topspin20 wrote:Very nice, Brent.
At the risk of coming across as sadistic, does anybody have a purely algebraic solution to this?
Only since you asked for it ...
Sunny portion of the trip:
Rate = s
Time = t
Cloudy portion of the trip
Rate = (s+1)
Time = z
Total trip:
Rate = 2.8
Time = t + z
Since D = RT, 2.8(t+z) = st + (s+1)z, or
2.8t + 2.8z = st + sz + z, or
2.8t - st = sz + z - 2.8z, or
(2.8t - st)/(s - 1.8) = z
We want the ratio of sunny distance to total distance, so
(st)/(2.8(t+z)), or
(st)/(2.8(t + (2.8t - st)/(s - 1.8))), or
(st)/(2.8t/(s-1.8)), or
(st * (s-1.8)) / 2.8t
At this point, however, we're in trouble - there's no way to tell our algebra that s must be an integer. (Even if we knew that this equation was equal to 1/7, we'd still get two solutions for s, one of which is negative.)
One thing we do notice, though, is that we have (s - 1.8) in our numerator. Since that has to be a positive number, we can conclude that s is at least 2. Since the average rate was 2.8 and is somewhere between s and (s+1), s can't be greater than 3, so s = 2.
... but as many other experts have already explained, we can deduce that without all this algebra.
Funny note on this question: I encountered it when beta-testing our Veritas exams, and TRIED TO SOLVE ALGEBRAICALLY. Whoops!
