To begin with, the question itself is a little bit ambiguous on its assumption. However, if we intend to treat this problem as a mathematical problem, then the statements of JeffM and Dr.Peterson are equivalent ("if A, then B" and "whenever A occurs, then B occurs" are logically equivalent). Thus it is a reasonable interpretation for this problem.
You are asking to find a relationship between a and b and the only thing you know is that
[MATH]a_1 = 1.35a_2 \implies b_1 = 1.25b_2[/MATH].
What you're implicitly assuming here, and may be what is intended, is that the question means,
Whenever A increases by 35%, then B increases by 25%. Given that A increases 1%, find the percentage increase of B.
Now, allow me to elaborate further on the assumptions, which are not specifically given by the problem itself.
Since we are only given a relationship between A and B, it makes sense to describe this relation as a function, namely f(A) = B. Moreover, the given relation implies that f(1.35 A) = 1.25 f(A). Since the question talks about increment, it is reasonable to assume both A and B are real numbers, not just integers or positive numbers. In other words, we treat f as a function whose input and output values are real numbers. To sum up, f is a real-valued function with a single real variable, i.e. the following statement (a):
f: R -> R, subject to
condition (b): f(1.35 x) = 1.25 f(x), for every x in R.
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Under this assumption, we can make precise what JeffM's conclusion means as follows:
A moment of thought will lead one to see that any real valued function g defined on [1, 1.35] union [-1.35, -1], subject to 1.25 g(1) = g(1.35) and 1.25 g(-1) = g(-1.35), can be extended to a function G : R -> R with G(0) = 0, which satisfies statement (a). Because there are infinitely many such functions g, there are infinitely many solutions G that answer G(1.01 x) / G(x).
As a matter of fact, both methods suggested in vincent1's post (#1) are just two of infinitely many such functions g, yielding two possible answers to the question. The first method, however, leads to some interesting discussion.
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Not to offend anyone, but I have a few words about the method of using Log and the model assuming "(B
1/B
0 = A
1/A
0)^k" (I suppose that 1's should be k though).
- To follow deductive reasoning, I would avoid using Log because the input and output values could be zero and negative under general assumption. For the same reason, I will not assume a particular model or relation between A and B.
- I think by "continuous domain", vincent1 referred to "f is a continuous function" since the domain is already assumed to be the real number line or the positive reals. In the case when
continuity is assumed, in addition to statement (a), then as Dr.Peterson would have almost had it, the solution will be of
polynomial form in x. Here is a sketch of proof without "cheating" with Log and assuming additional relations between A and B.
1. By induction on condition (b) we have f(1.35^k x) = 1.25^k f(x), for every natural number k and every x in R. By setting x' = 1.35^(-1) x, via condition (b) we have 1.25^(-1) f(x) = f(1.35^(-1) x). By induction again, we conclude f(1.35^k x) = 1.25^k f(x), for every non-zero integer k and every x in R.
2. By making use of the real domain, we further deduce that f(1.35^t x) = 1.25^t f(x), for every non-zero rational number t and every x in R because every such t = k/m for non-zero integers k and m.
3. Now, by
continuity of f and the fact that the set of rational numbers is dense in reals, we deduce that f(1.35^t x) = 1.25^t f(x) for all t in reals (including t = 0), and all x in reals.
4. To this end, any arbitrary non-zero x in R can be
uniquely determined either by x = 1.35^t or x = -1.35^t. Therefore, simple algebra gives us
f(x) = x^(Log 1.25 / Log 1.35) f(1), when x > 0,
f(x) = x^(Log 1.25 / Log 1.35) f(-1), when x <0, and
f(0) = 0 by continuity.
In conclusion, under the additional assumption of continuity (i.e. relation between A and B is continuous), then f is necessarily of
piecewise polynomial form that is uniquely determined by its values at x = 1 and x = -1 (I guess some people would like to call these initial values?). This answers the original question regarding f(1.01 x) / f(x) = 1.01^(Log 1.25 / Log 1.35) as a consistent value for all non-zero x, which also matches our immediate intuition toward this problem in the first sight.
Nevertheless, we cannot take continuity for granted if it is not given. Hence, I think JeffM made the most general conclusion to this problem; whereas the "natural" method would have been true otherwise. If this problem is meant to be an elementary or secondary or high school math problem, I would have taken the "natural" method without any justification, just assume everything and use the formula, easy-to-go, no-brainer. If it is a problem in a proof-based math course or analysis course, I would say the "natural" method must be justified very carefully in a deductive reasoning manner. Finally, I found this problem and its discussions demonstrated very well how our natural belief in everyday math (i.e. things are continuous, be it function or domain) might have failed easily in rigorous mathematics and lead to some very counter-intuitive and interesting observations.
