Tom and his brother Harry had a foot race from their home to school, which was an integral number of yards away. Harry ran faster than Tom, but less than twice as fast, and arrived at the school when Tom was still a 2-digit integer of yards behind.

The next day they ran the race again, but this time Harry started farther away from the school by the same amount as the winning margin the previous day, while Tom still started the same time as Harry, but again from home. Of course, since Harry still runs faster, the same ratio as the day before, he finished the difference in distances in less time than it would have taken Tom, but this time the gap at the end was reduced so that when Harry reached school, the amount by which Tom was behind had the two digits reversed from the preceding day.

How far was it from their home to school, and what was their gap at the end of each run when Harry arrived at school?

Given D = distance from home to school
Given r = ratio Harry's speed to Tom's speed
Given x as the 2-digit integer of distance the first day
Given y as the 2-digit integer of distance the second day

The equations for each day can be expressed as following:
1st Day : x = D - Dr = D(1 - r)
2nd Day: y = D - (D + (D - Dr))r = D(1 - r + r²) 

By subtracting the Day 2 equations from the Day 1 equations ...
D(1 - r) - D(1 - r + r²) = (x - y)
...can then be reduced, finding the ratio, r, ...
r = (x - y)/x
...then distance, D, can then be found from the re-written 1st Day equation ... 
D = x/(1 - r)

There are 36 possible 2-digit numbers where the tens-digit and ones-digit, when swapped, is a smaller (with no-leading zero) 2-digit number:

x    y    r(atio)       D(istance) 
21   12   0.428571429   36.75
31   13   0.580645161   73.92307692
32   23   0.28125       44.52173913
41   14   0.658536585   120.0714286
42   24   0.428571429   73.5
43   34   0.209302326   54.38235294
51   15   0.705882353   173.4
52   25   0.519230769   108.16
53   35   0.339622642   80.25714286
54   45   0.166666667   64.8
61   16   0.737704918   232.5625
62   26   0.580645161   147.8461538
63   36   0.428571429   110.25
64   46   0.28125       89.04347826
65   56   0.138461538   75.44642857
71   17   0.76056338    296.5294118
72   27   0.625         192        
73   37   0.493150685   144.027027
74   47   0.364864865   116.5106383
75   57   0.24          98.68421053
76   67   0.118421053   86.20895522
81   18   0.777777778   364.5
82   28   0.658536585   240.1428571
83   38   0.542168675   181.2894737
84   48   0.428571429   147
85   58   0.317647059   124.5689655
86   68   0.209302326   108.7647059
87   78   0.103448276   97.03846154
91   19   0.791208791   435.8421053
92   29   0.684782609   291.862069
93   39   0.580645161   221.7692308
94   49   0.478723404   180.3265306
95   59   0.378947368   152.9661017
96   69   0.28125       133.5652174
97   79   0.18556701    119.1012658
98   89   0.091836735   107.9101124

Only two result in integer distances -- (72, 27) and (84, 48). Of these, only (72, 27) has a ratio greater than .5. Thus, the distance is 192 yards, and the ending gaps between Harry and Tom for each day's run was 72 and 27.

Posted by Dej Mar on 2009-05-26 17:16:54