = 60. We do not agonize about making this starting guess. The important thing is to get the process started and then evaluate and refine our guesses to solve the problem. We will organize our guesses in the following chart.
Keep adjusting your guesses until you have a solution that fits all the conditions of the problem.
The administration at Center City Community College has formed a 26-person planning committee. There are five times as many administrators as there are students and five more faculty members than students. How many students are there on the committee?
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