Word Problem Discussion

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Often we find that we can measure the world in seemingly very different ways, only to find that, as we vary the objects being measured, the measurements themselves vary in related and predictable ways. Such predictions are exactly what interests us in this course.Let us Consider three examples.

  1. For objects in a store, that are for sale, the price of the object and the amount of sales tax charged for the sale of the item are proportional. In fact, the price uniquely determines the sales tax charged for the item. This is as strong as a relationship can be. The sales tax rate here, will be 5%.
  2. For people, a person's height and weight are closely related measurements; on the average, a taller person will will weigh more, although height is not the sole factor determining weight.

Let us examine the organization of possible data for the above example. As we recorded heights and weights, we wrote the person's name on one line and then recorded that person's height and weight on the same line as we measured them. It is vital that the heights and weights be thus paired. If someone took the list and randomly mixed up the heights and weights of people, no relationship could then be found between height and weight. The fact that a pair of numbers, like a person's height and weight, are different aspects of a single object or event is the reason that we consider representing such a pair as a single point in a two dimensional x,y plane. Similarly, recognizing this inseparable pairing in a word problem crucial to beginning the problem, much less solving it.

A third example is driving down the road at a constant speed. Then, the distance the car travels is related to the amount of time it travels. The speed relates these two measurements through the formula (rate)(time) = distance. If we define the time to be x and the distance to be y, then the formula becomes y = (rate)x or y = mx. In other words, the most concise description of a linear relationship is given by the slope. The slope is more difficult to recognize in a word problem than the coordinates of points. The major clue is the units associated with the slope. In the relationship between time and distance, the units are miles per hour. The units always represent the amount of y that corresponds to a single unit of x. The tax rate is given as the number of dollars charged in tax for each dollar in the item's price. This is a good example. Many examples can be expressed as "so many" units of y per unit fo x, but every example can be expressed as "so many" units of y for each unit of x.

The proceedure necessary to attack a linear word problem then, is to first identify the units of the two important measurements in the problem. In the last sample, the rate problem, the units are hours and miles. More specifically, the measurements are the number of hours that we drove the car and the number of miles that the car was driven. In example 1 above, both units are dollars. One is the number of dollars that the item costs and the other is the number of dollars that we must charge for sales tax.

Test yourself by chosing the units of example 2 from below:

height and weight

feet and pounds

height in feet and weight in pounds

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