Ratios With Three Factors?

Question

How do I compare a different objects with three rates? For instance, if I were to compare the price of apartments I would look at the price per square foot, a rate with two factors. But if I wanted to add a third value, let's say distance from town, which would increase gas consumption by a certain amount, how would I go about this?

For the two factor situation, I would divide the monthly cost of the apartment by the square footage and rank them. To add the third factor, would I divide the rate by the distance from town? Could a rate have more than two variables?

Answer 1

Rates are, by definition, a comparison of two different variables. However, if one wants do a comparison of more than two variables, we can take a lesson from multivariable calculus and compare multiple different rates (in your case, price vs. square feet, square feet vs. distance from town, and price vs. distance from town). Alternatively, you could assign different "weights" to different factors and compare it that way. For instance you could assign a score to each of the three factors, perhaps in the interval $[0,1]$, and conclude that the higher score is better at the end. The problem with this method, however, is that it introduces a certain subjectivity that math likes to pride itself on avoiding.

Answer 2

In your case, it's very natural to look at distance as an indirect price increase.

Let's look at a specific example. The standard mileage reimbursement is (in round numbers) about $50$ cents per mile; this is supposed to cover both the fixed and variable costs of driving. If you also value your time at $\$50$ per hour, and drive at an average of $50$ miles per hour between your house and wherever you're going, then the cost of driving a given distance is

$$ 50 ¢ / \text{mile} + \frac{$50 / \text{hour}}{50 \, \text{miles}/\text{hour}} =$1.50 / \text{mile} $$

If you plan on driving to and from your apartment about once every day, this would suggest that every mile farther from town corresponds to roughly a $\$90$/month increase in rent (which you can then divide by the square footage of the apartment).

Of course, all of the numbers I used will be very different for different people! So this method won't produce anything like an objective ranking of apartments — but it might tell you which one looks best for your specific situation.